Solutions to Exercises in "Discrete Mathematics Tutorial"

Transcription

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5 1.1 (1) {2} (2) {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196} (3) {1, 8, 27, 64} (4) {0, 1, 2, } (5) {2, 3} (6) {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z} 1.2 (1) {(x, y) x, y R x 2 + y 2 < 1} (2) {θ k(k Z θ = π 4 + kπ)} (3) {x x N x < 8} (4) {(x, y, z) x, y, z N x 2 + y 2 = z 2 } (5) {x x R x 2 + 5x + 6 = 0} 1.3 (1), (4), (5), (6), (8), (9) 1.4 (1) : A B B C A B x(x B x C) ( ) = A B (A B A C) (x/a) = A C ( ) (2) A = {a}, B = {{a}}, C = {{a}, {b}} A B B C A C (3) A = {a}, B = {a, b}, C = {{a, b}, {b, c}} A B B C A / C (4) A = {a}, B = {a, b}, C = {{a, b}, {b, c}} A B B C 5

6 A C 1.5 A = {a}, B = {{a}}, C = {{{a}}} A B B C A / C 1.6 (1) 0 1 {a}, {b}, {c} 2 {a, b}, {a, c}, {b, c} 3 {a, b, c} {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} (2) 0 1 {1}, {{2, 3}} 2 {1, {2, 3}} {, {1}, {{2, 3}}, {1, {2, 3}}} (3) 0 1 { }, {{ }} 2 {, { }} {, { }, {{ }}, {, { }}} (4) 0 1 {{1, 2}} {, {{1, 2}}} (5) 0 1 {{, 1}}, {1} 2 {{, 1}, 1} {, {{, 1}}, {1}, {{, 1}, 1}} 1.7 A B A C B (A B) A ( B C) A B A B C A (B C) C (A B C) (A B C) 1.8 (1) {4} (2) {1, 3, 5} 6

7 (3) {2, 3, 4, 5} (4) {2, 3, 4, 5} (5) {, {4}} (6) {{1}, {1, 4}} 1.9 (1) { 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 16, 18, 21, 24, 27, 30, 32, 64} (2) (3) { 7, 6, 5, 4, 3, 2, 1, 4, 5} (4) { 7, 6, 5, 4, 3, 2, 1, 0, 3, 4, 5, 6, 9, 12, 15, 18, 21, 24, 27, 30} 1.10 P(A) = {, {a}} PP(A) = {, { }, {{a}}, {, {a}}} (1), (2), (4), (5) 1.11 : A B = A A B = (A B) B (A B = A) = (A B) B = A ( B B) ( ) = A ( ) = ( ) A B = A = A E ( ) = A (B B) ( ) = (A B) (A B) ( ) = (A B) (A B = ) = A B ( ) = A B A B = A A B = A B A B = A B : A B = x(x (A B)) x (x (A B)) x (x A x / B) x (x A x B) x( x A x B) x(x A x B) A B ( ) ( ) (/ ) ( ) ( ) 7

8 (1) (A B) (A C) = A A B C = : (A B) (A C) = A A (B C) = A ( ) A (B C) = ( 1.11 ) A B C = ( ) (2) (A B) (A C) = A (B C) : (A B) (A C) = A (B C) = ( ) A (B C) ( 1.1) (3) (A B) (A C) = A (B C) : (A B) (A C) = A (B C) = ( ) A (B C) ( 1.1) (4) (A B) (A C) = A A (B C) = : (A B) (A C) = A A (B C) = A ( ) A (B C) = ( 1.11 ) 1.13 (1) 1.2 A B A B A A B B : x x A B x A x B = x A ( ) A B A A B B 1.3 A B A A B B A B : x x A = x A x B ( ) x A B A A B B A B : (A B) C = (A B) C A B ( 1.2) 8

9 (A B) (A C) ( 1.3) = A ( B C) ( ) = A (B C) ( ) = A (B C) (2) A C = (1) : A C = (A B) C = (A B) C = (A C) B ( ) = (A C) B = A B ( 1.11 ) = (A B) ( ) = (A B) (A C) (A C = ) = A ( B C) ( ) = A (B C) ( ) = A (B C) x x A x C x B x / (A B) C x A (B C) (A B) C = A (B C) 1.14 : B = E B ( ) = (A A) B ( ) = (A B) ( A B) ( ) = (A C) ( A C) ( ) = (A A) C ( ) = E C ( ) = C ( ) 1.15 A = B = D = G C = F = H 1.16 (1) {3, 4, {3}, {4}} (2) (3) {, { }} 1.17 (1) {, {{ }}, {{{ }}}, {{ }, {{ }}}} (2) {, { }, {{ }}, {, { }}} 9

10 (3) {{ }, {{ }}} 1.18 (1) {, 1, 2, 3} (2) (3) (4) 1.19 (1) A B (2) A (3) B A, B, C, D A B C D A C B D : x x A C x A x C (x A x C) (x A x B x C x D) ( ) = x B x D x B D 1.5 A, B, C, D A B C D A C B D : x x A C x A x C = x B x C ( ) = x B x D ( ) x B D : A = A E ( ) = A (C C) ( ) = (A C) (A C) ( ) (B C) (B C) ( 1.4) = B (C C) ( ) = B E ( ) = B ( ) 1.21 (1) A B = A A B 10

11 : A B = A x((x A x B) x A) x(((x A x B) x A) (x A (x A x B))) x(( (x A x B) x A) ( x A (x A x B))) x(( x A x B x A) ( x A (x A x B))) x(( x A x A x B) ( x A (x A x B))) x(( x A x A x B) (( x A x A) ( x A x B))) x((1 x B) (1 ( x A x B))) x(1 (1 ( x A x B))) x( x A x B) x(x A x B) A B (2) A B = A B A : A B = A x((x A x B) x A) x(((x A x B) x A) (x A (x A x B))) x(( (x A x B) x A) ( x A x A x B)) x((( x A x B) x A) ( x A x A x B)) x((( x A x A) ( x B x A)) ( x A x A x B)) x((1 ( x B x A)) (1 x B)) x((1 ( x B x A)) 1) x( x B x A) x(x B x A) B A ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (3) A B = A B = : B = A B = A (B = ) = A ( 1.7(4)) A B = A B = B ( 1.7(4)) = (A A) B ( 1.7(5)) 11

12 = A (A B) ( 1.7(2)) = A A (A B = A) = ( 1.7(5)) (4) A B = A B A = B : A = B A B = A A (A = B) = A ( ) = A A ( ) = A B (A = B) A B = A B A = A (A B) ( ) = A (A B) (A B = A B) = (A A) B ( ) = A B ( ) = A (B B) ( ) = (A B) B ( ) = (A B) B (A B = A B) = B ( ) 1.22 (1) (2) A = {a}, C = {b}, B = D = {a, b} A B C D A B C D A = C = {a, b}, B = {a, b, c}, D = {a, b, d} A B C D A B C D 1.23 : x B x / C x / B x C x B x / C x A x / A B x A C A B = A C x / A x A B x / A C A B = A C x / B x C 1.24 (A B) C = (A C) B : (A B) C = (A B) C = A ( B C) ( ) = A ( C B) ( ) = ((A C) B) ( ) = (A C) B 12

13 (A B) C = A (B C) : (A B) C = (A B) C = A ( B C) ( ) = A (B C) ( ) = A (B C) (A B) C = (A C) (B C) : (A B) C = (A B) C = (A B C) ( ) = (A B C) (A ) ( ) = (A B C) (A C C) ( ) = (A C B) (A C C) ( ) = (A C) ( B C) ( ) = (A C) (B C) ( ) = (A C) (B C) 1.25 (1) A (2) A B (3) B A 1.26 (1) : A C B C x x A B x A x B ( ) (x A x B) (x A x C) (x B x C) ( ) = (x C) (x C) = x C ( ) A B C x x A = x A x B ( ) = x C ( ) A C B C (2) 13

14 : C A C B x x C (x C) (x C) ( ) = (x A) (x B) ( ) x A B C A B x x C = x A B ( ) x A x B 1.27 : A A ( 1.1) P(A) ( ) { } P(A) P(A) ( 1.1) { } PP(A) PP(A) ( ) {, { }} PP(A) ( ) {, { }} PPP(A) ( ) {, { }} PPP(A) A A P(A) {, { }} PPP(A) 1.28 (1) (2), (1) (3), (1) (4), (1) (5) (1) (2) A B B A : A B x(x A x B) x( (x B) (x A)) x(x / B x / A) x(x B x A) B A (1) (3) A B A B = E : A B x(x A x B) x( (x A) (x B)) x(x / A x B) x(x A x B) x(x A B) A B = E ( ) ( ) (/ ) ( ) ( ) (/ ) ( ) 14

15 1 (1) (4) A B A B A : A B A x(x A B x A) x((x A x / B) x A) x( (x A x / B) x A) x(( x A x / B) x A) x(( x A x / B) x / A) x(( x A x B) x A) x(( x A x B) x A) x( x A x A x B) x( x A x B) x(x A x B) A B (1) (5) A B A B B : A B B x(x A B x B) x((x A x / B) x B) x( (x A x / B) x B) x(( x A x / B) x B) x(( x A x B) x B) x(( x A x B) x B) x( x A x B) x(x A x B) A B ( ) ( ) (/ ) ( ) ( ) ( ) ( ) ( ) ( ) (/ ) ( ) ( ) ( ) 1.29 : x x ( A ) ( B) x ( A ) x ( B) z(z A x z) z(z B x z) z((z A x z) (z B x z)) ( ) = z(z A x z) ( ) = z((z A x z) (z B x z)) ( ) z(( (z A ) (x z)) ( (z B) (x z))) 1 BBS chouxiaoya (1) (4) (1) (5) 15

16 z(( (z A ) (z B)) (x z) (x z)) z(( (z A ) (z B)) (x z)) z( (z A z B) x z) z(z A z B x z) z(z A B x z) x (A B) 1.30 (1) : x x P(A) P(B) x P(A) x P(B) x A x B x A B x P(A B) ( ) ( ) ( ) ( ) ( 1.26(2) ) ( ) (2) : x x P(A) P(B) x P(A) x P(B) x A x B y(y x y A) y(y x y B) ( ) ( ) = y((y x y A) (y x y B)) ( ) y(( y x y A) ( y x y B)) y( y x (y A y B)) y(y x (y A B)) x A B x P(A B) lim A k = lim A k = [0, 1] k k 1.34 lim B k = [0, 1], lim B k = k k 1.35 lim A k = [0, ], lim A k = {0} k k ( ) ( ) ( ) ( )

17 1.6 F G n 0 (n 0 N + k(k N + k n 0 (F (k) G(k)))) n 1 (n 1 N + k(k N + k n 1 F (k))) n 2 (n 2 N + k(k N + k n 2 G(k))) : x(a(x) B(x)) = xa(x) xb(x) n 0 = max(n 1, n 2 ) k(k N + k n 0 (k n 1 k n 2 )) (1) lim A k lim B k lim (A k B k ) k k k : x x lim A k lim B k k k n 0 (n 0 N + k(k N + k n 0 x A k )) n 0 (n 0 N + k(k N + k n 0 x B k )) n 0 ((n 0 N + k(k N + k n 0 x A k )) ( ) (n 0 N + k(k N + k n 0 x B k ))) ( ) n 0 (n 0 N + ( k(k N + k n 0 x A k )) ( k(k N + k n 0 x B k ))) = n 0 (n 0 N + k((k N + k n 0 x A k ) ( ) (k N + k n 0 x B k ))) ( ) n 0 (n 0 N + k( (k N + k n 0 ) x A k ) ( (k N + k n 0 ) x B k )) n 0 (n 0 N + k( (k N + k n 0 ) (x A k x B k ))) n 0 (n 0 N + k(k N + k n 0 (x A k x B k ))) ( ) x lim (A k B k ) ( ) k lim (A k B k ) lim A k lim B k k k k lim (A k B k ) lim A k lim B k k k k : x lim (A k B k ) k (1) x A k n 0 (x) k n 0 (x) x A k x lim A k k 17

18 (2) (1) {A k } x x lim (A k B k ) k k x / (A k B k ) k x B k x lim B k k lim (A k B k ) lim A k lim B k k k k lim (A k B k ) lim A k lim B k k k k (1) : lim A k lim B k lim A k lim B k k k k k lim A k lim B k lim A k lim B k k k k k lim (A k B k ) = lim A k lim B k k k k lim (A k B k ) lim A k lim B k k k k x lim (A k B k ) k k x A k x B k x / lim A k lim B k x / lim A k x / lim B k {A k } {B k } k k k k x A k x B k x lim (A k B k ) k lim (A k B k ) lim A k lim B k k k k lim A k lim B k lim (A k B k ) k k k x x lim k A k lim k B k n(n N + ( k(k N + k n x A k ))) n(n N + ( k(k N + k n x B k ))) = n(n N + ( k(k N + k n x A k ) k(k N + k n x B k ))) n(n N + k((k N + k n x A k ) ( ) ( ) (k N + k n x B k ))) ( ) n(n N + k(k N + k n (x A k x B k ))) x lim k (A k B k ) ( ) ( ) 18

19 (2) lim A k lim B k = lim (A k B k ) k k k : x x lim A k lim B k k k n 1 (n 1 N + k(k N + k n 1 x A k )) n 2 (n 2 N + k(k N + k n 2 x B k )) n 0 (n 0 N + k(k N + k n 0 x A k B k )) ( 1.6) ( ) x lim (A k B k ) ( ) k (1) lim A k lim B k lim A k lim B k k k k k lim A k lim B k lim A k lim B k k k k k lim A k lim B k lim (A k B k ) k k k lim A k lim B k lim (A k B k ) k k k : x lim A k lim B k x lim A k n 0 N + k n 0 k k k k x A k n N + n = max(n, n 0 ) x lim B k k k N + k n n B k n k n n 0 x A k x A k B k lim A k lim B k lim (A k B k ) k k k lim A k lim B k lim (A k B k ) k k k lim (A k B k ) lim A k lim B k k k k : x x lim k (A k B k ) n(n N + k(k N + k n x A k x B k )) n(n N + k(k N + k n x A k ( ) 19

20 k N + k n x B k )) = n(n N + ( k(k N + k n x A k ) k(k N + k n x B k ))) n( n N + ( k(k N + k n x A k ) k(k N + k n x B k ))) n(( n N + k(k N + k n x A k )) ( n N + k(k N + k n x B k ))) n( n N + k(k N + k n x A k )) n( n N + k(k N + k n x B k )) n(n N + k(k N + k n x A k )) n(n N + k(k N + k n x B k )) x lim k A k lim k B k ( ) ( ) ( ) ( ) ( ) (3) : E = (A k B k ) k N + lim (A k B k ) = lim (A k B k ) k k lim k A k lim k ( B k) = lim k A k lim k (E B k) ( (1) ) = lim A k (E lim B k ) ( 1.5(1)) k k = lim A k ( lim B k ) k k = lim A k lim B k k k (4) : E = (A k B k ) k N + lim (A k B k ) = lim (A k B k ) k k = lim A k lim ( B k ) k k ( (2) ) = lim A k lim (E B k ) k k = lim A k (E lim B k) ( 1.5(2)) k k = lim A k ( lim B k) k k = lim A k lim B k k k 20

21 2.1 a, b, c = a, b, c = {{{{a}, {a, b}}}, {{{a}, {a, b}}, c}} 2.2 (1) a, b c, d = {{a}, {a, b}} {{c}, {c, d}} = {{a}, {a, b}, {c}, {c, d}} (2) a, b c, d = {{a}, {a, b}} {{c}, {c, d}} = (3) a, b c, d = {{a}, {a, b}} {{c}, {c, d}} = {{a}, {a, b}, {c}, {c, d}} (4) a, b = {{a}, {a, b}} = {a} {a, b} = {a} (5) { a, b } = a, b = {{a}, {a, b}} (6) a, b, c = a, b, c = { a, b } = {{{a}, {a, b}}} (7) { a, b } = a, b = {a} (8) { a, b } 1 = { b, a } = b, a = {b} = b 2.3 a, b, c = {{a}, {a, {{b}, {b, c}}}} = a, b, c = {{{{a}, {a, b}}}, {{{a}, {a, b}}, c}} 2.4, = {{ }, {, }} = {{ }} a, {a} = {{a}, {a, {a}}} (3), (5), (7) 2.5 (1) A = B = (2) A = B A = B = (3) A = B = C = 2.6 (1) : x, y x, y (A C) (B D) (x A y C) (x B y D) ( ) (x A x B) (x A y D) (y C x B) (y C y D) ( ) = (x A x B) (y C y D) ( ) x (A B) (C D) ( ) (A C) (B D) (A B) (C D) 21

22 (2) : x, y x, y (A B) (C D) x A x / B y C y / D ( ) x A y C x / B y / D ( ) = x A y C x / B ( ) = (x A y C x / B) (x A y C y / D) ( ) (x A y C) (x / B y / D) ( ) (x A y C) ( (x B) (y D)) (/ ) (x A y C) (x B y D) ( ) ( x, y A C) ( x, y B D) ( x, y A C) ( x, y / B D) (/ ) x, y (A C) (B D) (A B) (C D) (A C) (B D) 2.7 (1) : x, y x, y (A B) C x (A B) y C x A x / B y C x A x B y C (x A x B y C) 0 (x A x B y C) (x A 0) (x A x B y C) (x A y C y C) (x A y C) ( x B y C) (x A y C) (x B y C) ( x, y A C) ( x, y B C) ( x, y A C) ( x, y / B C) x, y (A C) (B C) (A B) C = (A C) (B C) (/ ) ( ) ( ) ( ) ( ) ( ) (/ ) (2) : (A B) C =((A B) (B A)) C =((A B) C) ((B A) C) =((A C) (B C)) ((B C) (A C)) ( (1) ) 22

23 =(A C) (B C) 2.8 A = B = A B A A = 2.9 A B A B A B A B 2 A B = 2 mn : A B R 1 = R 2 = { a, 1 } R 3 = { b, 1 } R 4 = { c, 1 } R 5 = { a, 1, b, 1 } R 6 = { a, 1, c, 1 } R 7 = { b, 1, c, 1 } R 8 = { a, 1, b, 1, c, 1 } B A R 1 1 = R 1 2 = { 1, a } R 1 3 = { 1, b } R 1 4 = { 1, c } R 1 5 = { 1, a, 1, b } R 1 6 = { 1, a, 1, c } R 1 7 = { 1, b, 1, c } R 1 8 = { 1, a, 1, b, 1, c } a R S(S R a S) S S(S R S S a S) {{x}, {x, y}} S({{x}, {x, y}} R S {{x}, {x, y}} a S) (R ) {{x}, {x, y}} S({{x}, {x, y}} R (S = {x} S = {x, y}) a S) ( ) {{x}, {x, y}}({{x}, {x, y}} R (S = {x} a S) (S = {x, y} a S)) ( ) {{x}, {x, y}}({{x}, {x, y}} R (a = x (a = x a = y))) {{x}, {x, y}}({{x}, {x, y}} R (a = x a = y)) a dom R a ran R ( ) ( ) ( ) a fld R fld R = R 1 ([email protected]) ( ) 23

24 2.11 (1) R 1 R 2 = { a, b, a, c, b, d, c, c, c, d, d, b, d, d } R 1 R 2 = { b, d } R 1 R 2 = (R 1 R 2 ) (R 1 R 2 ){ a, b, a, c, c, c, c, d, d, b, d, d } (2) dom R 1 = {a, b, c} dom R 2 = {a, b, d} dom(r 1 R 2 ) = dom R 1 dom R 2 = {a, b, c, d} (3) ran R 1 = {b, c, d} ran R 2 = {b, c, d} ran R 1 ran R 2 = {b, c, d} (4) R 1 A = { a, b, c, c, c, d } R 1 {c} = { c, c, c, d } (R 1 R 2 ) A = { a, b, a, c, c, c, c, d } R 2 A = { a, c } (5) R 1 [A] = {b, c, d} R 2 [A] = {c} (R 1 R 2 )[A] = (6) R 1 R 2 = { a, c, a, d, d, d } R 2 R 1 = { a, d, b, b, b, d, c, b, c, d } R 1 R 1 = { a, d, c, c, c, d } 2.12 (1) R 1 = { {, { }},,, { },, } (2) R R = {,,, {, { }}, { },, { }, {, { }} } (3) R = R { } = {, {, { }},, } R {{ }} = { { }, } R {, { }} = R = {, {, { }}, { },,, } (4) R[ ] = R[{ }] = {{, { }}, } R[{{ }}] = { } R[{, { }}] = ran R = {{, { }}, } (5) dom R = {, { }} ran R = {{, { }}, } fld R = dom R ran R = {, { }, {, { }}} 2.13 (1) : R R R R R R 1 R R 1 R x, y x, y R R 1 x, y R x, y R 1 y, x R 1 y, x R 24

25 y, x R 1 R y, x R R 1 R R 1 R R x, y x, y R R 1 x, y R x, y R 1 x, y R y, x R ( ) = x, y R y, x R (R R ) ( x, y R x, y R ) ( y, x R y, x R ) ( x, y R x, y R 1) ( y, x R y, x R 1) ( ) ( ) ( x, y R ( x, y R ( x, y R y, x R ))) ( y, x R ( y, x R ( y, x R x, y R ))) (R ) = ( x, y R y, x R ) ( x, y R y, x R ) ( ) x, y R y, x R ( ) = x, y R ( ) R R 1 R R R 1 R (2) : R R R R R 1 R R R 1 R x, y x, y R R 1 x, y R x, y R 1 y, x R 1 y, x R y, x R 1 R y, x R R 1 R R 1 R R R x, y x, y R x, y R y, x R ( ) (R ) = x, y R y, x R (R R) x, y R x, y R 1 x, y R R 1 R R R 1 R R 1 R

26 (1) R = { 1, 2, 2, 3 } (2) : R 2 R = x z( x, z R 2 x, z R) ( ) x z ( x, z R 2 x, z R) ( ) x z( x, z R 2 x, z R) ( ) x z( ( y( x, y R y, z R)) x, z R) ( ) x z( y( ( x, y R y, z R)) x, z R) ( ) x z y( ( x, y R y, z R) x, z R) ( ) x y z( ( x, y R y, z R) x, z R) ( 2 ) x y z(( x, y R y, z R) x, z R) R 2.15 A R x x x x, x / R x x, x / R, A P(A) A A x, y P(A) x, y R y, x R x, y R y, x R x = y S A A P(A) A A A = A A, A / S P(A) =, S, A S A, S A A, S, A S A, A / S T, / T A P(A) A A = A A, A T, A T A, T A, A T A, T, / T A

27 R x x, x R x, y P(A) x, y R xry yrx x, y P(A) x, y R y, x R x, y R y, x R x = y R x, y, z x, y R y, z R x, y R y, z R x, z R S, S, S x y( x, y S y, x S x = y = x = y) x y z( x, y S y, z S x = y = z = x, z S) T = = A, T, T x y( x, y T y, x T x = y = x = y) x y z( x, y T y, z T x = y = z = x, z T ) 2.16 (1) R = { 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 0 } S = { 0, 4, 3, 3, 6, 2, 9, 1, 12, 0 } (2) R 0, 0 / R 5, 5 R 0, 10 R 10, , 10 R 10, 0 0, 0 / R S 0, 0 / S 3, 3 S 0, 4 S 4, 0 / S x, y A x, y S y, x S 27

28 12, 0 S 0, 4 S 12, 4 / S M(R) = G(R) R 1, 1 R 1, 2 R 2, 1 R 1 2 2, 0 R 0, 3 R 2, 3 / R 2.18 R 1 = { a, a, a, b, b, b, c, b, c, c } R 2 = { a, b, b, c } R 3 = { a, b, a, c, c, a, c, b } R 4 = { a, a, b, b, c, c } M(R 1 ) = 0 1 0, M(R 2) = 0 0 1, M(R 3) = 0 0 0, M(R 4) = R 1 a, a R 1 a, b R 1 b, a / R 1 R 2 a, a / R 2 a, b R 2 b, a / R 2 a, b R 2 b, c R 2 a, c / R 2 R 3 a, a / R 3 a, b R 3 b, a / R 3 28

29 a, c R 3 c, a R 3 a c a, c R 3 c, a R 3 a, a / R 3 R 4 a, a R R 1 = { a, a, a, b, b, a, b, b, b, c, c, a, c, c } R 2 = { a, a, a, b, b, b, c, a, c, b } R 3 = { a, b, a, c, b, a, b, c, c, a, c, b } R 4 = { a, a, a, b, a, c, b, c, c, a } a a a a b c b c b c b c G(R 1 ) G(R 2 ) G(R 3 ) G(R 4 ) R 1 a, a R 1 c, a R 1 a, c / R 1 a, b R 1 b, a R 1 a b c, a R 1 a, b R 1 c, b / R 1 R 2 c, c / R 2 a, a R 2 a, b R 2 b, a / R 2 R 3 a, a / R 3 a, b R 3 b, a R 3 a b a, b R 3 b, a R 3 a, a / R 3 R 4 29

30 b, b / R 4 a, a R 4 a, b R 4 b, a / R 4 a, c R 4 c, a R 4 a c c, a R 4 a, b R 4 c, b / R M(R 1 ) = a b c d G(R 1 ) M(R 1 ) = M(R 2 ) = M(R 2 R 1 ) = M(R 1 ) M(R 2 ) = = R 2 R 1 = { 1, β } 2.22 : R 2.19(1) R = r(r) = R I A R (R R) R ( 1.3) = (R R) (R I A ) (R I A = R) = R (R I A ) ( 2.6(1)) = R R (R = R I A ) R 2.14 R R R R R = R : A = {a, b} R = { a, a } R R = { a, a } = R b, b / R R R R = R (R R ) 2.23 : R S x, y x, y R S 30

31 = y, x R S (R S ) z( y, z S z, x R) ( ) = z( z, y S x, z R) (R S ) z( x, z R z, y S) ( ) x, y S R ( ) R S S R S R R S R S R S = S R R S = S R x, y x, y R S x, y S R (R S = S R) z( x, z R z, y S) ( ) = z( z, x R y, z S) (R S ) z( y, z S z, x R) ( ) y, x R S ( ) 2.24 R 1 = R 2 = { 1, 1 } R 3 = { 2, 2 } R 4 = { 1, 2 } R 5 = { 2, 1 } R 6 = { 1, 1, 1, 2 } R 7 = { 1, 1, 2, 1 } R 8 = { 1, 1, 2, 2 } R 9 = { 1, 2, 2, 1 } R 10 = { 1, 2, 2, 2 } R 11 = { 2, 1, 2, 2 } R 12 = { 1, 1, 1, 2, 2, 1 } R 13 = { 1, 1, 1, 2, 2, 2 } R 14 = { 1, 1, 2, 1, 2, 2 } R 15 = { 1, 2, 2, 1, 2, 2 } R 16 = { 1, 1, 1, 2, 2, 1, 2, 2 } R 8, R 13, R 14, R 16 R 1, R 4, R 5, R 9 R 1, R 2, R 3, R 8, R 9, R 12, R 15, R 16 R 1, R 2, R 3, R 4, R 5, R 6, R 7, R 8, R 10, R 11, R 13, R 14 31

32 R 1, R 2, R 3, R 4, R 5, R 6, R 7, R 8, R 10, R 11, R 13, R 14, R 16 R 1 R 8 R 16 R 13 R 4 R 14 R 5 R I dom R R 1 R : x, y x, y I dom R x = y x dom R ( ) x = y z( x, z R) x = y z( x, z R z, x R 1 ) x = y x, x R 1 R ( ) x, y = x, x x, x R 1 R ( 2.1) = x, y R 1 R ( ) I ran R R R 1 : x, y x, y I ran R x = y x ran R ( ) x = y z( z, x R) ( ) x = y z( z, x R x, z R 1 ) x = y z( x, z R 1 z, x R) ( ) x = y x, x R R 1 ( ) x, y = x, x x, x R R 1 ( 2.1) = x, y R R 1 ( ) 2.26 (1) M(R) = M(R 2 ) = M(R 3 ) = R 2 = { a, a, a, c, b, b, b, d } R 3 = { a, b, a, d, b, a, b, c } (2) m = 2, n = 4 (3) R 2 = R 4 R 1 = R 3 R 2 = I A ( ) A (2) m n 32

33 N R 1 = { x, y x, y N y = x + 1} k N R1 k = { x, y x, y N y = x + k} m, n N m n R1 m = R1 n R 1, R 2 dom(r 1 R 2 ) dom R 2 ran(r 1 R 2 ) ran R 1 m N + dom(r1 m ) dom R 1 ran(r1 m ) ran R 1 : x, y x, y R 1 R 2 z( x, z R 2 z, y R 1 ) ( ) = z( x, z R 2 ) z( z, y R 1 ) ( ) x dom R 2 y ran R 1 (dom ran ) dom(r 1 R 2 ) dom R 2 ran(r 1 R 2 ) ran R 1 R 2 = R 1 m m N +, dom(r1 m ) dom R 1 ran(r1 m ) ran R R 1, R 2 fld R 1 fld R 2 = m, n N + (R1 m R2 n = ) : R 1, R 2 ((fld R 1 fld R 2 = ) (R 1 R 2 = )) x, y (R 1 R 2 ) z( x, z R 2 z, y R 1 ) z ran R 2 fld R 2 z dom R 1 fld R 1 z fld R 1 fld R 2 R 1, R 2 ((fld R 1 fld R 2 = ) (R 1 R 2 = )) 2.1 dom(r1 m ) dom R 1 ran(r1 m ) ran R 1 fld(r1 m ) = dom(r1 m ) ran(r1 m ) (fld ) dom R 1 ran R 1 ( ) = fld R 1 (fld ) fld(r2 n ) fld R 2 fld(r1 m ) fld(r2 n ) fld R 1 fld R 2 ( 1.5) = ( ) R1 m R2 n = : m = 0 R 1 R 2 A (R 1 R 2 ) 0 = I A = I A I A ( ) = R1 0 R2 0 m = k (k N) (R 1 R 2 ) k = R1 k R2 k m = k + 1 (R 1 R 2 ) k+1 = (R 1 R 2 ) k (R 1 R 2 ) = (R1 k R2) k (R 1 R 2 ) ( ) = (R1 k R2) k R 1 (R1 k R2) k R 2 ( 2.6(1)) 33

34 2.28 m = 0, n = 15 3 = (R k 1 R 1 ) (R k 2 R 1 ) (R k 1 R 2 ) (R k 2 R 2 ) ( 2.6(2)) = R k+1 1 (R k 2 R 1 ) (R k 1 R 2 ) R k+1 2 = R k+1 1 R k+1 2 ( 2.2) = R k+1 1 R k+1 2 ( ) 2.29 r(r) = { a, a, a, b, b, b, c, c, c, d, d, d } s(r) = { a, a, a, b, b, a, b, b, c, d, d, c } t(r) = { a, a, a, b, b, b, c, d } a a a b d b d b d c r(r) c s(r) c t(r) 2.30 (1) : R + = t(r) 2.19(3) (R + ) + = t(r + ) = R + (2) : R = rt(r) 2.25(3) R 2.19(3) trt(r) = rt(r) 2.19(1) 2.25(3) rtrt(r) = trt(r) (R ) = rtrt(r) = trt(r) = rt(r) = R (3) : R R = R i=0 R i ( ) = R R i ( 2.6(1)) i=0 = R i+1 ( 2.17(1)) i=0 = R i (i := i + 1) i=1 = t(r) ( 2.24) = R + ( ) R + = R R 3 m, n(m n) m, n(m < n) m = n = 0 34

35 2.31 a d g e a c d b r(r) f a d g e g e c b f c b f s(r) t(r) 2.32 : A = I A I B = P A Q A = P 1 B Q 1 B = P A P 1 A = P 1 B (P 1 ) 1 B = P A P T A = P 1 B (P T ) 1 = P 1 B (P 1 ) T B = P 1 A Q 1 C = P 2 B Q 2 C = P 2 P 1 A Q 1 Q 2 B = P A P 1 C = Q B Q 1 C = Q P A P 1 Q 1 = Q P A (Q P ) 1 B = P A P T C = Q B Q T C = Q P A P T Q T = Q P A (Q P ) T 2.33 : a + bi C a a 0 a 2 > 0 a + bi, a + bi R a + bi, c + di R a + bi C c + di C ac > 0 c + di C a + bi C ca > 0 c + di, a + bi R a 1 + b 1 i, a 2 + b 2 i, a 2 + b 2 i, a 3 + b 3 i R a 1 + b 1 i C a 3 + b 3 i C a 1 a 2 > 0 a 2 a 3 > 0 ab > 0 sgn(a) = sgn(b) 4 a 1 a 2 > 0 a 2 a 3 sgn(a 1 ) = sgn(a 2 ) sgn(a 2 ) = sgn(a 3 ) ( ) = sgn(a 1 ) = sgn(a 3 ) a 1 a 3 > 0 ( ) a 1 + b 1 i, a 3 + b 3 i R R C /R = {{a + bi a + bi C a > 0}, {a + bi a + bi C a < 0}} R y C /R y y 2.34 (1) 1 x < 0 4 sgn (signum) sgn(x) = 0 x = 0 1 x > 0 35

36 : x(x A x, x R 1 ) x(x A x, x / R 1 ) A x A x, x / R 1 R 1 (2) : x(x A ( x, x R 1 x, x R 2 )) x(x A x, x / R 1 R 2 ) A x A x, x / R 1 R 2 R 1 R 2 R 2 R 1 (3) : A = {a, b, c}, R 1 = E, R 2 = { a, c, c, a } I A R 1 R 2 r(r 1 R 2 ) = { a, b, b, a, b, c, c, b } I A a, b r(r 1 R 2 ) b, c r(r 1 R 2 ) a, c / r(r 1 R 2 ) r(r 1 R 2 ) r(r 2 R 1 ) R 1 = R 2 = I A R 1 = R 2 = r(r 1 R 2 ) = r(r 2 R 1 ) = I A R 1 R 2 r(r 1 R 2 ) r(r 2 R 1 ) (4) : A = {a, b, c}, R 1 = { a, b, b, a } I A, R 2 = { a, c, c, a } I A R 1 R 2 R 1 R 2 = { a, b, a, c, b, a, c, a, c, b } I A ( c, b R 1 R 2 b, c / R 1 R 2 ) R 2 R 1 R 1 = R 2 = I A R 1 = R 2 = R 1 R 2 = R 2 R 1 = I A R 1 R 2 R 1 R 2 R 2 R : R x, y A x, y R x, y R 1 x, y R x, x R ( ) (R ) = y, x R ( (2)) R x, y, z A x, y R y, z R = y, x R y, z R (R ) = x, z R ( (2)) R R A 2.36 : (1) B ik / π 2 (2) j, k {1, 2,..., m} j k B ij B ik = (A ij B) (A ik B) ( ) = A ij A ik B B ( ) 36

37 = B B (π 1 ) = ( ) (3) A B = π 1 B (π 1 A ) ( n ) = A i B i=1 n = (A i B) ( ) i=1 m = (A ik B) (A i B m ) k=1 m = ( ) k=1 B ik = π 2 π 2 π 2 A B 2.37 : 2.38 A/R = {{1, 6, 11, 16}, {2, 7, 12, 17}, {3, 8, 13, 18}, {4, 9, 14, 19}, {5, 10, 15, 20}} : (1) A / A (2) A i1 B j1, A i2 B j2 A A i1 B j1 A i2 B j2 x(x A i1 B j1 A i2 B j2 ) x(x A i1 x B j1 x A i2 x B j2 ) x(x A i1 x A i2 x B j1 x B j2 ) ( ) ( ) ( ) = x(x A i1 x A i2 ) x(x B j1 x B j2 ) ( ) x(x A i1 A i2 ) x(x B j1 B j2 ) ( ) A i1 A i2 B j1 B j2 ( ) = A i1 = A i2 B j1 = B j2 (π 1, π 2 ) = A i1 B j1 = A i2 B j2 ( ) A i1 B j1, A i2 B j2 R A i1 B j1 A i2 B j2 A i1 B j1 A i2 B j2 = (3) A = A A ( ) = ( π 1 ) ( π 2 ) (π 1, π 2 ) ( m ) ( n ) = A i B j (π 1, π 2 ) i=1 = 1 i m 1 j n j=1 (A i B j ) ( ) = A (A ) 37

38 A A π 1 π 2 A i B j A 1.2 A i B j A i A i B j B j A π 1 π 2 A π 1 π (1) R π = { 1, 2, 1, 3, 2, 1, 2, 3, 3, 1, 3, 2 } I A A/R π = π = {{1, 2, 3}, {4}} (2) π 1 = A/R π1 = {{1}, {2}, {3}, {4}} R π1 = I A π 2 = A/R π2 = {{1, 2}, {3}, {4}} R π2 = { 1, 2, 2, 1 } I A π 3 = A/R π3 = {{1, 3}, {2}, {4}} R π3 = { 1, 3, 3, 1 } I A π 4 = A/R π4 = {{1}, {2, 3}, {4}} R π4 = { 2, 3, 3, 2 } I A π 5 = A/R π5 = π R π5 = R π = { 1, 2, 1, 3, 2, 1, 2, 3, 3, 1, 3, 2 } I A 2.40 : A (A A/R 1 B(B A/R 2 A B)) x, y A x, y R 1 A (A A/R 1 x A y A ) ( ) = A A/R 1 x A y A ( ) A A/R 1 x A y A B(B A/R 2 A B) B(A A/R 1 x A y A B A/R 2 A B) ( ) ( ) = A A/R 1 x A y A B A/R 2 A B ( ) = B A/R 2 x A y A A B ( ) = B A/R 2 x B y B ( ) = B(B A/R 2 x B y B) ( ) x, y R 2 ( ) A x y(a A/R 1 x A y A B(B A/R 2 x B y B)) A, x, y A A/R 1 x A y A x, y R 1 x, y R 1 R 1 R 2 ( ) ( ) = x, y R 2 ( ) B(B A/R 2 x B y B) ( ) 38

39 2.41 : R 3 x, y x, y A B x A y B = x, x R 1 y, y R 2 (R 1, R 2 ) x, y, x, y R 3 R 3 x 1, x 2, y 1, y 2 x 1, y 1, x 2, y 2 R 3 x 1, x 2 R 1 y 1, y 2 R 2 (R 3 ) (R 3 ) = x 2, x 1 R 1 y 2, y 1 R 2 (R 1, R 2 ) x 2, y 2, x 1, y 1 R 3 R 3 x 1, x 2, x 3, y 1, y 2, y 3 x 1, y 1, x 2, y 2 R 3 x 2, y 2, x 3, y 3 R 3 x 1, x 2 R 1 y 1, y 2 R 2 x 2, x 3 R 1 y 2, y 3 R 2 x 1, x 2 R 1 x 2, x 3 R 1 y 1, y 2 R 2 y 2, y 3 R 2 (R 3 ) (R 3 ) ( ) = x 1, x 3 R 1 y 1, y 3 R 2 (R 1, R 2 ) x 1, y 1, x 3, y 3 R 3 R 3 (R 3 ) 2.42 { 4 2} = = 7 R 1 = { b, c, b, d, c, b, c, d, d, b, d, c } I A R 2 = { a, c, a, d, c, a, c, d, d, a, d, c } I A R 3 = { a, b, a, d, b, a, b, d, d, a, d, b } I A R 4 = { a, b, a, c, b, a, b, c, c, a, c, b } I A R 5 = { a, b, b, a, c, d, d, c } I A R 6 = { a, c, c, a, b, d, d, b } I A R 7 = { a, d, d, a, b, c, c, b } I A 2.43 { } { } { } { } { } ( 5 = 1 + (2 4 1) { } { }) 4 + C = 1 + (2 4 1) + (3 C ) + C = = 52 39

40 (1) R R R 3 1 R 4 (2) R e b c d a 1 a b d 2 c e (1) A, 1 e a (2) A, 2 a, d, e a, b, c, e 2.46 B {k [1, 2,..., 10] k N + } [1, 2,..., 10] = B A B 1 = 1, 2, 6, 18, 54 B 2 = 1, 3, 9, 27, 54 B 3 = 1, 3, 6, 18, 54 B 4 = 1, 3, 9, 18, (2) A 5 A (1) 40

41 : R B x x B = x A (B A) = x, x / R (R ) x, x R (/ ) = x, x R x, x B B ( ) ( x, x R x, x B B) ( ) ( x, x R B B) ( x, x R B) (R B ) x, x / R B (/ ) R B x, y, z x, y R B y, z R B x, y R B B y, z R B B (R B ) x, y R x, y B B y, z R y, z B B x, y R x B y B y, z R y B z B x, y R y, z R x B y B z B ( ) = x, z R x B y B z B (R ) = x, z R x B z B ( ) x, z R x, z B B x, z R B (R B ) R B (2) : R B x x B x B x B ( ) = x A x B (B A) = x, x R (R ) x, x R x, x B B x, x R B B x, x R B (R B ) R B x, y 41

42 x, y R B y, x R B x, y R B B y, x R B B (R B ) x, y R x, y B B y, x R y, x B B x, y R x B y B y, x R y B x B x, y R y, x R ( ) = x = y (R ) R B ( (1) ) R B (3) : (2) R B x, y B R B x, y x B y B x B y B x B y B ( ) = x A y A x B y B (B A) = ( x, y R y, x R) x B y B (R ) ( x, y R x B y B) ( y, x R y B x B) ( ) x, y R B y, x R B (R B ) R B (4) : (3) R B C B R B C B B A C A R A y(y C x(x C y, x R)) C B x, y x B y B y, x R B R B B 2.49 : R x, y x, y A B x A y B = y, y R 2 x, x R 1 (R 1, R 2 ) = y, y R 2 ( x, x R 1 y = y) ( ) y, y R 2 ( x, x R 1 y = y) ( ) ( y, y R 2 ( x, x R 1 y = y)) ( ) ( x, y, x, y R) (R ) x, y, x, y / R (/ ) R x 1, x 2, x 3, y 1, y 2, y 3 42

43 x 1, y 1, x 2, y 2 R x 2, y 2, x 3, y 3 R ( y 1, y 2 R 2 ( x 1, x 2 R 1 y 1 = y 2 )) ( y 2, y 3 R 2 ( x 2, x 3 R 1 y 2 = y 3 )) (R ) ( y 1, y 2 R 2 y 2, y 3 R 2 ) ( y 1, y 2 R 2 x 2, x 3 R 1 y 2 = y 3 ) ( x 1, x 2 R 1 y 1 = y 2 y 2, y 3 R 2 ) ( x 1, x 2 R 1 y 1 = y 2 x 2, x 3 R 1 y 2 = y 3 ) ( ) 4 1 R 2 y 1, y 3 R 2 2 y 1, y 2 R 2 y 2 = y 3 y 1, y 3 R 2 3 y 2, y 3 R 2 y 1 = y 2 y 1, y 3 R 2 4 R 1 = x 1, x 3 R 1 y 1 = y 3 4 x 1, y 1, x 3, y 3 R R R 2.50 : R x, y x, y A B x A y B = x, x R 1 y, y R 2 (R 1, R 2 ) x, y, x, y R (R ) R x 1, x 2, y 1, y 2 x 1, y 1, x 2, y 2 R x 2, y 2, x 1, y 1 R x 1, x 2 R 1 y 1, y 2 R 2 x 2, x 1 R 1 y 2, y 1 R 2 (R ) x 1, x 2 R 1 x 2, x 1 R 1 y 1, y 2 R 2 y 2, y 1 R 2 ( ) = x 1 = x 2 y 1 = y 2 (R 1, R 2 ) x 1, y 1 = x 2, y 2 ( 2.1) x 1, y 1, x 2, y 2 = x 2, y 2, x 1, y 1 ( 2.1) R x 1, x 2, x 3, y 1, y 2, x 3 x 1, y 1, x 2, y 2 R x 2, y 2, x 3, y 3 R x 1, x 2 R 1 y 1, y 2 R 2 x 2, x 3 R 1 y 2, y 3 R 2 (R ) x 1, x 2 R 1 x 2, x 3 R 1 y 1, y 2 R 2 y 2, y 3 R 2 ( ) = x 1, x 3 R 1 y 1, y 3 R 2 (R 1, R 2 ) x 1, y 1, x 3, y 3 R (R ) R

44 8, 6 8, 3 8, 2 4, , 1 4, 3 4, 2 2, , 1 2, 3 2, 2 1, R 1 R 2 2, 1 1, 3 1, 2 1, 1 R : 2.40 x, y X(x y R x R y ) R x R y x y R X A

45 3.1 R 2, R 3, R 6, R 7 A B R 2, R 6 A B f g : f, g A B x, y, z x, y f g x, z f g x, y f x, y g x, z f x, z g = x, y f x, z f ( ) = y = z (f ) f g 2 f g A B f = g : x, y f x, y / g x, y / f x, y g x, y f x, y / g x / dom(f g) ( z x, z f g x, z g x, y / g z y x, y f x, z f g f y z x, y f x, z f f ) f g 3 f g f g A B f = g f g f g A B : f g A B f g f g dom(f g) = dom f dom g ( 2.3(1)) = A A (f, g A B) = A ( ) f g A B f g f g A B f g f g A B f g A B f = g 45

46 : f g x, y f x, y / g x, y / f x, y g x, y f x, y / g g z x, z g x, y / g z y x, y (f g) x, z (f g) z y f g 3.3 (1), (2), (6), (10) (1), (4), (5), (6), (9), (10) (1), (6), (10) 3.4 f = { S, F S, F A B x(x A (x S F (x) = 1))} f A B f 1 B A A B = A B = : A = B A = B A = A B A B B a B f = { x, a x A} A ( A f A B ) f A B A B A = B A B A B = A B = A B = : 3.1 A B = B A = A B = B A = A B = B A A B B A f A B A B = B A f B A A = dom f (f A B) = B (f B A) 3.6 : f f C A x y z(x dom f y ran f z ran f xfy xfz y = z) dom f = C ran f A = x y z(x dom f y ran f z ran f xfy xfz y = z) dom f = C ran f B f C B (A B ) 46

47 3.7 ( 10 ) 3.2 f, g A B f g dom g dom f f = g : f g g f x, y x, y g = x dom g (dom ) = x dom f (dom g dom f) z( x, z f) z( x, z f x, z f) (dom ) = z( x, z f x, z g) (f g) ( ) = z( x, z f z = y) (g x, y g) = x, y f ( ) A = N, f : N N, f(x) = x + 1, g : N N, g(x) = x/2 f f ( 0 N 0 / ran f ) k N 2k, k g g k N 2k, k g 2k + 1, k g 2k 2k + 1 g : 1 y(y dom g g(y) ) : y y dom g y B (dom g = B) = x(x A x, y f) (f ) x(x g(y)) g(y) g y 1, y 2 B, s P(A) y 1, s g y 2, s g = x(x s x, y 1 f) x(x s x, y 2 f) s = g(y 1 ) (g ) x((x s x, y 1 f) (x s x, y 2 f)) s = g(y 1 ) 47 (g ) ( ) ( )

48 = x((x s x, y 1 f) (x s x, y 2 f)) s ( 1) x((x s x, y 1 f) (x s x, y 2 f)) x(x s) x(( x s x, y 1 f) ( x s x, y 2 f)) x(x s) x( x s ( x, y 1 f x, y 2 f)) x(x s) x(x s ( x, y 1 f x, y 2 f)) x(x s) ( ) ( ) = ( x(x s) x( x, y 1 f x, y 2 f)) x(x s) ( ) = x( x, y 1 f x, y 2 f) ( ) = y 1 = y 2 (f ) g g 3.12 : x R x, 0, x f x, 1, x g f, g x R x, 0, x f x 1, 1, x f x x 1 x, 0, 0 g x + 1, 0, 0 g x x 1 f, g 3.13 : f : E F, f(x) = A/x f(x) f A R, S E f(r) = f(s) f A/R = A/S x, y x, y R B(x B y B B A/R) B(x B y B B A/S) ( ) (A/R = A/S) x, y S ( ) f(r) = f(s) R = S f f A A F R A = { x, y x, y A B(x B y B B A )} R A E 1 f(r A ) = A R A E x x A x A B(x B B A ) B(x B x B B A ) x, x R A x, y (A ) ( ) (R A ) x, y R A (2) 48

49 B(x B y B B A ) B(y B x B B A ) y, x R A x, y, z (R A ) ( ) (R A ) x, y R A y, z R A B(x B y B B A ) B(y B z B B A ) (R A ) = x B 1 y B 1 B 1 A y B 2 z B 2 B 2 A ( ) = x B 1 B 1 A z B 2 B 2 A y B 1 y B 2 ( ) = x B 1 B 1 A z B 2 B 2 A y B 1 B 2 = x B 1 B 1 A z B 2 B 2 A B 1 B 2 ( ) = x B 1 B 1 A z B 2 B 2 A B 1 = B 2 (B 1 B 2 B 1 = B 2 2 ) = x B 1 z B 2 B 1 A ( ) = B(x B z B B A ) ( ) x, z R A R A E R A f(r A ) = A f f 3.14 : S f f A (R A ) = x(x [0, 1] (f(x) f(x)) = 0) (f [0, 1] R) = x(x [0, 1] (f(x) f(x)) 0) ( ) f, f S S f, g (S ) f, g A g, f A x(x [0, 1] (f(x) g(x)) 0) x(x [0, 1] (g(x) f(x)) 0) (S ) x((x [0, 1] (f(x) g(x)) 0) (x [0, 1] (g(x) f(x)) 0)) ( ) x(( x [0, 1] (f(x) g(x)) 0) ( x [0, 1] (g(x) f(x)) 0)) x( x [0, 1] ((f(x) g(x)) 0 (g(x) f(x)) 0)) ( ) 2 (2) 49

50 x( x [0, 1] ((f(x) g(x)) = 0)) ( ) x(x [0, 1] ((f(x) g(x)) = 0)) f = g ( ) S f, g, h f, g A g, h A x(x [0, 1] (f(x) g(x)) 0) x(x [0, 1] (g(x) h(x)) 0) (S ) x((x [0, 1] (f(x) g(x)) 0) (x [0, 1] (g(x) h(x)) 0)) ( ) x(( x [0, 1] (f(x) g(x)) 0) ( x [0, 1] (g(x) h(x)) 0)) x( x [0, 1] ((f(x) g(x)) 0 (g(x) h(x)) 0)) ( ) x( x [0, 1] ((f(x) h(x)) = (f(x) g(x)) + (g(x) h(x)) 0)) ( ) x(x [0, 1] ((f(x) h(x)) 0)) f, g A (S ) S S f : [0, 1] R, f(x) = x g : [0, 1] R, g(x) = 1 x 0, 1 [0, 1] f(0) g(0) < 0 g(1) f(1) < 0 f, g / S g, f / S S 3.15 (1) N/R 1 = {{x} x N} N/R 2 = {{2k + j k N} j {0, 1}} N/R 3 = {{3k + j k N} j {0, 1, 2}} N/R 4 = {{6k + j k N} j {0, 1, 2, 3, 4, 5}} (2) N/R 2 N/R 3 N/R 4 N/R 1 (3) f 1 (H) = H f 2 (H) = {0} f 3 (H) = {0, 1, 2} f 4 (H) = {0, 2, 4} 3.16 g f(x) = x f g(x) = x 2 + 4x + 14 f g, h 50

51 g 1 (x) = x 4; h 1 (x) = 3 x A A R f : A A/R R = I A : f f x, y A x, y R = [x] R = [y] R ( 2.27(2)) f(x) = f(y) x = y (f ) (f ) x, y I A R I A R I A R R = I A 2 R = I A f f 1 : A/R A, f 1 ([x]) = x : (1) dom f = R {0} = (, 0) (0, + ) ran f = R {0} = (, 0) (0, + ) dom g = R ran g = {0} R + = R R = [0, + ) dom h = {0} R + = R R = [0, + ) ran h = {0} R + = R R = [0, + ) (2) dom f, dom g, dom h f, g, h 3.19 (1) f(a 1 ) = {1, 2, 3}; f 1 (B 1 ) = {0, 4, 5, 6} (2) g(a 2 ) = N; g 1 (B 2 ) = {2k + 1 k N} {6} (3) f g 3.20 (1) : f g 3.5(2) g g g g g 1 g g f = f I B = f g g 1 f g g 1 3.4(2) f = f g g 1 (2) : f g 3.5(1) f f f f f 1 f f 51

52 g = I B g = f 1 f g f 1 f g 3.4(1) g = f 1 f g f, g A B f = g x(x A f(x) = g(x)) : f = g x x A y( x, y f) (f A B) = x, a f ( ) f(x) = a (f(x) ) f(x) = a f(x) = a ( ) f(x) = a x, a f (f(x) ) f(x) = a x, a g (f = g) f(x) = a g(x) = a (g(x) ) f(x) = g(x) f = g x(x A f(x) = g(x)) x(x A f(x) = g(x)) x, y x, y f f(x) = y (f(x) ) g(x) = y (f(x) = g(x)) x, y g (g(x) ) x(x A f(x) = g(x)) f = g : f h 1 = g h dom(f h 1 ) = dom(g h 1 ) = dom h 1 = A 3.3 f h 1 = g h 1 x(x A f h 1 (x) = g h 1 (x)) A x A x X f(x) = g(x) x A, f h 1 (x) = f(h 1 (x)) ( 3.3) = f(x) (h 1 (x) = x) = g(x) (f(x) = g(x)) = g(h 1 (x)) (h 1 (x) = x) = g h 1 (x) ( 3.3) B A 52

53 3.3 f h 2 = g h 2 x(x B f h 2 (x) = g h 2 (x)) x x B = f h 2 (x) = g h 2 (x) x X ( ) f(h 2 (x)) = g(h 2 (x)) x X ( 3.3) f(x) = g(x) x X (h 2 (x) = x) x A (A ) B A 3.22 f, g (X X)(h f = h g f = g) h : h x 1, x 2 X x 1 x 2 h(x 1 ) = h(x 2 ) f : X X, f(x) = x 1 g : X X, g(x) = x 2 h f = h g f g h f = h g f = g h f = h g f = g h h 3.10(1) h h h f, g (X X) h f = h g f = I X f ( 3.6) = h h f (h h ) = h h g (h f = h g) = I X g (h h ) = g ( 3.6) f, g (X X) h f = h g f = g h f, g (X X)(f h = g h f = g) h : h a X x(x X h(x) a) ( h(a) a ) x x a f : X X, f(x) = x g : X X, g(x) = f h = g h f g h(a) x = a f h = g h f = g f h = g h f = g h h 3.10(2) h h h f, g (X X) f h = g h f = f I X ( 3.6) = f h h (h h ) = g h h (h f = h g) = g I X (h h ) = g ( 3.6) f, g (X X) f h = g h f = g h

54 3.4 A f A A f n A A (n N) : n n = 0 f n = f 0 = I A A A n = k (k 0) n = k + 1 f k+1 = f k f A A ( 3.3) : I A I A f n = f n 1 f = f f n 1 = I A n n 1 n 1 0 ( ) (n 1) N 3.4 f n 1 A A f n 1 f = I A 3.5(3) f f f n 1 = I A 3.5(3) f f 3.24 : 2.9(3) f 1 (A B) f 1 (A) f 1 (B) f 1 (A) f 1 (B) f 1 (A B) x x f 1 (A) f 1 (B) x f 1 (A) x f 1 (B) x X f(x) A x X f(x) B x X f(x) A f(x) B x X f(x) A B x f 1 (A B) f 1 (A B) = f 1 (A) f 1 (B) ( ) ( ) ( ) 54

55 4.1 (1) (2) (3) (4) a = 4.2 (1) 2 3 = 3; (2) 2 3 = 2; (3) 5 = 4; (4) 6 = 0; (5) 7 = 5; 4.3 : S = {n n N n 0 m(m N n = m + )} S = S {0} S N (1) S = 0 S (2) n S ( S n N ) N n + N n n {n} = n + n + 0 n + n m(m N n = m + ) n + S S S N S = N S = S {0} : S = {n m N(m m + n + )} (1) 0 S m N, m m {m} = m + = (m + 0) + = m (2) n S m N, m m + n + (m + n + ) {(m + n + )} = (m + n + ) + = (A m (n + )) + = A m (n ++ ) = m + (n + ) + n + S S = N 4.5 : A x x A + x A {A} ( ) 55

56 x A x {A} x A x = A ( ) = x A x = A ( 4.10) = x A x A (x = A x A) x A ( ) = x A x {A} ( ) x A {A} x A + ( ) 4.10 A (1) : A x, x A y(y A x y) = y(y A x y) (y 4.10) = y(y A x y) = x A ( ) 4.10 A (2) : A A x, x A y(y A x y) = y(y A x y) (y 4.10) y(y A z(z x z y)) ( ) y z(y A (z x z y)) ( ) y z( y A ( z x z y)) y z( z x ( y A z y)) ( ) y z(z x (y A z y)) z(z x y(y A z y)) ( ) z(z x z A ) x A ( ) 4.10 A 4.7 : S = {n n N m(m N m n h(m) = h(n))} 0 / S 0 S S h m N, m 0 56

57 h(m) = h(0) = a m ( 0 ) m n h(m) = h(n + ) = f(h(n)) = a a A ran f 0 / S S N S n 0 S m 0 N, m 0 n 0 h(m 0 ) = h(n 0 ) m 0 n 0 N, n 0 m 0 h(n 0 ) = h(m 0 ) m 0 S m 0, n 0 S 0 / S m 0 0, n m p, n p N m 0 = m + p, n 0 = n + p h f(h(m p )) = h(m + p ) = h(m 0 ) = h(n 0 ) = h(n + p ) = f(h(n p )) f f(h(m p )) = f(h(n p )) h(m p ) = h(n p ) 4.4 m 0 n 0 m p n p n p S n 0 = n + p > n p n 0 S S n, m N, n m h(n) = h(m) h 57

58 5.1 : f : A B x A, f(x) = x I A f R 1, R 2 A f(r 1 ) = f(r 2 ) (I A R 1 ) ((R 1 I A ) R 1 ) ( 1.2) (I A R 1 ) ((R 1 I A ) R 1 ) (I A R 1 ) ((R 2 I A ) R 1 ) (f(r 1 ) = f(r 2 )) = (I A (R 2 I A )) R 1 R 1 ( 1.4) (I A (R 2 I A )) R 1 ( ) (I A (R 2 I A )) R 1 (I A R 2 ) (I A I A ) R 1 ( ) (I A R 2 ) E R 1 ( ) (I A R 2 ) R 1 ( ) R 2 R 1 (I A R 2 ) R 2 R 1 R 1 R 2 f(r 1 ) = f(r 2 ) R 1 = R 2 f R B 2.29(3) R I A A f(r I A ) = R f f A B A B 5.2 (1) (2) : f : ((A A)/R) (P(A) ), x (A A)/R, f(x) = ran(x) (A A)/R f S P(A) / P(A) S a S x, x S g : A A, x A, g(x) = g (A A) f([g] R ) = S a, x / S f f (A A)/R P(A)

59 : 5.1 [0, 1] R [0, 1] [a, b] f : [0, 1] [a, b], x [0, 1], f(x) = (b a)x + a f [0, 1] [a, b] 5.4 : I A A A 3.9 f : A B f 1 : B A A B B A 3.4(3) g : A B f : B C f g : A C A B B C A C 5.5 : S = {n n N x(x n m(m n x m))} (1) 0 S x x 0 (2) n S n + x 1 x = n x n n + 2 x n m n n + x m n + 3 n x x {n} n ( n x {n} n + = n {n} x {n} {n} = x x n ) m n x {n} m x m + f(y), y n ( f : x {n} m g : x m +, y x, g(y) = g m, y = n f ) 4.4 m n x m + n + x n + x n + = n {n} n / x x n x n 1 2 n x 3 m n + x m n S n + S S = N 5.6 : I A : A A A A 3.4(2) g : A B, f : B C f g f g : A C A B B C A C 5.7 : A A N A N A N A A n N n N 5.5 2(2) N A 5.7 (2) A N = A N A A N A A A N 5.14 N A Schröder-Bernstein A N 59

60 : n = 2 A B κ = card A λ = card B κ ℵ 0, λ ℵ 0 κ λ κ ℵ 0 ( 5.22(2)) = ℵ 0 κ ( 5.21(1)) ℵ 0 ℵ 0 ( 5.22(2)) = ℵ 0 ( 5.9(4)) card(a B) = κ λ ℵ 0 n = k(k 2) n = k + 1 k S n = 2 S k + 1 n = k : P(A) card P(A) ℵ ℵ 0 card A card P(A) ℵ 0 ℵ 0 card A ℵ 0 ℵ 0 card P(A) ℵ 0 Schröder-Bernstein card A = card P(A) = ℵ 0 A P(A) P(A) 5.11 (1) f : N A, x N, f(x) = (x + 1) 7 f card A = card N = ℵ 0 (2) f : N B, x N, f(x) = (x + 1) 109 f card B = card N = ℵ 0 (3) 5.7 N A A B N A B A B N 5.7 (1) A B N Schröder-Bernstein card(a B) = card N = ℵ 0 (4) C = {n 763 n N n 0} f : N C, x N, f(x) = (x + 1) 763 f N C C A B 5.7 N A B A B N 5.7 (1) A B N Schröder-Bernstein A B N card(a B) = card N = ℵ : 5.20 card P(A) = 2 card A, card P(B) = 2 card B 2 card A = 2 card B card A = card B 5.7 card A card B card B card A 5.22(4) 2 card A 2 card B 2 card B 2 card A Schröder-Bernstein 2 card A = 2 card B card P(A) = 2 card A = 2 card B = card P(B) 5.13 (1) 60

61 : n N S = {{{0}}, {1}, {2},..., {n 1}} card S = n, S N = {{0}}, x = 0 f : N S N, x N, f(x) = {x}, 0 < x < n x n, x n f n + ℵ 0 = card(s N) = card N = ℵ 0 (2) : 1 n N + S = {nm m N} f : N S, x N, f(x) = nx f S N g : (n S) N, x, y (n S), g( x, y ) = x + y g n S N n ℵ 0 = card(n S) = card N = ℵ 0 (3) : 5.1 N N N N N =, N N = N ℵ 0 + ℵ 0 = card(n N ) = card N = ℵ 0 (4) 5.1(2) 5.14 (1) : K κ card = 0, K =, K = K κ + 0 = card(k ) = card K = κ (2) : K κ K = κ 0 = card(k ) = card = 0 (3) : K κ card({ }) = 1, K K { } ( f : K K { }, x K, f(x) = x, ) κ 1 = card(k { }) = card K = κ (4) : K κ ( K) = { } κ 0 = card( K) = card({ }) = 1 (5) : K κ κ 0 K (K ) = κ 1 = card( K ) = card = 0 (6) : K 1, K 2 κ K 1 K 2 = K 1, f(x) x K 1 f : K 1 K 2 g : K 1 K 2 {K 1, K 2 } K 2, x K 1 K 2, g(x) = K 2, x, x K 2 1 n 0 card( N) = card = 0 ℵ 0 61

62 K 1 K 2 = g g x, y K 1 K 2, x y (1) x K 1, y K 2 x K 2, y K 1 K 1 K g(x) g(y) (2) x, y K 1 f f(x) f(y) 2.1 g(x) g(y) (3) x, y K g(x) g(y) g K 1 K 2 {K 1, K 2 } K 2 κ + κ = card(k 1 K 2 ) = card({k 1, K 2 } K 2 ) = 2 κ (7) : K κ f : K ({ } K), x K, f(x) = {, x } f K ({ } K) κ 1 = card({ } K) = card K = κ (8) : n N card({n}) = 1, n {n} = n + 1 = card(n {n}) = card(n + ) = n + 62

63 6.1 A, A, B, B (1) (2) A = {1, 2, 3, 4, 12} A B = {1, 2, 3, 4} B f(1) = 1, f(2) = f(3) = 2, f(4) = 3, f(12) = 4 A, A, B, B f f(2) = f(3) f f(3) B f(4) 3 A 4 f(x) B f(y) x A y A, A, B, B (1) (2) : (1) f x, y A x y f(x) = f(y) A, A x A y y A x f(x) B f(y) f(y) B f(x) f(x) = f(y) f (2) x, y A f(x) B f(y) x A y A, A x = y y A x x = y f f(x) = f(y) f(x) B f(y) y A x f(y) B f(x) f(x) B f(y) x, y A, x A y f(x) B f(y) (2) (5) 6.3 (1) : x, y A x y x, y y, x R x y x, y y, x R R I A x, y Cn 2 = n(n 1)/2 R I A R I A n(n 1)/2 R I A R R = I A + R I A I A (R I A ) = n + n(n 1)/2 = n(n + 1)/2 (2) x, y A x y x, y y, x R x A, x, x / R R = Cn 2 = n(n 1)/2 6.4 A, A, 6.1 A, A B A, : A, x, y A x y x y y x B = {x, y} A B x y x y x y B A, A, 63

64 : A Z + B = f(a) = {f(x) x A} f B N N B < b C = A f 1 (b) = {x x A f(x) = b} C Z + C < c c A R x A x c f(c) = b f(a) f(c) < f(x) f(c) = f(x) f(c) < f(x) R crx f(x) = f(c) = b x, c f 1 (b) x, c A x, c C = A f 1 (b) c C x c c < x R crx c A R A Z Z +, R 6.5 : B = {x x A f(x) x} B = B B A t B f(t) t f f(f(t)) f(t) f(t) B f(t) t t 6.6 (1) F (0) = A ( ran(f (seg 0))) (γ ) = A ( {F (x) x seg 0}) ( ) = A ( ) (seg 0 = ) = A F (1) = A ( ran(f (seg 1))) (γ ) = A ( {F (x) x seg 1}) ( ) = A ( {F (0)}) (seg 1 = {0}) = A ( {A}) (F (0) = A) = A ( A) F (2) = A ( ran(f (seg 2))) (γ ) = A ( {F (x) x seg 2}) ( ) = A ( {F (0), F (1)}) (seg 2 = {0, 1}) = A ( {A, A ( A)}) (F (0) = A, F (1) = A ( A)) = A ( (A (A ( A)))) = A ( (A ( A))) ( ) = A ( A) ( A) ( (A B) = ( A) ( B)) n N, F (n + ) = A ( F (n)) : n N F (n + ) = A ( F (n)) F (n) F (n + ) S = {x x N F (x + ) = A ( F (x)) F (x) F (x + )} F (0) = A A ( A) = A ( F (0)) = F (1) 0 S n N n 1 x N, x < n x S F (n + ) = A ( ran(f (seg(n + )))) (γ ) 64

65 = A ( {F (x) x seg(n + )}) ( ) = A ( {F (0), F (1),, F (n)}) (seg(n + ) = {0, 1,, n}) = A ( (F (0) F (1) F (n))) = A ( (F (n))) ( F (0) F (1) F (n)) F (n + ) = A ( F (n)) n 1 n 1 N n 1 S F (n) = A ( F (n 1)) F (n 1) F (n) 1.8(1) F (n 1) F (n) F (n 1) F (n) F (n) = A ( F (n 1)) A ( F (n)) = F (n + ) x N, x < n x S n + S N S = N n N F (n + ) = A ( F (n)) (2) : (1) F (n + ) = A ( F (n)) a F (n) = a F (n) ( 1.8(2)) = a A ( F (n)) (F (n) A ( F (n))) = a F (n + ) (F (n + ) = A ( F (n))) (3) : ran F = {F (0), F (1), F (2), } B = ran F = F (n) x B n N x F (n) (2) x F (n + ) F (n + ) ran F x F (n + ) ran F = B x B x B 4.10 B A = F (0) ran F A ran F = B 6.7 (1) : Z A S = A N S N S s x A x S x s x / S x Z N s N s x s A S = A (Z N) f : (Z N) N, x (Z N), f( x) = x Z f x y f(x) < f(y) N 6.3(3) Z N, A (Z N) Z 6.1 Z, (2) E(3) = {0, 1, 2}; E( 1) = N; E( 2) = N { 1}; E( n) = N { m m N + m < n}, n N 6.8 : f, g : A B A, A B, B f, g 3.9 f 1 : B A f f 1 = I B x, y A x A y g(x) B g(y) (g ) 65 n N

66 I B g(x) B I B g(y) ( 3.6) (f f 1 ) g(x) B (f f 1 ) g(y) (f f 1 = I B ) f(f 1 g(x)) B f(f 1 g(y)) ( ) f 1 g(x) A f 1 g(y) (f ) x, y, x A y f 1 g(x) A f 1 g(y) 6.5 x A x A f 1 g(x) f(x) B f(f 1 g(x)) = g(x) g(x) f(x), x A f(x) = g(x), x A f = g A, A B, B : F a, b A a b a b ( b a ) a F (b) ( b F (a) ) b / F (a) ( a / F (b) ) F (a) F (b) a b b a a F (a) b / F (a) F (a) F (b) F a, b A a b x x F (a) x a x = a (F (a) ) = x b (a b ) = x F (b) (F (b) ) F (a) F (b) b F (b) b / F (a) F (a) F (b) F (a) F (b) a F (a) F (b) a b F F (a) F (b) F (a) F (b) a b a b a b F (a) F (b) F A, S, 6.10 : α β α β A, α, α f : A α B, 0 β, β g : B β B A g 1 : β A α β f : A β f g 1 : β β f g 1 x, y β x y I β (x) I β (y) (I β ) g g 1 (x) g g 1 (y) (g g 1 = I β ) g(g 1 (x)) g(g 1 (y)) ( 3.3) g 1 (x) 0 g 1 (y) (g ) = g 1 (x) g 1 (y) ( 0 ) f(g 1 (x)) f(g 1 (y)) (f ) f g x f g 1 (x), x β α β α f g 1 (α) g 1 g 1 (α) B A f f g 1 (α) α 1 F A, S, F A, S, 66

67 67

68 = G (1) 9 6 (2) (3) (4) (5) (a) G G ( 1) G i v i, v j V (G i ) v i v j d(v i ) = d(v j ) (b) V (G) 2 G v i, v j V (G) v i v j d(v i ) = d(v j ) : (a) G G i V (G i ) = n i G v V (G i ), d(v) n i 1 G i d(v) 1 v V (G i ) d(v) n i 1 G i n i (a) (b) G (a) G 0 G G : u v (u, v) xbz 68

69 G d n, d n 1,, d 2, d 1 1 d 1 1 d 1 < d 2 < < d n d i (i = 1, 2,, n) d n n G 7.5 G G 1, G 2 G 1 = G2 G 1 = G2 : f 7.3 r n 1, n 2,..., n r (r 1) r K n1,n 2,...,n r : r G = V 1, V 2,..., V r, E G = V 1, V 2,..., V r, E V i = V i = n i(i = 1, 2,..., r) f : V (G) V (G ) f(x) V i x V i(i = 1, 2,..., r) f G = G G G n 1, n 2,..., n r r K n1,n 2,...,n r r K n1,n 2,...,n r G 3-2m = 3n n = G G G 6 3- G ( G 1 G 2 ) a f a f b e b e c d c d G 1 G 2 : 6 2- G G v 1 V (G ) f(v 1 ) = a v 1 ( G 2 ) b, c, d, e, f G = G 1 G G 2 3 V (G) = 6 G K G = G G 1 G G 1 G (1) (2) 69

70 a f a f a f b e b e b e c d c d c d G 1 G (1) G (1) (6, 6, 5, 5, 3, 3, 2) (5, 4, 4, 2, 2, 1) (3, 3, 1, 1, 0) (2, 0, 0, 0) (2) (5, 3, 3, 2, 2, 1) (2, 2, 1, 1, 0) (1, 0, 1, 0) (1, 1, 0, 0) ( ) (3) (3, 3, 2, 2, 2, 2) (2, 1, 1, 2, 2) (2, 2, 2, 1, 1) (1, 1, 1, 1) a f a f a f b e b e b e c d G 7.7(2) c G 1 d 7.7(3) c G 2 d 7.8 a d a d a d a d a d b c b c b c b c b c G 1 G 2 G 3 G 4 G 5 a d a d a d a d a d b c b c b c b c b c G 6 G 7 G 8 G 9 G 10 a d a a a a b c b c b c b c b c G 11 G 12 G 13 G 14 G 15 a a a b b G 16 G 17 G 18 G 19 70

71 G 1 G 11 K 4 G 6, G 18, G a a a a a b c b c b c b c b c G 1 G 2 G 3 G 4 G 5 a a a a a b c b c b c b c b c G 6 G 7 G 8 G 9 G 10 a a a a a b c b c b c b c b c G 11 G 12 G 13 G 14 G 15 a a a a a b c b b b G 16 G 17 G 18 G 19 G 20 G 21 G 1 G 16 G 7, G 9, G 10, G 18, G 20, G : G 5, G 6, G G 3 4 (1) δ(g) 1 G ( ) 3, 1, 1, 1 2, 2, 1, 1 G 7 G 6 (2) δ(g) = 0 G ( ) 2, 2, 2, 0 G G G 5, G 6, G 7 5 ( G i = Gj = Gk G i = Gj G k = Gm i, j, k, m 5 ) 71

72 7.11 : 2 G E(G) = E(G) E(G) + E(G) = E(K n ) = n(n 1)/2 E(G) = n(n 1)/4 E(G) 4 n(n 1) n n n 4 n 1 n = 4k n = 4k : G v 1 G 5 3 v 1 3 v 1 ( G v 1 ) 3 G v 1 3 v 2, v 3, v 4 (1) 3 3 G 3 (2) 3 v 1 G : G G G 1 G 2 G 1 d(v) v G G u, v, w V (G)((u, v), (v, w) E(G) (u, w) E(G)) ( ) ( ) : (u, v V (G), u v u v (u, v) E(G)) Γ = w 0 w 1 w k 1 w k u v ( w 0 = u w k = v) (u, v) / E(G) k 2 u, v, w V (G)((u, v), (v, w) E(G) = (u, w) E(G)) (w 0, w 1 ), (w 1, w 2 ) E(G) (w 0, w 2 ) E(G) Γ = w 0 w 2 w k 1 w k : G 7.4 G 7.15 : Γ = v 0, v 1,..., v l Γ v 0 Γ δ(g) δ(g) v i1, v i2,..., v iδ (G)(i 1 < i 2 <... < i δ (G)) v 0 δ(g) 2 δ(g) 1 Γ v i1 v iδ (G) (v iδ (G), v 0 ) Γ v 0, v 1,..., v iδ (G), v 0 δ(g) : Γ = v 0, v 1,..., v l Γ v 0 Γ δ(g) 3 v 1 v i, v j (2 i < j l) v 0 2 G G 9 3 G 7 2 G 18 72

73 v 0, v 1,..., v i, v 0 i + 1 v 0, v 1,..., v j, v 0 j + 1 v 0, v i,..., v j, v 0 j i + 2 d G d i + 1 d j + 1 d j i + 2 d (i + 1) + (j i + 2) (j + 1) = 2 (d d ) d 2 d : G κ(g) = (1) n 2 δ(g) < n 2 2n 4 < n n < 4 G p(g) 2 G G 2 2 n 2p(G) 4 n < 4 G δ(g) = κ(g) = 0 G δ(g) (G) n 1 δ(g) n 1 n 2 δ(g) = n 1 G κ(g) = δ(g) = n 1 δ(g) = n κ(g) δ(g) = n κ(g) 2δ(G) n + 2 = n 2 κ(g) = δ(g) = n (1) : v V (G) v d(v) + 1 δ(g) + 1 G G p(g) = k 2 V 1, V 2,, V k k V (G) = V i ( V (G) ) i=1 k(δ(g) + 1) ( V i δ(g) + 1) k( n 2 + 1) ( ) 2( n 2 + 1) (k 2 n ) = n + 2 > n k = 1 G (2) : V 1 G t = V 1 1 G G G δ(g V 1 ) δ(g) V 1 n + k 1 2 G V 1 (1) δ(g V 1 ) < V (G V 1) = n t 2 2 n + k 1 2 t < n t 2 t > k 1 t k G k- t 73

74 (1) : v 1 V (G) v 2 N G (v 1 ) N G (v 1 ) N G (v 2 ) = ( v 1 v 2 3 G 4 ) N G (v 1 ) = N G (v 2 ) = k G N G (v 1 ) N G (v 2 ) = N G (v 1 ) + N G (v 2 ) N G (v 1 ) N G (v 2 ) = 2k (2) : G K k,k (1) v 1, v 2 N G (v 1 ), N G (v 2 ) (1) N G (v 1 ) N G (v 2 ) = N G (v 1 ) = N G (v 2 ) = k N G (v 1 ) N G (v 2 ) = N G (v 1 ) + N G (v 2 ) = 2k = V (G) N G (v 1 ) N G (v 2 ) G u 1, u 2 N G (v i )(i = 1, 2) u 1, u 2 N G (v 1 ) N G (v 1 ) v 1 v 1, u 1, u 2, v 1 3 G 4 N G (v i )(i = 1, 2) G k- k N G (v 1 ) N G (v 2 ) G K k,k 7.3 G 7.20 : v G (G) = n 2 G v u d(g) = 2 G u v ( u v ) V (G) {u, v} v u G = G v G ( V (G ) {u} u ) 7.9 E(G ) V (G ) 1 = n 2 G G n 2 m = E(G) = E(G ) + n 2 2n n G E(G) = n p(g) p(g) G : n n = 1 n = i n = i + 1 V (G) = i + 1 G x = E(G) + p(g) x = i + 1 v V (G) G = G I G (v) G 7.18 G x = E(G) + p(g) = E(G ) + p(g ) G = G v v G p(g ) = p(g ) 1 E(G ) = E(G ) V (G ) = i E(G ) = i p(g ) x = E(G )

75 p(g ) = E(G ) + p(g ) + 1 = i + 1 E(G) = (i + 1) p(g) n = i + 1 : G G 7.5 G n 1 m < n m n G 7.22 : 2 K 2 3 B 7.20 B C B C B e = (u, v ) / E(C) C e = (u, v) E(C) 7.20 e e C (u, v) E(C) E(C ) V (C) V (C ) 2 e C V (C) V (C ) s t ( s t V (C) V (C ) 2 ) C Γ = s t Γ C s t C Γ 1 Γ 2 G E(Γ 1 ) + E(Γ 2 ) = E(C) E(Γ 1 ) E(Γ 2 ) C 1 = Γ 1 Γ, C 2 = Γ 2 Γ G B C C v v e e / C V (B) = V (C), E(B) = E(C) B = C 7.23 n = 4r, δ = s, λ = r, κ = (1) 7.24 (1) 7.12 G G (2) 7.12 (3) : 6 v K n 6 H H = K 6 (1) H K 3 K 3 K 3 K 3 3 v 1 K : 7.4 D D D 75

76 8.1 : G C = e i1 e i2 e im G C e E(G) 1 k m e ik = e C e ik Γ = e ik+1 e im e i1 e ik 1 G e G e e λ(g) 2 G G B = {B 1, B 2,, B k } G V i = V (B i ) E i = E(B i ) i = 1, 2,, k B i, B j B B i B j G (1) B i = G[V i ] (2) V i V j (3) V i V j 1 (4) G 2 H = V, E H H (5) E i E j = {(v, v) v V i V j (v, v) E(G)} (6) G = B : (1) (2) V i V j (1) B i = G[V i ] G[V j ] = B j B i B j B i B j B j B i 2- B i (3) v 1, v 2 V i V j v 1 v 2 u V i {v 1, v 2 } w V j {v 1, v 2 } u w v 1 Γ 1 u w v 2 Γ ( 7.20 V i 3 V j 3 V i 2 V i V j 2 V i = V i V j V j (2) ) B i u v 2 v 1 B j v 2 w v 1 Γ 1 Γ 2 B i B j 2- ( B i B j B i B j v 1 v 2 B i B j B i B j ) (2) V i V j V i V j ( 1.32 (2) V i V j = V i V j V i V i V j = V j V i V j (2) ) B i B j B i B j 2- B i, B j (4) H H G[V ] v V (G) V G[V {v}] v V v G[V ] G H G[V ] B s = G[V ] 76

77 B t B B t B s H B s B t V (H) 2 V (B s ) V (B t ) 2 (3) H (5) e = (x, y) E i E j x, y V i V j (3) V i V j 1 x = y e = (x, x) v V i V j (1) (v, v) E(G) (v, v) E i E j (6) B G G B G B e = (u, v) E(G) u v (u, v) H = G[{(u, v)} 2 (4) H G B s e E(B s ) E( B) V (G) V ( B) G G v B v B V (G) V ( B) G B (v, v) E(G) G v V ( B) B s B v V (B s ) (1) (v, v) E(B s ) E( B) G B G = B : G k B 1, B 2,, B k G 8.1 G B i S i = {C i1, C i2,, C ini } B i = S i S = n S i 8.1(6) G = S S i=1 C ij, C st S C ij C st S i = s C ij C st i s 8.1(5) C (x, x) C C = (x, x) ( x C ) C ij C st e e = (x, x) C ij = C st = G[{e}] C ij C st G = S 8.1 G G 8.1 G S = {C 1, C 2, C r } B i G S i = {C j C j S E(C j ) E(B i ) } S i B i = S i B i S i B i e E(B i ) C j ( G = S) C j S i e E(C j ) E( S i ) E(B i ) S i B i v e E(B i ) E( S i ) v V ( S i ) V (B i ) V ( S i ) B i S i S i B i C j S i C j B i S i e = (u, v) E(C j ) E(B i ) e ( e C j = e C j B i e E(B i ) {e} = C j B i ) u v 8.1(6) C j B s ( B s B i ) (u, v) E(B i ) E(B s ) u, v V (B i ) V (B s ) 8.1(3) B i = S i B i 8.1 G 77

78 8.3 : 2k v 1, v 2,, v 2k E = {(v 2i 1, v 2i ) i = 1, 2,, k} G = G E G 8.1 G C C E C k ( E C k ) t P 1, P 2,, P t t E(G) = t i=1 (1 t k) E(P i ) P i n i 1 s n i P i s ( P i s 1 s 1 n s + 1 s ) t s(t s m) m G G G 2k k 1 m n 1 2k 1 k t k P 1, P 2,, P k E(G) = k E(P i ) G G 1 = G[V 1 ] G v V 1 d G1 (v) = d G (v) : E(G 1 ) E(G) d G1 (v) d G (v) d G1 (v) < d G (v) (u, v) E(G) (u, v) / E(G 1 ) u v v V 1 u V 1 G 1 = G[V 1 ] V 1 (u, v) E(G 1 ) : G v 0 v 0 G 8.1 G G v 0 C 0 G = G E(C 0 ) 2 v i V (G) v i V (C 0 ) v i G d G (v i ) = d G (v i ) 2 v i / V (C 0 ) d G (v i ) = d G (v i ) G v 0 / V (C 0 ) d G (v 0 ) = d G (v 0 ) > 0 G 1 v 0 G 8.2 G G 1 C 1 G 1 C 1 v 0 G v 0 C 1 C 1 v 0 G ( C 0 ) C 1 G v 0 v 0 G v 0 v 0 v 0 ( v 0 v i v 0 v 0 v i Γ v i t t 1 v i Γ 2 v i Γ 1 v i Γ E(G) v i ) C G = G E(C) G G 1 G 8.2 G G 1 ( C 1, C 2, C k ) C 1 v 0 C 1 ( v 0 G ) C 1 G v 0 i=1 78

79 8.5 ( ) ( 000 ) β α αα α αβ β β α γ γ γα β β ββ α βα α γ β γ γ α β γβ γ β βγ αγ γ β α γ α γγ α γ ( αα ) αβγαβαγβαβββααγγγββγβγγαγαα

80 B v 1 v 5 A B A B A v 2 v 3 v 4 B A A B A B : (a) (a) 4 ( v 1, v 2,, v 5 ) (b) X A Y B G = X, Y, E G 13 G : G n m = 1 (n 1)(n 2) + 2 G (n 1)(n 2) + 2 = m 1 n(n 1) n 3 2 G u, v V (G) u v (u, v) / E(G) d(u) + d(v) n d(u) + d(v) n 1 K n u v 2n 3 d(u) + d(v) n 1 2n 3 (2n 3) (n 1) = n 2 G m E(K n ) (n 2) = 1 2 n(n 1) (n 1)+1 = 1 2 (n 1)(n 2)+1 < 1 2 (n 1)(n 2)+2 G 3 G u, v d(u) + d(v) n G m = 1 (n 1)(n 2) + 1 G n = 3 m = 2 1 (n 1)(n 2) + 1 = 2 G = T 3 2 G m = 1 2 (n 1)(n 2) + 1 G n = 4 m = 1 (n 1)(n 2 2) + 1 = 4 G = C 4 G G 8.10 m = 1 (n 1)(n 2) + 1 G 2 : C = v i1 v i2 v ik V (G) V (C) C N(C) = {v s v s V (G) V (C) v ij (v ij A (b) V (C) (v ij, v s ) E(G))} N(C) ( N(C) V (C) V (G) V (C) G ) v s N(C) N(C) v ij V (C) (v ij, v s ) E(G) 80

81 Γ = v ij+1 v ik v i1 v ij v s k C (v ij, v ij+1 ) k 1 Γ = v ij+1 v ik v i1 v ij G 8.11 {a, b, c, d, e, f, g} a b g c f d e C = abdfgec k G = V, E V = {v 1, v 2,, v 2k } k 2 G = 2k 4 v i V d(v i ) = k G C = v i1 v i2 v i2k j = 1, 2,, k v i2j 1 v i2j k 8.13 : n n G n v i V (G) d(v i ) n 2 d(v i ) n 3 v G v j, v k V (G) (v i, v j ), (v i, v k ) / E(G) v j v k v i n 3 v i, v j V (G) d(v i ) + d(v j ) 2n 4 n G Γ Γ n ( ) n 4 v i V (G) d(v i ) n 2 n G 2 C C n 8.14 G v i, v j V (G) (v i, v j ) E(G) v i v j G a, b, c, d 4 G c, d ( v c v d ) a, b G {v c, v d } G b c d a 81

82 G 1 G 2 v 11 v 12 v 18 v 21 v ( 1990) 68 2 xbz 82

83 G Hamilton Hamilton C d(v 11 ) = 2 C v 11 v 23 v 32 v 11 v 11, v 23 v 32, v 11 G v 11 v 23 v 32 Hamilton v 18, v 81, v 88 v 11 C {v 11, v 18, v 81, v 88 } 4 {v 23, v 26, v 32, v 37, v 62, v 67, v 73, v 76 } 1 2 G G 90, 180, 270 G C {v 11, v 18, v 81, v 88 } 4 C 8 3 G {v 11, v 18, v 81, v 88 } v 21 90, 180, 270 v 17, v 78 v

84 a b a G b 6 3 G v 21 v G

85 Hamilton {6 1, 4 3, 4 5, 4 2, 8 3, 4 4, 3 1, 6 2, 8 1, 3 2, 4 1, 6 3, 8 2, 8 4, 6 4 } Hamilton

86 90, 180, v 11, v 18, v 81, v 88 2 C 1, C 2 7 (w 11, w 12 ) C 1, (w 21, w 22 ) C 2 (w 11, w 21 ) G, (w 12, w 22 ) G (C 1 (w 11, w 12 )) (C 2 (w 21, w 22 )) (w 11, w 21 ) (w 21, w 22 ) : G G (u, v) C G (u, v) (u, v) E(C) ( C G G ) C (u, v) Γ = v 1 v 2 v n u = v 1, v = v n v i (2 i n 1) v i u v i 1 v ( v 1 v 2 v i 1 v n v n 1 v i G 86

87 ) u d(u) ( (u, u), (u, v) / E(G) d(u) v 2, v 3,, v n 1 ) v d(u) d(u) d(u) v v v d(u) + 1 d(v) n (d(u) + 1) d(u) + d(v) n 1 d(u) + d(v) n

88 9.1 (1) (2) (3) (4) (5) (6) (7) (4) (5) 2 (6) T x 4 2( x 1) = 2(n 1) = 2m = d(v) = x x = v V (T ) 88

89 4 ( 3 )

90 T n 1 k k 2 n i 2 = 2(n 1) = 2m = i n i i=1 i=1 k n 1 = (i 2)n i + 2 i = 2 (i 2)n i = 0 k n 1 = (i 2)n i + 2 i=2 i= : k k = 1 T K 2 δ(g) 1 G e E(G) {e} G[{e}] T k = t(t 1) k = t + 1 v V (T ) T u V (T ) T ( ) v T = T v G T H ϕ : V (T ) V (H ) u = ϕ(u) u V (G) V (H ) ( u V (H ) u V (H ) H d(u ) V (H ) 1 = t < t + 1 = δ(g) ) v V (G) V (H ) u V (H ) H = V (H ) {v }, E(H ) {(u, v )} ϕ : V (T ) V (H) x V (T ) ϕ(x), x V (T ) ϕ(x) = v, x = v ϕ T H k = t : T 3 ( T T ) T 3 u, v V (T 3 ) u, v V (T 1 ) u, v V (T 2 ) T 1 T 2 T 1 T 2 Γ 1 Γ 2 u, v Γ 1 Γ 2 T T T Γ 1 = Γ 2 T 1 T 2 Γ 1 T 3 u, v T 3 u, v T : G G 7.5 E(G) = n p(g) n 1 E(G) = n p(g) n(n 1) n 1 = E(K n ) = E(G G) 2n 2 n 2 5n n 4 2 n 5 90

91 : n n = 2 d 1 = d 2 = 1 n = t(t 2) n = t + 1 n 3 n < n i=1 d i = 2n 2 < 2n 1 i, j n d i > 1 d j < 2 d i 2 d j = 1 i = 1 j = n t d 1 1, d 2,, d t t d i = 2t 2 d 1 1, d 2,, d t T T i=1 v v T d 1 1 T d 1, d 2,, d n n = t τ(g) = 8 v 1 v 2 v 1 v 2 v 1 v 2 v 1 v 2 v 1 v 2 v 1 v 2 v 1 v 2 v 1 v 2 v 4 v 3 v 4 v 3 v 4 v 3 v 4 v 3 v 4 v 3 v 4 v 3 v 4 v 3 v 4 v C a = aegdj, C b = bgdji, C c = cdg, C f = fge, C h = hij {C a, C b, C c, C f, C h } S d = {a, b, c, d}, S e = {a, e, f}, S g = {a, b, c, f, g}, S i = {b, h, i}, S j = {a, b, h, j} {S d, S e, S g, S i, S j } 9.11 : v V (T ) d(v) k T t T v n t 1 2 2n 2 = 2m = d(v i ) k + 2(n t 1) + t = 2n + k t 2 t k 9.12 v i V (G) 9.1 G k G 1, G 2,, G k (k 1) u V (G i ) v V (G j ) (1 i, j k) e = (u, v) / E(G) G = G e (1) i = j p(g ) = p(g) G G 1, G 2,, G i e,, G k (2) i j p(g e) = p(g) 1 G G i G j e, G 1, G 2,, G i 1, G i+1,, G j 1, G j+1,, G k : x, y V (G) x, y G G x, y G G x, y G x V (G i ) y V (G j ) x V (G j ) y V (G i ) x V (G i ) y V (G j ) x V (G j ) y V (G i ) x, y G x, y G Γ = v 0 v 1 v 2 v t G v 0 = x, v t = y e E(Γ) ( G ) e Γ e Γ (v r, v r+1 ) (v s, v s+1 ) 0 r s t 1 ( e r = s) Γ 1 = v 0 v 1 v r Γ 2 = v s+1 v s+2 v t Γ 1 Γ 2 e G v r v s+1 e u v 91

92 u v ( v r = v s+1 Γ 1 Γ 2 G x y ) G x u ( v) Γ 1 y v ( u) Γ 2 x V (G i ) y V (G j ) x V (G j ) y V (G i ) (1) i = j x, y G G ( G i e) (2) i j x, y G G G G i G j e G i G j E G p(g E ) = p(g) + 1 : E G p(g E ) p(g)+2 e E E = E {e} E p(g E ) = p((g E ) e) 9.1 p((g E ) e) p(g E ) 1 p(g) C G S E(G) G S E(C) S E(C) 2 : S E(C) = 1 e = (u, v) S E(C) G = G S S = S {e} p(g ) > p(g e) = p(g S ) = p(g) G e 9.1 u v G u, v C C e u v e S E(C) C G C e G u v G : C G 1 C = C e 1 C 9.1 G 1 T C T T e 2 S e2 e 2 S e2 E(C) 9.2 S e2 C e S e2 e 2 T e / E(T ) C T e E(C) e / E(T ) e 1 e 1 = e S e2 S e G 1 G 2 G V (G 1 ) = V (G 2 ) = V E(G 1 ) E(G 2 ) (1) 1, 2 V V G 1 G 2 V / 1 V / 2 (2) p(g 2 ) p(g 1 ) : G 2 G E(G 2 ) E(G 1 ) 9.4 G E E(G) e = (u, v) E G 1 G E G E u E = (V (G 1 ), V (G 1 )) : G = G E S = (V (G 1 ), V (G 1 )) G 2 G E v G 1 G 2 ( 9.1 G (u, v) E ) E = S S E (x, y) S (x, y) / E x G 1 y / G 1 (x, y) E(G ) x G 1 x y G y G 1 92

93 (x, y) S ( y / G 1 ) E S (x, y) E (1) x, y V (G 1 ) (2) x, y / V (G 1 ) (1) 9.1 x, y V (G 1 ) p(g (E {(x, y)})) = p((g E ) {(x, y)}) = p(g E ) < p(g) E (2) G = G {(x, y)} E {(x, y)} E p(g ) = p(g {(x, y)}) = p(g (E {(x, y)})) = p(g) x, y G G 1 G G e = (u, v) E(G {e}) = E(G) (E {(u, v), (x, y)}) E(G) 9.3 p(g {e}) p(g) 9.1 p(g {(u, v)}) = p(g ) 1 = p(g) 1 < p(g) E S E = S : T 1 e 1 S e1 e 1 = (u, v) u G S e1 V S e1 = (V 1, V 1 ) T 1 [V 1 ] T 1 [V 1 ] T 1 e 1 ( E(T 1 e 1 ) E(G S e1 ) 9.3 T 1 [V 1 ] T 1 [V 1 ] p(t 1 ) = p(t 1 e 1 ) 2 T 1 [V 1 ] T 1 e 1 ) S e S e1 T 2 e 2 S e1 E(T 2 ) e 2 e 1 ( e 1 / E(T 2 )) e 2 / E(T 1 ) ( S e1 e 1 T 1 ) e 2 S e1 = (V 1, V 1 ) e 2 V 1 V 1 ( T 1 [V 1 ] T 1 [V 1 ] ) 9.1 p((t 1 e 1 ) {e 2 }) = p(t 1 e 1 ) 1 = 1 (T 1 e 1 ) {e 2 } T 1 (T 1 e 1 ) {e 2 } = T = T 1 = n (T 1 e 1 ) {e 2 } ( G ) (T 2 e 2 ) {e 1 } G 9.14 : G = K n H = G e ( e E(G)) G T G H e / E(T ) τ(h) G e 9.7 G n n 2 n 1 G (n 1)n n 2 G K n (n 1)nn 2 e n(n 1)/2 = 2nn 3 e G 2n n 3 e τ(h) = τ(g) 2n n 3 = n n 2 2n n 3 = (n 2)n n : G i, G j Ω a, b {0, 1} (1) G i G j Ω Ω (2) 1.7 (3) Ω G i = G i = G i Ω (4) 1.7 G i G i = Ω (5) a = b = 1 (ab)g i = 1 G i = 1 (1 G i ) = a(b G i ) a = 0 b = 0 (ab)g i = 0 G i = = a(b G i ) (6) 1 {0, 1} 1 G i = G i 1 93

94 (7) a + b = 1 a = 1 b = 0 a = 0 b = 1 (a + b)g i = 1 G i = 1 G i = 0 G i 1 G i = a G i b G i a + b = 0 a = b (a + b)g i = 0 G i = = a G i a G i = a G i b G i F (8) a = 1 a(g i G j ) = G i G j = a G i a G j a = 0 a(g i G j ) = = = a G i a G j Ω {0, 1} M G i = G[{e i1, e i2,, e it }] Ω G i = g i1 g i2 g it M Ω 9.16 T T = G[{a, f, g, b}] T C c = acbgf, C d = adbgf, C e = efg C = {C c, C d, C e } C c C d = cd C c C e = acbe C d C e = adbe C c C d C e = cd C e C = {, C c, C d, C e, cd, acbe, adbe, cd C e } T S a = {a, c, d}, S b = {b, c, d}, S f = {c, d, e, f}, S g = {c, d, e, g} S a S b = {a, b} S a S f = {a, e, f} S a S g = {a, e, g} S b S f = {b, e, f} S b S g = {b, e, g} S f S g = {f, g} S a S b S f = {a, b, c, d, e, f} S a S b S g = {a, b, c, d, e, g} S a S f S g = {a, c, d, f, g} S b S f S g = {b, c, d, f, g} S a S b S f S g = {a, b, f, g} S = {, S a, S b, S f, S g, S a S b, S a S f, S a S g, S b S f, S b S g, S f S g, S a S b S f, S a S b S g, S a S f S g, S b S f S g, S a S b S f S g } 9.17 : V (T ) = {v 1, v 2,, v n } d (v 1 ) = 0 d (v i ) = 1(i = 2, 3,, n) v i = 2, 3,, n d (v i ) 1 v 1 1 ( v 2 ) d (v 2 ) 2 m = n d (v i ) = d (v 1 ) + d (v 2 ) + n d (v i ) (n 2) = n i=1 T m = n 1 < n i=3 94

95 T k (k 1) T t i t = (k 1)i + 1. : T ki t + i 1 ki = d + (v i ) = d (v i ) = t + i 1 v i V (T ) v i V (T ) t = (k 1)i + 1 : T k k = 0 L = I = 0 k < t(t 1) k = t k 2 T v 0 T T l T 1 T 2 v 0 i 1, l 1, I 1, L 1 i 2, l 2, I 2, L 2 T 1 T 2 L 1 = I 1 + 2i 1 L 2 = I 2 + 2i 2 v 0 T V (T 1 ) V (T 2 ) V (T 1 ) V (T 2 ) = i = i 1 + i T t 1 v 0 T V (T 1 ) V (T 2 ) l = l 1 + l 2 V (T 1 ) V (T 2 ) T 1 ( T 2 ) T 1 v 0 T 0 I = I 1 + i 1 + I 2 + i 2 L = L 1 + l 1 + L 2 + l 2 L = L 1 + l 1 + L 2 + l 2 = (I 1 + 2i 1 ) + l 1 + (I 2 + 2i 2 ) + l 2 ( ) = I + i 1 + i 2 + l 1 + l 2 (I = I 1 + I 2 + i 1 + i 2 ) = I + i 1 + i 2 + l (l = l 1 + l 2 ) = I + i 1 + i 2 + i + 1 ( 9.5) = I + 2i (i = i 1 + i 2 + 1) k = t 95

96 9.20 : 2 T n = t + i 9.5 t = i + 1 i = t 1 m = n 1 (T ) = t + i 1 (n = t + i) = t + (t 1) 1 (i = t 1) = 2t a bcde+fg hij abc d +e fg+ hi j + 96

97 10.1 (a) (b) e 6 G e 6 G e G e 6 M = v 4 M f = M f 3 F = {0, 1} C5 3 = 10 97

98 (1) (4) (7) (10) = 1 (2) = 1 (5) = 1 (8) = 1 = 1 (mod 2) = 0 (3) = 1 (6) = 0 (9) = 1 = 1 = 1 (mod 2) = 1 = 1 (mod 2) 3 e 1, e 2, e 4 e 2, e 3, e 5 G e 6 G e 6 G e 6 e 6 G K 4 M = v 1 M f = M f 3 F = {0, 1} C6 3 = (1) = 1 (2) = 1 (3) = (4) = 1 (5) = 1 (6) = 1 = 1 (mod 2) (7) = 1 (8) = 0 (9) = 1 = 1 (mod 2)

99 (10) (13) (16) (19) = 1 (11) = 1 (12) = 1 (14) = 1 (15) = 1 (17) = 0 (18) = 1 (20) = 1 = 1 (mod 2) = 1 = 1 (mod 2) = 1 = 1 (mod 2) = 2 = 0 (mod 2) 3 e 1, e 2, e 5 e 1, e 4, e 6 e 3, e 4, e 5 e 2, e 3, e 6 K 4 3 K A 1 = A 2 = A 3 = A 4 = (1) v 1 v v 1 v v 1 v v 1 v (2) v 1 v (3) v 1 v v 1 v v 1 v v 1 v (4) v 4 v (5) D 4 33 (6) D 4 11 (7) D 4 88 D

100 (8) P (D) = A 1 = A 4 = A 2 = A 5 = A 3 = A 6 = k a (k) 22 = 0 k a (k) 22 = 2k/2 100

101 14.1 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v /v /v /v /v /v /v /v v 1 v 9 v 1 v 4 v 3 v 8 v 9 12 v 2 6 v 5 4 v v 1 v 3 v v 4 4 v 6 4 v

102 v 1 v 2 v 3 v 4 v 5 v /v /v /v /v /v v 1 v 2 v 1 v 2 1 v 1 v 3 v 1 v 2 v 3 3 v 1 v 4 v 1 v 2 v 3 v 5 v 4 7 v 1 v 5 v 1 v 2 v 3 v 5 4 v 1 v 6 v 1 v 2 v 3 v 5 v 4 v v 2 7 v 4 2 v v v 3 1 v v 1 v 2 v 3 v 4 v 5 v 6 v /v /v /v /v /v /v v 1 v 7 v 1 v 3 v 4 v

103 2 v 2 8 v 5 v v v 7 10 v 3 3 v v i v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 T E(v i ) T L(v i ) T S(v i ) v 1 v 2 v 3 v 5 v 6 v 8 v 9 v v 2 v v v v 5 v v 8 v v i v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v 12 T E(v i ) T L(v i ) T S(v i ) v 1 v 4 v 8 v 7 v 10 v 12 v v 2 6 v 4 2 v 2 5 v v v 7 9 v 8 v v 10 6 v v

104 : u u P E u T E, w(u w) w T E w Γ (u) 14.1(2) D 0 v 1 v 1 P E T E T E T E u 1 u 2 (u 2 u 1 )u 2 T E u 2 Γ (u 1 ) u 3 (u 3 u 2 ) D u 3 u 1, u 3 T E u 3 Γ (u 1 ) D n D T E u TE u TE +1 u TE +1 T E u TE +1 u 1 u TE D 14.1 u 14.7 : u u P L u T L, w(u w) w T L w Γ + (u) V n P L D 0 w T L u 1 u 2 (u 2 u 1 ) u 2 T L u 2 Γ + (u 1 ) u 3, u 4, D T L u TL u TL +1 u 1, u 2, u TL 14.1(1) u 14.8 V = {v 2, v 4, v 6, v 8 } V = 4 Dijkatra v 2 v 4 v 2 v 1 v 4 7 v 2 v 6 v 2 v 5 v 6 10 v 2 v 8 v 2 v 5 v 8 10 v 4 v 6 v 4 v 5 v 6 7 v 4 v 8 v 4 v 5 v 8 7 v 6 v 8 v 6 v 5 v K 4 M = {(v 2, v 4 ), (v 6, v 8 )} v 7 2 v v 8 8 v 6 7 K 4 v 2 v 4 v 6 v 8 104

105 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v v 2 v 1 v 4 v 7 v 8 v 5 v 8 v 9 v 6 v 5 v 6 v 3 v 2 v 5 v 4 v 1 v (1) (2)

106 (3) (4)

107 : m = i r n = i + t 9.1 m = n 1 (r 1)i = t (1) (2)

108 (3) w i = 100p i, i = a, b, c, d, e, f, g, h a, b, c, d, e, f, g, h a 10 b 01 c 110 d 111 e 001 f 0001 g h

109 14.13 v 2 v v v 3 v 4 9 (1) v 1 v 1 v 2 v 5 v 4 v 3 v 1 33 (2) T v 2 v v T v 3 v 4 9 v v 2 v v 3 9 v 4 v 1 2 E v1,1 = v 1 v 2 v 5 v 3 v 5 v 4 v 5 v 2 v 1 H v1,1 = v 1 v 2 v 5 v 3 v 4 v 1 W (H v1,1) = 31 E v1,2 = v 1 v 2 v 5 v 4 v 5 v 3 v 5 v 2 v 1 H v1,2 = v 1 v 2 v 5 v 4 v 3 v 1 W (H v1,2) = 33 v

110 E v2,1 = v 2 v 1 v 2 v 5 v 4 v 5 v 3 v 5 v 2 H v2,1 = v 2 v 1 v 5 v 4 v 3 v 2 W (H v2,1) = 32 E v2,2 = v 2 v 1 v 2 v 5 v 3 v 5 v 4 v 5 v 2 H v2,2 = v 2 v 1 v 5 v 3 v 4 v 2 W (H v2,2) = 33 E v2,3 = v 2 v 5 v 3 v 5 v 4 v 5 v 2 v 1 v 2 H v2,3 = v 2 v 5 v 3 v 4 v 1 v 2 W (H v2,3) = 33 E v2,4 = v 2 v 5 v 4 v 5 v 3 v 5 v 2 v 1 v 2 H v2,4 = v 2 v 5 v 4 v 3 v 1 v 2 W (H v2,4) = 33 (3) T v 2 v v v 3 v 4 9 T V = {v 1, v 3, v 4, v 5 } G[V ] v 2 v v v 3 v 4 9 M = {(v 1, v 4 ), (v 3, v 5 )} M T G v 2 v v 5 v 3 9 v 4 v 1 2 E v1,1 = v 1 v 2 v 5 v 3 v 5 v 4 v 1 H v1,1 = v 1 v 2 v 5 v 3 v 4 v 1 W (H v1,1) =

111 E v1,2 = v 1 v 4 v 5 v 3 v 5 v 2 v 1 H v1,2 = v 1 v 4 v 5 v 3 v 2 v 1 W (H v1,2) = 31 (4) 12 v 1 v 2 v 3 v 4 v 5 v 1 32 v 1 v 2 v 3 v 5 v 4 v 1 31 v 1 v 2 v 4 v 3 v 5 v 1 33 v 1 v 2 v 4 v 5 v 3 v 1 34 v 1 v 2 v 5 v 3 v 4 v 1 31 v 1 v 2 v 5 v 4 v 3 v 1 33 v 1 v 3 v 2 v 4 v 5 v 1 37 v 1 v 3 v 2 v 5 v 4 v 1 35 v 1 v 4 v 2 v 3 v 5 v 1 35 v 1 v 4 v 2 v 5 v 3 v 1 36 v 1 v 5 v 2 v 3 v 4 v 1 34 v 1 v 5 v 2 v 4 v 3 v v 1 v 2 v 3 v 5 v 4 v 1 v 1 v 2 v 5 v 3 v 4 v

112 112

113 A 4 f 1 = { 0, 0, 1, 1 } f 2 = { 0, 1, 1, 0 } f 3 = { 0, 0, 1, 0 } f 4 = { 0, 1, 1, 1 } f 1 f 3 f 4 f 1 f 2 f 3 f 4 f 1 f 1 f 2 f 3 f 4 f 2 f 2 f 1 f 4 f 3 f 3 f 3 f 3 f 3 f 3 f 4 f 4 f 4 f 4 f i τ(i) τ τ(i) = (i mod n) + 1 A

114 Z Z 1 N 1 M n(r) 0 0 GL n (R) nz 0 0 R + ab a b {a i } a b = b R(A) I A Z + gcd(a, b) lcm 1 lcm(a, b) gcd 1 n = 1 nz (1) a, b, c ( a = b = c = 0 ) (2) A = { 1, 0, 1} 15.6 (1) 0 (2) ( 0 ) (3) 1 0 (4) g(x) 0 f/g / S S 15.7 (1) 1 (2) 1 (3) 1 (4) ab a + b a/b + b/a / Z + Z p = q : a = 1, b = 0 a b = b a pa + qb + r = qa + pb + r ( ) p + r = q + r (a = 1, b = 0) p = q 114

115 2 p, q {0, 1} (p = q r = 0) : a, b, c R (a b) c = a (b c) p(pa + qb + r) + qc + r = pa + q(pb + qc + r) + r ( ) p 2 a + pqb + pr + qc + r = pa + pqb + q 2 c + qr + r p 2 a + qc + pr = pa + q 2 c + qr a, c p 2 = p q 2 = q pr = qr p 2 = p q 2 = q p, q {0, 1} pr = qr p = q r = 0 3 p + q = 1 r = 0 : x R a x = x (p + q)x + r = x p + q = 1 r = 0 ( ) 4a q = 1 (p 0 r = 0) : e l x R e l x = x pe l + qx + r = x ( ) q = 1, pe l + r = 0 p 0 r = 0 pe l + r = 0 ( p = r = 0 ) 4b p = 1 (q 0 r = 0) : 4a 4c p = q = 1 : p = q = 1 r 4a 4b 5a q = 0 ((p = 1 r = 0) p 1) : p = 1, q = r = 0 q = 0, p = 1 r/(1 p) θ l x R θ l x = θ l 115

116 pθ l + qx + r = θ l q = 0, pθ l + r = θ l r = 0 p = 1 θ l p 1 θ l = r/(1 p) r 0, p = 1 5b p = 0 ((q = 1 r = 0) q 1) : 5a 5c p = q = 0 : p = q = 0 r 5a 5b ( ) x x 1 = x/(x 1) x a b x a b x a b x a b 1 x a a 2 x a b 3 x b a 4 x b b 16 1 a b 2 a b 3 a b 4 a b 5 a b 6 a b a a a a a a a a a a a a a a b a a b b a a b a b b b a b b b b a a b a b 7 a b 8 a b 9 a b 10 a b 11 a b 12 a b a a b a a b a b a a b a a b a a b a b b a b b b b a a b a b b b a b b b 13 a b 14 a b 15 a b 16 a b a b b a b b a b b a b b b a a b a b b b a b b b 3, 5, 11, 12, 13, , 2 3, 4 = 1 3, = 3, 6 3, 4 1, 2 = 3 1, = 3, 10 1, 2 3, 4 = 3, 4 1, 2 ( a, b c, d ) e, f = ac, ad + b e, f = ace, acf + ad + b a, b ( c, d e, f ) = a, b ce, cf +d = ace, acf +ad+b a, b, c, d, e, f A ( a, b c, d ) e, f = a, b ( c, d e, f ) 1, 0 0, x ( x Q ) d a, b Q, ad + b = d a, b A a 0 1/a, b/a ( ) a = 0 c 0c = (1) ϕ : A Z 3, ϕ(a) = 0, ϕ(b) = 1, ϕ(c) = 2 A,, a ϕ = Z3,, 0 116

117 3 3 a (2) x y = y, x, y A A (3) ϕ : A Z 3, ϕ(a) = 1, ϕ(b) = 2, ϕ(c) = 0 A,, a ϕ = Z3,, b b = a a c (4) {B,, 1} B = {1, 4, 6} 10 ϕ : A B, ϕ(a) = 1, ϕ(b) = 6, ϕ(c) = 4 A,, a ϕ = B,, 1 10 c c = b a : θ l θ l θ r = θ l θ r θ l θ r = θ r θ l = θ l θ r = θ r θ ( ) θ ( ) θ θ = θ ( θ θ = θ ) θ θ θ = θ θ = θ θ = θ θ = θ ( θ = θ θ = θ) V 1 = Z 6, V V V 2 = {0, 2, 4}, V V 3 = {0, 3}, V V 4 = {0}, V (1) V 1 V 2 { 1, 5, 1, 6, 2, 5, 2, 6, 3, 5, 3, 6 },, 1, 6 1, 5 1, 6 2, 5 2, 6 3, 5 3, 6 1, 5 1, 5 1, 5 2, 5 2, 5 3, 5 3, 5 1, 6 1, 5 1, 6 2, 5 2, 6 3, 5 3, 6 2, 5 2, 5 2, 5 2, 5 2, 5 3, 5 3, 5 2, 6 2, 5 2, 6 2, 5 2, 6 3, 5 3, 6 3, 5 3, 5 3, 5 3, 5 3, 5 3, 5 3, 5 3, 6 3, 5 3, 6 3, 5 3, 6 3, 5 3, 6 1, 6 3, 5 1, 6 (2) V 1 = {1, 2, 3},, 1 V V V 2 = {1, 2},, 1 V V 3 = {1, 3},, 1 V V 4 = {1},, 1 V (1) V 1 V 2 Z 3 Z 2, 117

118 0, 0 0, 1 1, 0 1, 1 2, 0 2, 1 0, 0 0, 0 0, 1 1, 0 1, 1 2, 0 2, 1 0, 1 0, 1 0, 0 1, 1 1, 0 2, 1 2, 0 1, 0 1, 0 1, 1 2, 0 2, 1 0, 0 0, 1 1, 1 1, 1 1, 0 2, 1 2, 0 0, 1 0, 0 2, 0 2, 0 2, 1 0, 0 0, 1 1, 0 1, 1 2, 1 2, 1 2, 0 0, 1 0, 0 1, 1 1, 0 (2) V 1 V 2 0, 0 Z 3 Z 2 0, 0 0, 0 0, 1 0, 1 1, 0 2, 0 1, 1 2, : + C C ϕ : C B, a + bi C, ϕ(a + bi) = a b ϕ b a ϕ a + bi, c + di C ϕ((a + bi) + C (c + di)) = ϕ((a + c) + (b + d)i) = a + c b + d (b + d) a + c = a b + c d b a d b ( ) (ϕ ) ( ) = ϕ(a + bi) + ϕ(c + di) (ϕ ) ϕ((a + bi) C (c + di)) = ϕ((ac bd) + (bc + ad)i) ac bd bc + ad = (bc + ad) ac bd = a b c d b a d b ( ) (ϕ ) ( ) = ϕ(a + bi) ϕ(c + di) (ϕ ) ϕ C, + C, C B, +, C, + C, C ϕ = B, +, : V 1 V 2 V 2 V 1 A B, 1, 2 B A, 1, 2 ϕ : A B B A, x, y A B, ϕ( x, y ) = y, x ϕ ϕ x 1, y 1, x 2, y 2 A B, i 1, 2 ϕ( x 1, y 1 i x 2, y 2 ) = ϕ( x 1 i x 2, y 1 i y 2 ) 118

119 = y 1 i y 2, x 1 i x 2 (ϕ ) = y 1, x 1 i y 2, x 2 = ϕ( x 1, y 1 ) i ϕ( x 2, y 2 ) (ϕ ) ϕ V 1 V 2 V 2 V 1 V 1 V 2 ϕ = V2 V f, g : A B B = 2 f = g b 0 (b 0 B x(x A (f(x) = b 0 g(x) = b 0 ))) : f g x A f(x) g(x) B = 2 f(x) g(x) f(x) g(x) b 0 x(x A (f(x) = b 0 g(x) = b 0 )) : ϕ ( x P({a, b}) a x a / x ) {a} P({a, b}), ϕ({a}) = 1 {b} P({a, b}), ϕ({b}) = 0 ϕ ϕ x, y A ϕ(x y) = 1 a x y (ϕ ) a x a y ϕ(x) = 1 ϕ(y) = 1 (ϕ ) ϕ(x) + ϕ(y) = 1 ϕ(x y) ϕ(x) + ϕ(y) P(a, b) {0, 1} {0, 1} = 2, 1 {0, 1} 15.1 x, y A, ϕ(x y) = ϕ(x) + ϕ(y) x, y A ϕ(x y) = 1 a x y (ϕ ) a x a y ϕ(x) = 1 ϕ(y) = 1 (ϕ ) ϕ(x) ϕ(y) = 1 x, y A, ϕ(x y) = ϕ(x) ϕ(y) x A ϕ( x) = 1 a x (ϕ ) a / x ϕ(x) = 0 (ϕ ) ϕ(x) = 1 x A, ϕ( x) = ϕ(x) a /, a {a, b} ϕ( ) = 0, ϕ({a, b}) = 1 ϕ V 1 V 2 ϕ ϕ V 1 V (1) 119

120 : 1 i x, y B ϕ a, b A ϕ(a) = x, ϕ(b) = y x i y = ϕ(a) i ϕ(b) (ϕ(a) = x, ϕ(b) = y) = ϕ(a i b) (ϕ ) = ϕ(b i a) ( i ) = ϕ(b) i ϕ(a) (ϕ ) = y i x (ϕ(a) = x, ϕ(b) = y) i 2 i x, y, z B ϕ a, b, c A ϕ(a) = x, ϕ(b) = y, ϕ(c) = z (x i y) i z = (ϕ(a) i ϕ(b)) i ϕ(c) (ϕ(a) = x, ϕ(b) = y, ϕ(c) = z) = (ϕ(a i b)) i ϕ(c) (ϕ ) = ϕ((a i b) i c) (ϕ ) = ϕ(a i (b i c)) ( i ) = ϕ(a) i ϕ(b i c) (ϕ ) = ϕ(a) i (ϕ(b) i ϕ(c)) (ϕ ) = x i (y i z) (ϕ(a) = x, ϕ(b) = y, ϕ(c) = z) i 3 i x B ϕ a A ϕ(a) = x x i x = ϕ(a) i ϕ(a) (ϕ(a) = x) = ϕ(a i a) (ϕ ) = ϕ(a) ( i ) = x (ϕ(a) = x) i (2) : i j x, y, z B ϕ a, b, c A ϕ(a) = x, ϕ(b) = y, ϕ(c) = z x i (y j z) = ϕ(a) i (ϕ(b) j ϕ(c)) (ϕ(a) = x, ϕ(b) = y, ϕ(c) = z) = ϕ(a) i ϕ(b j c) (ϕ ) = ϕ(a i (b j c)) (ϕ ) = ϕ((a i b) j (a i c)) ( i j ) = ϕ(a i b) j ϕ(a i c) (ϕ ) = (ϕ(a) i ϕ(b)) j (ϕ(a) i ϕ(c)) (ϕ ) = (x i y) j (x i z) (ϕ(a) = x, ϕ(b) = y, ϕ(c) = z) (y j z) i x = (y i x) j (z i x) i j (3) : i, j i j (1) 120

121 i j x, y B ϕ a, b A ϕ(a) = x, ϕ(b) = y x i (x j y) = ϕ(a) i (ϕ(a) j ϕ(b)) (ϕ(a) = x, ϕ(b) = y) = ϕ(a) i ϕ(a j b) (ϕ ) = ϕ(a i (a j b)) (ϕ ) = ϕ(a) ( i, j ) = x (ϕ(a) = x) x j (x i y) = x i, j (4) : 1 e V 1 i x B ϕ a A ϕ(a) = x x i ϕ(e) = ϕ(a) i ϕ(e) (ϕ(a) = x) = ϕ(a i e) (ϕ ) = ϕ(a) (e i ) = x (ϕ(a) = x) ϕ(e) i x = x ϕ(e) V 2 i 2 θ V 1 i x B ϕ a A ϕ(a) = x x i ϕ(θ) = ϕ(a) i ϕ(θ) (ϕ(a) = x) = ϕ(a i θ) (ϕ ) = ϕ(θ) (θ i ) ϕ(θ) i x = θ ϕ(θ) V 2 i (5) : x 1 x i ϕ(x) i ϕ(x 1 ) = ϕ(x i x 1 ) (ϕ ) = ϕ(e) (x 1 x ) (4) ϕ(e) V 2 i ϕ(x 1 ) ϕ(x) i ϕ(x 1 ) ϕ(x) i ϕ(x 1 ) ϕ(x) i : 3.3 ϕ 2 ϕ 1 : A C x A, ϕ 2 ϕ 1 (x) = ϕ 2 (ϕ 1 (x)) x, y A ϕ 2 ϕ 1 (x y) = ϕ 2 (ϕ 1 (x y)) ( 3.3) = ϕ 2 (ϕ 1 (x) ϕ 1 (y)) (ϕ 1 V 1 V 2 ) = ϕ 2 (ϕ 1 (x)) ϕ 2 (ϕ 1 (y)) (ϕ 2 V 2 V 3 ) = ϕ 2 ϕ 1 (x) ϕ 2 ϕ 1 (y) ( 3.3) ϕ 2 ϕ 1 V 1 V 3 121

122 15.23 (1) : V 1 A I A V 1 V 1 V 1 = V1 (2) : V 1 = A, 1, 2,..., k, V 2 = B, 1, 2,..., k ϕ : A B i, i ϕ( i (x 1, x 2,..., x ki )) = i (ϕ(x 1 ), ϕ(x 2 ),..., ϕ(x ki )), x 1, x 2,..., x ki A 3.9 ϕ 1 : B A i, i y 1, y 2,..., y ki B ϕ 1 ( i (y 1, y 2,..., y ki )) =ϕ 1 ( i (ϕ(ϕ 1 (y 1 )), ϕ(ϕ 1 (y 2 )),..., ϕ(ϕ 1 (y ki )))) (ϕ ϕ 1 = I B ) =ϕ 1 (ϕ( i (ϕ 1 (y 1 ), ϕ 1 (y 2 ),..., ϕ 1 (y ki )))) (ϕ V 1 V 2 ) = i (ϕ 1 (y 1 ), ϕ 1 (y 2 ),..., ϕ 1 (y ki )) (ϕ 1 ϕ = I A ) ϕ 1 V 2 V 1 ϕ 1 V 2 = V1 (3) : V 1 = A, 1, 2,..., k, V 2 = B, 1, 2,..., k, V 3 = C, 1, 2,..., k ϕ 1 : A B ϕ 2 : B C i, i, i ϕ( i (x 1, x 2,..., x ki )) = i (ϕ(x 1 ), ϕ(x 2 ),..., ϕ(x ki ), x 1, x 2,..., x ki ) A i (ϕ(y 1 ), ϕ(y 2 ),..., ϕ(y ki ) = ϕ( i(y 1, y 2,..., y ki )), y 1, y 2,..., y ki ) B 3.4(3) ϕ 2 ϕ 1 : A C i, i, i x 1, x 2,..., x ki A ϕ 2 ϕ 1 ( i (x 1, x 2,..., x ki )) =ϕ 2 (ϕ 1 ( i (x 1, x 2,..., x ki ))) ( 3.3) =ϕ 2 ( i (ϕ 1 (x 1 ), ϕ 1 (x 2 ),..., ϕ 1 (x ki ))) (ϕ 1 V 1 V 2 ) = i (ϕ 2 (ϕ 1 (x 1 )), ϕ 2 (ϕ 1 (x 2 )),..., ϕ 2 (ϕ 1 (x ki ))) (ϕ 2 V 2 V 3 ) = i (ϕ 2 ϕ 1 (x 1 ), ϕ 2 ϕ 1 (x 2 ),..., ϕ 2 ϕ 1 (x ki )) ( 3.3) ϕ 2 ϕ 1 V 1 V 3 ϕ 2 ϕ 1 V 1 = V (1) : 1 C ϕ(1 1) = ϕ(1) = = 2 ϕ(1) ϕ(1) = 2 2 = 4 ϕ(1 1) ϕ(1) ϕ(1) ϕ (2) R R, : a 1 e iθ1, a 2 e iθ2 C ϕ(a 1 e iθ1 a 2 e iθ 2 ) = ϕ(a 1 a 2 e i(θ 1+θ 2 ) ) ( ) = a 1 a 2 e i(θ 1+θ 2 ) (ϕ ) = a 1 a 2 = a 1 e iθ 1 a 2 e iθ 2 = ϕ(a 1 e iθ 1 ) ϕ(a 2 e iθ 2 )e (ϕ ) 122

123 (3) {0}, : x, y C ϕ(x y) = 0 (ϕ ) = 0 0 (0 0 = 0) = ϕ(x) ϕ(y) (ϕ ) (4) : 1 C ϕ(1 1) = ϕ(1) = 2 ϕ(1) ϕ(1) = 2 2 = 4 ϕ(1 1) ϕ(1) ϕ(1) ϕ I A V V : ϕ : A A V ϕ(5 1 ) = ϕ(5 k ) ϕ(5 2 ) = ϕ( ) = 5 k 5 k = 5 2k n n Z +, ϕ(5 n ) = ϕ(5 kn ) V ϕ {5 kn n Z + }, k / Z + ϕ(a) A ϕ : A A k = 1 ϕ = I A k > / {5 kn n Z + } ϕ I A V : ϕ ϕ x, y Z +, ϕ(x y) ϕ(x y) = 1 x y = 1 (ϕ ) x = 1 y = 1 ( ) ϕ(x) = 1 ϕ(y) = 1 (ϕ ) ϕ(x) ϕ(y) = 1 ( ) 1 Z 2, x, y Z + (ϕ(x y) = 1 ϕ(x) ϕ(y) = 1) 15.1 x, y Z + (ϕ(x y) = ϕ(x) ϕ(y)) ϕ V 1 V (1) ( 1)R( 3) 2R2 ( 1 + 2) R( 3 + 2) (2) R ( 1R5 5R9 1 R9 ) (3) ( 1)R1 1R1 ( 1 + 1) R(1 + 1) (4) R : g : A A/, g(a) = [a], a A V V/ g g V V/ (1) 123

124 : ϕ x, y Z, (x + y) mod 2 = ((x mod 2) + (y mod 2)) mod 2 x Z ϕ( x) = ϕ(x + 1) ( ) = (x + 1) mod 2 (ϕ ) = ((x mod 2) + (1 mod 2)) mod 2 ( ) = ((x mod 2) + 1) mod 2 (1 mod 2 = 1) = (ϕ(x) + 1) mod 2 (ϕ ) = ϕ(x) ( ) ϕ V 1 V 2 (2) {{2k k Z}, {2k + 1 k Z}} (1) : ϕ x A k A k x k nk m nx nk m ϕ A k A m x, y A k ϕ(x + y) = n(x + y) (ϕ ) = nx + ny = ϕ(x) + ϕ(y) (ϕ ) ϕ V 1 V 2 (2) 1 n 0 ϕ x, y A k, ϕ(x) = ϕ(y) x = y = I Ak A k / = {{x} x A k } V 1 / = {{x} x A k }, {x} {y} = {x + y}, x, y A k 2 n = 0 ϕ(x) = 0, x A k = E Ak A k / = {A k } V 1 / = {A k }, {A k } A k A k = A k (1) 13 : b V ϕ ϕ(b) ϕ(b) = ϕ(b b) (ϕ ) = ϕ(b) (b b = b) ϕ(b) A b V ϕ ϕ(b) = b ϕ(a) b = ϕ(a) ϕ(b) (ϕ(b) = b) = ϕ(a b) (ϕ ) = ϕ(a) (a b = a) ϕ(a) b = ϕ(a) A a b ϕ(a) a b ϕ(a) = a ϕ(a) = b 124

125 1 ϕ(a) = a ϕ(c) b = ϕ(c) ϕ(b) (ϕ(b) = b) = ϕ(c b) (ϕ ) = ϕ(b) (c b = b) = b (ϕ(b) = b) ϕ(c) a ( a b = a b) a ϕ(c) = ϕ(a) c (ϕ(a) = a) = ϕ(a c) (ϕ ) = ϕ(b) (a c = b) = b (ϕ(b) = b) ϕ(c) b ( a b = a b) ϕ(d) ϕ(d) a b c d 4 ϕ(a) = a, ϕ(b) = b ϕ 1 (a) = a, ϕ 1 (b) = b, ϕ 1 (c) = c, ϕ 1 (d) = d ϕ 2 (a) = a, ϕ 2 (b) = b, ϕ 2 (c) = d, ϕ 2 (d) = c ϕ 3 (a) = a, ϕ 3 (b) = b, ϕ 3 (c) = c, ϕ 3 (d) = c ϕ 4 (a) = a, ϕ 4 (b) = b, ϕ 4 (c) = d, ϕ 4 (d) = d 2 ϕ(a) = b ϕ(c) ϕ(d) a ϕ(c) ϕ(d) b ϕ(a) = b ϕ(a c) = ϕ(a) ϕ(c) = b ϕ(c) ϕ(c) = b ϕ ϕ(a) = ϕ(b) = b ϕ(c) ϕ(d) a ϕ ϕ(a) = ϕ(b) = b ϕ(c) ϕ(d) a a / ran(ϕ) b a, b x, y A, ϕ(x) ϕ(y) = b ϕ(a) = ϕ(b) = b x, y A ϕ(a), x = a y = b ϕ(x y) = ( ) ϕ(b), = b (ϕ(a) = ϕ(b) = b) x, y A, ϕ(x y) = ϕ(x) ϕ(y) ϕ 9 ϕ(a) = ϕ(b) = b ϕ 5 (a) = b, ϕ 5 (b) = b, ϕ 5 (c) = b, ϕ 5 (d) = b ϕ 6 (a) = b, ϕ 6 (b) = b, ϕ 6 (c) = b, ϕ 6 (d) = c ϕ 7 (a) = b, ϕ 7 (b) = b, ϕ 7 (c) = b, ϕ 7 (d) = d ϕ 8 (a) = b, ϕ 8 (b) = b, ϕ 8 (c) = c, ϕ 8 (d) = b ϕ 9 (a) = b, ϕ 9 (b) = b, ϕ 9 (c) = c, ϕ 9 (d) = c ϕ 10 (a) = b, ϕ 10 (b) = b, ϕ 10 (c) = c, ϕ 10 (d) = d ϕ 11 (a) = b, ϕ 11 (b) = b, ϕ 11 (c) = d, ϕ 11 (d) = b ϕ 12 (a) = b, ϕ 12 (b) = b, ϕ 12 (c) = d, ϕ 12 (d) = c ϕ 13 (a) = b, ϕ 13 (b) = b, ϕ 13 (c) = d, ϕ 13 (d) = d V 125

126 (2) V V 15.2 V = A, 1, 2,, n V V : V g : A A/, x A, g(x) = [x] g V V/ f : A/ A x A/, f(x) x f V/ V ( 5.23) f g : A A V f g f g V V V 13 i ϕ i A 1 1 = 2 = I A A 2 3 = 4 = { c, d, d, c } I A {{a}, {b}, {c, d}} 3 5 = E A A 4 6 = 7 = { a, b, a, c, b, a, b, c, c, a, c, b } I A {{a, b, c}, {d}} 5 8 = 11 = { a, b, a, d, b, a, b, d, d, a, d, b } I A {{a, b, d}, {c}} 6 9 = 13 = { a, b, b, a, c, d, d, c } I A {{a, b}, {c, d}} 7 10 = 12 = { a, b, b, a } I A {{a, b}, {c}, {d}} V : V 1 V 2 = A B,,, k, k ϕ : A B A, a, b A B, ϕ( a, b ) = a R ϕ A ϕ R V 1 V 2 V 1 V 2 /R = V 1 ϕ a, b, c, d A B ϕ( a, b c, d ) = ϕ( a c, b d ) = a c (ϕ ) = ϕ( a, b ) ϕ( c, d ) (ϕ ) ϕ( a, b ) = ϕ( a, b ) = a (ϕ ) ϕ V 1 V 2 V 1 R ϕ A R V 1 V 2 B ( ) a A b B a, b A B, ϕ( a, b ) = a ϕ(a B) = A V 1 V 1 V 2 ϕ V 1 V 2 /R = V 1 126

127 16.1 (1) : R a, b, c R (a b) c = (a + b + ab) c ( ) = (a + b + ab) + c + (a + b + ab)c ( ) = a + b + c + ab + ac + bc + abc ( ) = a + (b + c + bc) + a(b + c + bc) ( ) = a (b + c + bc) ( ) = a (b c) ( ) R, (2) : (1) R, a R a 0 = a a 0 ( ) = a (0 ) = a (0 ) 0 a = 0 + a + 0 a ( ) = 0 + a + 0 (0 ) = a (0 ) 0 R, 16.2 : x = a u 0, v 0 S a u 0 = v 0 a = a v 0 x S u, v S a u = v a = x v 0 x = v 0 a u (a u = x) = a u (v 0 a = a) = x (a u = x) v 0 u u 0 = v 0 = e 127

128 V 16.3 (1) : x, y, z S (x y) z = x z ( ) = x ( ) = x (y z) ( ) S, y, x = e (2) S = S {e} : S S S, x y = 16.2 S,, e x, 16.4 : (a b) c = a (b c) ( ) = a (c b) (b, c ) = (a c) b ( ) = (c a) b (a, c ) = c (a b) ( ) 16.5 (1) : a b = a (a a) (a a = b) = (a a) a ( ) = b a (a a = b) (2) : V {a, b} a, b a b = a a b = b 1 a b = a b b = (a a) b (a a = b) = a (a b) ( ) = a a (a b = a) = b (a a = b) 2 a b = b b b = (a a) b (a a = b) = a (a b) ( ) = a b (a b = b) 128

129 = b b b = b 16.6 (1) : a a a a (a a) (a a) a V (a b = b) (2) : a b a a a (a b a) (a b a) a a (a b a) = (a a) (b a) ( ) = a (b a) ( (1) ) = a (b a a) ( (1) ) = (a b a) a ( ) (3) : a b a a c (a c) (a b c) (a b c) (a c) (a c) (a b c) = (a c a) (b c) ( ) = a (b c) ( (2) ) = a (b (c a c)) ( (2) ) = (a b c) (a c) ( ) 16.7 : V (a b) (a b) = a (b a) b ( ) = a (a b) b ( ) = (a a) (b b) ( ) = a b (a, b ) 16.8 : x, y S (x θ l ) y = x (θ l y) ( ) = x θ l (θ l ) 16.9 : V = S, S a S S i, j N +, i < j a i = a j p = j i a i = a j = a i+p a i+2p = a j+p = a j a p = a i a p = a j = a i n n N, a i+np = a i a i+ip = a i (a ip ) (a ip ) = a 2ip = a (i+ip)+(ip i)) = a i+ip a ip i = a i a ip i = a ip a ip 129

130 = 0, 3 3 = ( S, V S {0}, S {1}, V ) 2 3 = 2 ( ) V V 1 = {0}, ; V 2 = {1}, ; V 3 = {0, 1}, ; V 4 = {0, 2}, ; V 5 = {1, 3}, ; V 6 = {0, 1, 2}, ; V 7 = {0, 1, 3}, ; V 8 = V = Z 4, ; 1 1 V V 1 1 V Z 4,, 1 V V 1 Z 4,, 1 1 V 2 2 = 2 0 = 0 V (1) 7 V 4 V 1 V 4 V [a] [b] [a] [a] [b] [b] [b] [a] : a, b, c, d S, arc brd 1 a, b, c, d / I R a = c b = d a b = c d R (a b)r(c d) 2 a, c b, d I a, c I IS I a b I c d I b, d I SI I a b I c d I arc brd (a b)r(c d) R V (2) R I S I S/R = {I} {{x} x S I} I I x = x I = I, x S/R {x y}, x y / I [x], [y] (S/R {I}) {x} {y} = [x y] = I, x y I ( [x], [y] S/R x I y I x y I [x] [y] = I) 1 S, S a a a = a a x S(a x = x a = x) 130

131 {x y}, x y / I V/R = S/R,, [x] [y] = I, [x], [y] S/R Q = Γ = {0, 1}; Σ = {0, 1, e}; δ 0 1 e λ 0 1 e (0) 1(1) 1(0) 0 1 0(1) e(0) e(1) Q = {1, 2, 3}; Σ = {a, b}; Γ = {x, y}; δ a b λ a b 1 y x 2 x x 3 x y δ

132 q 0 q 1, q 2 q 3, q 4 q 5 x Σ, λ(q 0, x) = λ(q 2, x) λ (q 0, 00) = λ(q 0, 0)λ(q 1, 0) = 00 λ (q 2, 00) = λ(q 2, 0)λ(q 4, 0) = 01 q 0 q 2 Q/ = {{q 0, q 1 }, {q 2, q 3 }, {q 4, q 5 }, {q 6 }} δ 0 1 [q 0 ] [q 0 ] [q 6 ] [q 2 ] [q 4 ] [q 0 ] [q 4 ] [q 4 ] [q 2 ] [q 6 ] [q 0 ] [q 4 ] λ 0 1 [q 0 ] 0 0 [q 2 ] 0 0 [q 4 ] 1 0 [q 6 ]

133 17.1 : G,, ( ), 1 Klein G,, e, G 17.2 : a, b G, a b = au 1 b G a, b, c G, (a b) c = au 1 bu 1 c = a (b c) G a G u a = uu 1 a = a a u = au 1 u = a u a G (ua 1 u) a = ua 1 uu 1 a = u a (ua 1 u) = au 1 ua 1 u = u G G 17.3 u = : G x G x x 1 = x 1 x = e, x 1 x = xx 1 = e G 17.5 G, : ( ) ( ) ( ) w 0 0 w w w ( ), ( ) ( 0 w 2 w 0, 0 w ) w : (ab) 2 = a 2 b 2 abab = aabb = ba = ab ( )

134 : k Z +, (x 1 yx) k = x 1 y k x k = 1 k = m k = m + 1 (x 1 yx) m+1 = (x 1 yx) m (x 1 yx) = x 1 y m x(x 1 yx) ( ) = x 1 y m yx (xx 1 = e) = x 1 y m+1 x k Z +, (x 1 yx) k = x 1 y k x 17.8 (2) : a, b G b 1 a 1 ab = b 1 b (a 1 a = e) = e (b 1 b = e) abb 1 a 1 = aa 1 (b 1 b = e) = e (a 1 a = e) (ab) 1 = b 1 a 1 (4) (ab) 1 = b 1 a G k a 1, a 2,, a k G (a 1 a 2 a k ) 1 = a 1 k a 1 2 a 1 1 k Z +, a G, (a k ) 1 = (a 1 ) k : k = 1 k = t k = t + 1 (a 1 a 2 a t a t+1 ) 1 = a 1 t+1 (a 1a 2 a t ) 1 ( 17.2(2)) = a 1 t+1 a 1 t a 1 2 a 1 1 ( ) a 1 = a 2 = = a k = a (a k ) 1 = (a 1 ) k : m, n 0 (a n ) m = a nm = e 1 m > 0 n > (2) 2 m > 0 n < 0 t = n (a n ) m = (a t ) m (n = t) = ((a 1 ) t ) m = (a 1 ) tm ( 16.1(2)) = a tm = a nm (n = t) 3 m < 0 n > 0 s = m (a n ) m = (a n ) s (m = s) = ((a n ) 1 ) s 134

135 = ((a 1 ) n ) s ( 17.1) = (a 1 ) ns ( 16.1(2)) = a ns = a nm (m = s) 4 m < 0 n < 0 s = m, t = n (a n ) m = (a t ) s (m = s, n = t) = (((a 1 ) t ) 1 ) s = (((a t ) 1 ) 1 ) s ( 17.1) = (a t ) s ( 17.2(1)) = a ts = a nm (ts = nm) m, n Z, (a n ) m = a nm (5) : n N n = 0 n = k n = k + 1 (ab) k+1 = (ab) k ab = a k b k ab ( ) = a k ab k b (G Abel ) = a k+1 b k+1 n N, (ab) n = a n b n n < 0 t = n t > 0 (ab) n = (ab) t (n = t) = ((ab) 1 ) t = (b 1 a 1 ) t ( 17.2(2)) = (b 1 ) t (a 1 ) t ( n N, (ab) n = a n b n ) = b t a t = b n a n (n = t) = a n b n (G Abel ) 17.9 (1) : k Z (b 1 ab) k = e (b 1 ab) k+1 = (b 1 ab) a k+1 = a ( ) ( 17.7 ) a k = e ( ) {k k Z + (b 1 ab) k = e} = {k k Z + a k = e} 135

136 b 1 ab = a (2) : a, b G (ab) k a = a(ba) k, k Z k Z (ab) k = e (ab) k a = a ( ) a(ba) k = a ((ab) k a = a(ba) k ) (ba) k = e ( ) {k k Z + (ab) k = e} = {k k Z + (ba) k = e} ab = ba (3) : G = 2k H = V, E V = G G E = {(x, y) x, y G xy = e} x G y (x, y) E V = G 2, x 1 = x; x d H (x) = d H (e) = 2 G 1,. e H 2 H G 1 = 2k : 16.6 a G, aa = a a G, a = e G = {e} G : (p, q) = 1 m, n Z mp + nq = 1 mp = nq + 1 u mp 1 = u mp 2 = v nq 1 = v nq 2 = e (e m = e, e n = e) = u mp 1 v nq 1 = u mp 2 v nq 2 = e (ee = e) u 1 u nq 1 v nq 1 = u 2 u nq 2 v nq 2 = e (mp = nq + 1) u 1 (u 1 v 1 ) nq = u 2 (u 2 v 2 ) nq (u 1 v 1 = v 1 u 1 u 2 v 2 = v 2 u 2 ) u 1 = u 2 (u 1 v 1 = u 2 v 2 ) (1) (2) (3) S M n (R) 0 A =

137 B = A = 1, B = 0 A, B S A+B = = A + B / S S (4) : ea = ae = a e H H x, y H xya = xay (ya = ay) = axy (xa = ax) xy H x x H xa = ax (H ) x 1 xa = x 1 ax ( x 1 ) x 1 xax 1 = x 1 axx 1 ( x 1 ) ax 1 = x 1 a (xx 1 = x 1 x = e) x 1 H (H ) H G (1) G {e} G G G G = {e} (2) Lagrange {e} G (3) Lagrange p G p 2 G 3 {e} G p : H 1 H 2 = H 2 H 1 H 1, H 2 e H 1, e H 2 e = ee H 1 H 2 H 1 H 2 H 1 H 2 H 1 H 2 ab a H 1, b H 2 x, y H 1 H 2 x = ab, y = cd a, c H 1, b, d H 2 xy 1 = ab(cd) 1 = abd 1 c 1 b, d H 2 H 2 bd 1 H 2 abd 1 H 1 H 2 = H 2 H 1 h 2 H 2, h 1 H 1 abd 1 = h 2 h 1 h 1, c H 1 H 1 h 1 c 1 H 1 xy 1 = abd 1 c 1 = h 2 h 1 c 1 H 2 H 1 = H 1 H 2 H 1 H 2 G H 1 H 2 x H 1 H 2 a H 1, b H 2 x = ab x H 1 H 2 = x 1 H 1 H 2 (H 1 H 2 ) a b(a H 1 b H 2 x 1 = ab) (H 1 H 2 ) 137

138 = a b(a 1 H 1 b 1 H 2 x 1 = ab) (H 1, H 2 ) = a b(b 1 a 1 H 2 H 1 x 1 = ab) (H 2 H 1 ) a b((ab) 1 H 2 H 1 x 1 = ab) ( 17.2(2)) = a b((x 1 ) 1 H 2 H 1 ) (x 1 = ab) a b(x H 2 H 1 ) ( 17.2(1)) H 1 H 2 H 2 H 1 H 2 H 1 H 1 H 2 ( H 2 H 1 H 2 H 1 H 1 H 2 ) x x H 2 H 1 a b(a H 1 b H 2 x = ba) (H 2 H 1 ) = a b(a 1 H 1 b 1 H 2 x = ba) (H 1, H 2 ) = a b(a 1 b 1 H 1 H 2 x = ba) (H 1 H 2 ) a b((ba) 1 H 1 H 2 x = ba) ( 17.2(2)) = a b((x) 1 H 1 H 2 ) (x = ba) = a b(((x) 1 ) 1 H 1 H 2 ) (H 1 H 2 ) a b(x H 1 H 2 ) ( 17.2(1)) H 2 H 1 H 1 H 2 H 1 H 2 = H 2 H 1 1 : H 1 H 2 H 1 H 2 (H 1 H 2)(H 1 H 2 ) x h 1 H 1, h 2 H 2 x = h 1 h 2 (H 1 H 2) h 2 H 2 H 2 h 1 2 H 2 h 1 = (h 1 h 2 )h 1 2 H 2 h 1 H 1 H 2 h 2 H 1 H 2 x = h 1 h 2 (H 1 H 2)(H 1 H 2 ) (H 1 H 2)(H 1 H 2 ) H 1 H 2 H 1 H 2 x (H 1 H 2)(H 1 H 2 ) a H 1 H 2, b H 1 H 2 x = ab a H 1, b H 2 x = ab H 1 H 2 H 1 H 1, H 2 H 2 a, b H 1 H 2 H 1, H H 1 H 2 x = ab H 1 H 2 (H 1 H 2)(H 1 H 2 ) H 1 H 2 H 1 H H 1 H 2 H 1 H 2 = (H 1 H 2)(H 1 H 2 ) (1) G Klein G G H 1 = {( )}; H 2 = {( ), ( ) }; H 3 = {( ), ( ) }; H 4 = {( ), ( ) }; H 5 = G. 1 H 1 H 2 x, y G xy H 1 H 2 h 1 H 1, h 2 H 2 xy = h 1 h 2 xy H 1 H 2 x H 1 y H 2 a H 1, b H 2 ab H 1 H 2 (ab) 1 H 1 H 2 b 1 a 1 H 1 H 2 b 1 H 1 a 1 H 2 b H 1 a H 2 ab H 2 H 1 b 1 a 1 H 1 H 2 b 1 H 1 a 1 H 2 138

139 H 5 H 2 H 3 H 4 H 1 (2) 6 G H 1 = {( )}; H 2 = {( ), )}; H 3 = {( ), ( ) 0 w 2 w 0 }; H 4 = {( ), ( ) 0 w w 2 0 }; H 5 = {( ), ( ) ( ) w 0 0 w, w w }; H 6 = G. G 6 Lagrange G 2 3 Lagrange G 3 2 H 2 H 3 H 4 3 H 5 6 G H 6 H 2 H 3 H 4 H 5 H (1) 17.12(2) G a, a 2, a 4, a 7, a 8, a 11, a 13, a 14 (2) 17.13(3) Lagrange G 3 5 G H 1 = {e}; H 2 = a 5 = {e, a 5, a 10 }; H 3 = a 3 = {e, a 3, a 6, a 9, a 12 }; H 4 = a = G. H 4 H 2 H 3 H

140 : c a b c e c c e c, e c c > 1 c a a c c a a = a = p Lagrange c = a c a a c = a k Z c k = a b c b a = c k b a / b : H 1 H 2 G G = a 17.13(3) H 1 = a s, H 2 = a r (r, s) = 1 m, n Z mr+ns = 1 x G x = a t x = a tmr+tns = a s(tm) a r(tn) H 1 H 2 G H 1 H G = H 1 H 2 : H = H 1 H 2 a d H 17.12(1) H G a d H H a d G a t H H = H 1 H 2 a t H 1 a t H 2 k 1, k 2 Z a k 1r = a k 2s = a t G k 1 r = t k 2 s = t ( k 1 r t a k 1r t = 0 k 1 r t 0 G = a k 1 r t G ) r t s t d = [r, s] t a t a d a t H a d G n H H 1 H H 2 Lagrange H H 1 = n (n, r) H H 2 = n (n, s) (n, r) n H (n, s) n H [(n, r), (n, s)] n H H n [(n, r), (n, s)] a d = a d = n (n, d) = n (n, [r, s]) (n, [r, s]) = [(n, r), (n, s)] H a d a d H a d H a d = H H G 5.5 a d = H (n, [r, s]) = [(n, r), (n, s)] 17.2 x, y, z R min(x, max(y, z)) = max(min(x, y), min(x, z)) : x max(y, z) min(x, max(y, z)) = x y, z ( y) min(x, y) = x min(x, z) x max(min(x, y), min(x, z)) = x x > max(y, z) x > y x > z min(x, y) = y min(x, z) = z max(min(x, y), min(x, z)) = max(y, z) x > max(y, z) min(x, max(y, z)) = max(y, z) (n, [r, s]) = [(n, r), (n, s)] n, r, s n = p a i i r = p b i i s = p c i i (n, [r, s]) = p min(n,max(r,s)) i (gcd, lcm ) = p max(min(n,r),min(n,s)) i ( 17.2) = [(n, r), (n, s)] (gcd, lcm ) 140

141 : G a a, a 2,, a k, (k Z) G G g G g = g S = { g g G} S G G = S G = x (G = S) x S x S x ( ) S x G x S (1) στ = ( ), τσ = ( ), σ 1 = ( ), τ 1 = ( ); (2) σ = (152)(34) = (12)(15)(34), τ = (14523) = (13)(12)(15)(14) : A = {(12), (13),, (1n)}, B = {(12), (23),, (n 1 n)} B S n S n A A B S n = A = B σ S n σ A σ A τ = (i 1 i 2 i k ) 1 1 j k i j = 1 τ = (i j i j+1 i k i 1 i 2 i j 1 ) i j = τ = (1i j 1 ) (1i 2 )(1i 1 )(1i k ) (1i j+1 ) τ A τ = (i 1 i 2 i k ) 1 i j 1(1 j k) τ = (1i 2 )(1i 2 i k )(1i 1 ) = (1i 2 )(1i k ) (1i 2 )(1i 1 ) τ A n σ S n A A σ A S N A A B n = 1, 2 A B n 3 k (1k) = (12)(23) (k 1 k) (2 k n) k = 2 k = t k = t + 1 (12)(23) (t 1 t)(t t + 1) = (1t)(t t + 1) (1t)(t t + 1) 1, t, t + 1 (1t)(t t + 1) = (1 t + 1) (1k) {(12), (13),, (1n)} (1k) = (12)(23) (k 1 k) B A A B A = B = S n (1) x = σ 1 τ = ( ) ( ) = ( ), y = τσ 1 = ( ) ( ) = ( ) ; (2) σ = (12354) = 5 1, τ = (15423) = H = {(1), (1234), (13)(24), (1432)}; H(1) = H(1234) = H(13)(24) = H(1432) = H; 141

142 H(12) = H(134) = H(1423) = H(243) = {(12), (134), (1423), (243)}; H(13) = H(14)(23) = H(24) = H(12)(34) = {(13), (14)(23), (24), (12)(34)}; H(14) = H(234) = H(1243) = H(132) = {(14), (234), (1243), (132)}; H(23) = H(124) = H(1342) = H(143) = {(23), (124), (1342), (143)}; H(34) = H(123) = H(1324) = H(142) = {(34), (123), (1324), (142)} r, s, t Q, r 0 ( 0 r 1 s ) ( t ) = ( ) r rt+s 0 1 H {{ ( ) r rt+s 0 1 t Q} r, s Q, r 0} r, s Q, r 0 {rt + s t Q} = Q r, s, t Q rt + s Q {rt + s t Q} Q t Q t = (t s)/r t Q t = rt + s {rt + s t Q} {rt + s t Q} = Q H {{( 0 r 1 t ) t Q} r Q, r 0} : H = H 1 H 2 e H 17.12(1) H H 1 H 2 Lagrange H r H s H (r, s) = 1 e H H 1 H = {e} : G p m m m = 1 G G p m < k m = k G = p k > 1 G a a = p k a = a = p k = G G a G G = G 5.5 G = a 17.13(3) G p a < p k Lagrange a = a = p t (1 t < k) a p p G m = k 2 G p m G 2 a G a e < a p m p a = p k k Z + b = a pk 1 b e ( a = p k ) b p = (a pk 1 ) p = a pk = e 1 < b p b = p G p b : G H G [G : H] = G H [G : K] K = K /[K : H] = [G : K][K : H] (Lagrange ) (Lagrange ) (1) 2 sunbird

143 : f : A B AB a, b A B f( a, b ) = ab f {f 1 [g] g AB} A B f 1 [g] = { a, b a, b A B f( a, b ) = g} {g} A B = f 1 [g] g AB = f 1 [g] g AB g AB f 1 [g] = A B g AB a A, b B g = ab S g = { ac, c 1 b c A B} S g ac, c 1 b ac, c 1 b A B f( ac, c 1 b ) = g S g f 1 [g] x, y f 1 [g] xy = ab (xy = g = ab) = x = aby 1 ( y 1 ) = xa 1 = by 1 ( a 1 ) c = by 1 b B, y B c B c = by 1 = xa 1 A c A B x, y = ac, c 1 b S g f 1 [g] = S g c A B ac = ac c = c f 1 [g] = S g = A B A B = A B = f 1 [g] = A B = AB A B g AB g AB (2) : A B = {e} (1) AB = AB A B = A B σ = (i 1 i 2 i k ) A k τ A τστ 1 = (τ(i 1 )τ(i 2 ) τ(i k )) A k : x A i j {i 1, i 2,, i k } x = τ(i j ) i j = τ 1 (i j ) τστ 1 (x) = τσ(i j ) (i j = τ 1 (i j )) τ(i j+1 ), j < k = (σ = (i 1 i 2 i k )) τ(i 1 ), j = k = (τ(i 1 )τ(i 2 ) τ(i k ))(x) ( ) i j {i 1, i 2,, i k } x = τ(i j ) τ 1 (x) / {i 1, i 2,, i k } σ(τ 1 (x)) = τ 1 (x) τστ 1 (x) = τ(τ 1 (x)) (σ(τ 1 ) = τ 1 ) = x (ττ 1 = (1)) = (τ(i 1 )τ(i 2 ) τ(i k ))(x) ( ) : σ, τ S n τστ 1 σ σ = σ 1 σ 2 σ t σ σ i = τσ iτ 1, i = 1, 2,, t i t σ i σ i 143

144 σ j = (i j1 i j2 i jm ) σ k = (i k1 i k2 i kr ) 1 j < k t 17.3 σ j = (τ(i j1 )τ(i j2 ) τ(i jm )) σ k = (τ(i k 1 )τ(i k2 ) τ(i kr )) τ σ j, σ k σ j σ k τστ 1 = τσ 1 σ 2 σ t τ 1 (σ = σ 1 σ 2 σ t ) = τσ 1 τ 1 τσ 2 τ 1 τσ t τ 1 (τ 1 τ = 1) = σ 1σ 2 σ t (σ i = τσ i τ 1 ) τστ 1 σ σ = (12354) 5 1 τ = (15423) S 3 σ 1 = σ 2 = τ 1 = (12), τ 2 = (13) σ 1 τ 1 σ 2 τ 2 σ 1 τ 1 = (1) σ 2 τ 2 = (123) (1) Abel x G xax 1 = xx 1 a = a x G, x = {x} G e = {e} a = {a} a 2 = {a 2 } a 3 = {a 3 } (2) Klein Abel x G, x = {x} Klein e = {e} a = {a} b = {b} c = {c} : y G y N(x 1 ax) yx 1 ax = x 1 axy (N(x 1 ax) ) yx 1 a = x 1 axyx 1 ( x 1 ) xyx 1 a = axyx 1 ( x) xyx 1 N(a) (N(a) ) y x 1 N(a)x (x 1 N(a)x ) : a = [G : N(a)] a n = [G : N(a n )] N(a) G N(a n ) G N(a) N(a n ) N(a) N(a n ) G a n = [G : N(a n )] [G : N(a n )][N(a n ) : N(a)] = [G : N(a)] = a n n N +, N(a) N(a n ) n = 1 n = k(k 1) n = k + 1 x N(a) xa k+1 = xa k a = a k xa ( ) = aa k x (xa = ax) = a k+1 x x N(a k+1 ) N(a) N(a k+1 ) 144

145 17.41 : C N(a) k = [G : N(a)] = [G : N(a)] [G : N(a)][N(a) : C] = [G : C] = n c : Abel Abel G H g G x gh h H x = gh G Abel x = hg Hg gh Hg Hg gh : g = ( 0 r 1 s ) G gh = {( 0 r 1 t ) t Q} Hg Hg = {( t ) ( 0 r 1 s ) t Q} = { ( ) r s+t 0 1 t Q} s, t Q s + t Q Hg = { ( ) r s+t 0 1 t Q} {( r t 0 1 ) t Q} = gh {( r t 0 1 )} gh t = t s {( 0 r 1 t )} = {( ) r s+t 0 1 } Hg gh = Hg g H : K H N H H KN H N H h H hn = Nh KN = NK KN G H KN H = KN : N = 2 e N N = {e, a} e C N G g G gng 1 = N {geg 1, gag 1 } = {a, e} g G geg 1 = gg 1 = e gag 1 = a ga = ag a C N = {e, a} C (1) : g G, h H g 0, h > 0 ghg 1 = g h g 1 = g g 1 h = g g 1 h = h > 0 ghg 1 H H G (2) a = a = 1 a 2 = E h H ah = h < ah / H ah H g G g H gh = H g / H g < 0 g = a(ag) ag = g > 0 ag H gh = ah (1) G H ah [G : H] = 2 : a, b G 1, ϕ(ab) = ab = a b = ϕ(a)ϕ(b) ϕ G 1 G 2 145

146 x (2) x Q A = A M n (Q), x = ϕ(a) ϕ(g 1 ) ϕ(g 1 ) = Q ϕ ker ϕ = SL n (Q) = {A A M n (Q) A = 1} Q : ϕ : Q Z Q, + Z, + x Q ϕ(x) = n 0 m = ϕ( x 2n ) Z n = ϕ(x) = ϕ( x 2n + x 2n + + x ) }{{ 2n } 2n = ϕ( x 2n ) + ϕ( x 2n ) + + ϕ( x 2n ) (ϕ ) }{{} 2n = m + m + + m }{{} 2n (ϕ( x 2n ) = m) = 2nm 2m = 1 m = 1 x m = ϕ( 2 2n ) Z : 3.4 ϕ 2 ϕ 1 : G 1 G 3 x, y G ϕ 2 ϕ 1 (xy) = ϕ 2 (ϕ 1 (xy)) ( 3.3) = ϕ 2 (ϕ 1 (x)ϕ 1 (y)) (ϕ 1 ) = ϕ 2 (ϕ 1 (x))ϕ 2 (ϕ 1 (y)) (ϕ 2 ) = ϕ 2 ϕ 1 (x)ϕ 2 ϕ 1 (y) ( 3.3) ϕ 2 ϕ 1 : G 1 G 3 G 1 G 3 : 3.9 ϕ 1 : G 2 G 1 x, y G (1) ϕ 1 (xy) = ϕ 1 (ϕ(ϕ 1 (x))ϕ(ϕ 1 (y))) (ϕ ϕ 1 = I G2 ) = ϕ 1 (ϕ(ϕ 1 (x)ϕ 1 (y))) (ϕ ) = ϕ 1 (x)ϕ 1 (y) (ϕ 1 ϕ = I G1 ) ϕ : G 2 G 1 G 2 G 1 : H e 1 ϕ 1 (e 2 ) ϕ 1 (H) ϕ 1 (H) a, b ϕ 1 (H) ϕ(a), ϕ(b) H ϕ(ab 1 ) = ϕ(a)ϕ(b 1 ) = ϕ(a)ϕ(b) 1 H ab 1 ϕ 1 (H) ϕ 1 (H) G 1 146

147 (2) : (1) ϕ 1 (H) g G 1, h ϕ 1 (H) ϕ(ghg 1 ) = ϕ(g)ϕ(h)ϕ(g 1 ) H ghg 1 ϕ 1 (H) ϕ 1 (H) G : ϕ b km = ϕ(a m ) = ϕ(e 1 ) = e (1) n mk n mk ϕ foralls, t Z, a s = a t a s t = e 1 ( ) m s t ( 17.8(1)) = mk ks kt ( k) = n ks kt (n mk) b ks kt = e 2 ( 17.8(1)) b ks = b kt ( b kt ) ϕ(a s ) = ϕ(a t ) (ϕ ) ϕ a s, a t G 1 ϕ(a s a t ) = ϕ(a s+t ) = b k(s+t) = b ks b kt = ϕ(a s )ϕ(a t ) ϕ ϕ G 1 G 2 H G 1 ϕ(h) ( G2, H ). : 17.35(1) ϕ(h) G 2 Lagrange ϕ(h) G 2 ϕ H : H ϕ(h) ϕ H H/ ker(ϕ H) = ϕ(h) ϕ(h) = H/(ker ϕ H) = [H : ker ϕ] H ϕ(h) ( H, G 2 ) : 17.4 ϕ(h) ( H, G 2 ) = 1 ϕ(h) = {e 2 } H ker ϕ : ϕ : G G/N g G ϕ(g) = Ng 17.4 ϕ(h) ( H, G/N ) = ( H, [G : N]) = 1 ϕ(h) = {N} h H ϕ(h) = hn = N h N, h H H N ϕ G 1 G 2 H G 1 ker ϕ H a G 1 a H ϕ(a) ϕ(h). : a H ϕ(a) ϕ(h) ϕ(a) ϕ(h) a H ϕ(a) ϕ(h) 147

148 b(b H ϕ(a) = ϕ(b)) (ϕ(h) ) b(b H a ker ϕ = b ker ϕ) ( 17.36(2)) b(b H a b ker ϕ) ( 17.22) = b(b H a bh) (ker ϕ H) b(b H a H) (b H) = a H ( ) a H ϕ(a) ϕ(h) : G 2 /ϕ(n) G 2 f : G 2 G 2 /ϕ(n) a G 2 f(a) = ϕ(n)a g = f ϕ : G 1 G 2 /ϕ(n) g ( ϕ f ) ker g a G 1 a ker g g(a) = ϕ(n) (ker g ) f(ϕ(a)) = ϕ(n) (g = f ϕ) ϕ(a) ϕ(n) (f ) a N ( 17.5) ker g = N G 1 /N = G 2 /ϕ(n) : HK G x HK, g G h H, k K x = hk gxg 1 = ghkg 1 = gh(g 1 g)kg 1 = (ghg 1 )(gkg 1 ) H, K G ghg 1 H, gkg 1 K gxg 1 = (ghg 1 )(gkg 1 ) HK HK G 17.46(1) H HK G C G H C (1) H G (2) G/H G Abel : (1) g G, h H h H C ghg 1 = hgg 1 = h H H G (2) G/H a G G/H = Ha x, y G m, n Z x Ha m, y Ha n h 1, h 2 H x = h 1 a m, y = h 2 a n xy = h 1 a m h 2 a n (x = h 1 a m, y = h 2 a n ) = h 2 h 1 a m a n (h 2 C) = h 2 h 1 a n a m (a m a n = a m+n = a n a m ) = h 2 a n h 1 a m (h 1 C) = yx G Abel 148

149 : G = C C G C G = p 2 C 1 p p 2 k C = G [G : N(a i )] i=1 a i (i = 1, 2,, k) 2 [G : N(a i )] p 2 [G : N(a i )] < p 2 [G : N(a i )] = p i = 1, 2,, k C = p 2 kp = (p k)p C = 1 C p C = p G/C = p G/C 17.6 G Abel C = G p 2 C = p 2 G = C Abel G H G G/H : H G G/H G G/H : G p a q b 17.6 ab = a b = pq G = ab 17.7 G/H : G = {e, a, b, c} Klein G = {(1), (12)(34), (13)(24), (14)(23)} S 4 ϕ : G G, ϕ(e) = (1), ϕ(a) = (12)(34), ϕ(b) = (13)(24), ϕ(c) = (14)(23) ϕ G G = G G S S n σ S 4, τ G στσ στσ 1 G G S : G G = n ϕ ϕ(g) = G G/ ker ϕ = G Lagrange G ker ϕ = [G : ker ϕ] = G G = 1 ker ϕ = {e} ϕ : (x 1 ) 1 = x ϕ a, b G ϕ(ab) = (ab) 1 (ϕ ) = b 1 a 1 ( 17.2(2)) = a 1 b 1 (G ) = ϕ(a)ϕ(b) (ϕ ) 149

150 ϕ ϕ a, b G ab = ((ab) 1 ) 1 ( 17.2(1)) = ϕ((ab) 1 ) (ϕ ) = ϕ(b 1 a 1 ) ( 17.2(2)) = ϕ(b 1 )ϕ(a 1 ) (ϕ ) = (b 1 ) 1 (a 1 ) 1 (ϕ ) = ba ( 17.2(1)) G (1) : n, t n nt ϕ (2) : (n, t) = 1 p, q Z pn + qt = 1 a i G a i = a i ipn == a iqt = ϕ(a iq ) ϕ(g) ϕ ϕ ϕ q Z ϕ(a q ) = a qt = a qt 1 n k Z qt 1 = kn p = k Z pn + qt = 1 (n, t) = : f : G Inn G g G f(g) = ϕ g f f x, y, a G ϕ xy (a) = xya(xy) 1 (ϕ xy ) = xyay 1 x 1 ( 17.2(2)) = ϕ x (yay 1 ) (ϕ x ) = ϕ x (ϕ y (a)) (ϕ y ) = ϕ x ϕ y (a) ( 3.3) f ker f = C g G, g C a(a G ga = ag) (C ) a(a G gag 1 = a) ( g 1 ) a(a G ϕ g (a) = a) ϕ g = I G g ker f ker f = C G/C = Inn G (ϕ ) (I G ) (ker f ) 150

151 17.64 : 10 G (1) G Abel G 2 a 5 b 17.6 ab = a b = 10 G = ab = Z 10 ( n n n ) (2) G Abel 5 G 2 x, y G (xy) 2 = e = x 2 y 2 (x, y, xy 2 1 ) = xyxy = xxyy ((xy) 2 = x 2 y 2 ) yx = xy G Abel ( ) a G 5 a = A a i (i = 1, 2, 3, 4) G 5 b / A 5 B = b G 5 H = A B H A Lagrange H 5 a b H 5 H = (1) AB G AB = A B = 25 > 10 A B G A 2 2 b ( b / A) G = A Ab G G = {e, a, a 2, a 3, a 4, ab, a 2 b, a 3 b, a 4 b} a i i = 1, 2, 3, 4 (a i b) 2 = e a i ba i b = e ba i b = (a i ) 1 = a 5 i G G a i b j i = 0, 1, 2, 3, 4 j = 0, 1 a i b j, a k b r G a i b j a k b r a s b t ( ba i b = (a i ) 1 = a 5 i b j r < j a i b j a k b r = a i b j a k b r e n = a i b j a k b r+2n r) 10 G = {a i b j i Z 5, j Z 2 } a = 5, b = 2, ba i b = a 5 i 10 D Z Inn G = 1 [G : C] = G/C = 1 C = G Inn G = {I G } G Abel : ϕ : G 1 G 2 G 2 G 1 g 1, g 2 G 1 G 2 ϕ( g 1, g 2 ) = g 2, g 1 ϕ ϕ G 1 G 2 = G2 G n (dihedral group) D n G = σ, τ σ n = τ 2 = I, τστ = σ 1 151

152 : e 1 H 1, e 2 H 2 e 1, e 2 H 1 H 2 H 1 H a, b, c, d H 1 H 2 a, b c, d 1 = a, b c 1, d 1 ( 15.6(5)) = ac 1, bd 1 H 1 H 2 (a, c H 1 b, d H 2 ) H 1 H 2 G 1 G 2 : ϕ : G G/H G/K g G ϕ(g) = Hg, Kg ϕ ϕ a, b G ϕ(a) = ϕ(b) Ha, Ka = Hb, Kb (ϕ ) Ha = Hb Ka = Kb ( 2.1) ab 1 H ab 1 K ( 17.22) ab 1 H K ab 1 = e (H K = {e}) a = b ( b) ϕ G ϕ(g) G ϕ(g) = G G/H G/K ϕ 1 : G 1 G 2 G 1 g 1, g 2 G 1 G 2 ϕ 1 ( g 1, g 2 ) = g 1 ϕ 2 : G 1 G 2 G 2 g 1, g 2 G 1 G 2 ϕ 2 ( g 1, g 2 ) = g 2 ϕ 1 ϕ 2 N 1 = ker / ϕ 1 = { e 1, g 2 g 2 G 2 }, N 2 = ker ϕ 2 = { g 1, e 2 g 1 G 1 } G 1 G 2 N 1 G 1 G 2 /N 2 152

153 18.1 : a 1, a 2, b 1, b 2 A (a 1 b 1 ) (a 1 b 2 ) (a 2 b 1 ) (a 2 b 2 ) =((a 1 b 1 ) (a 1 b 2 )) ((a 2 b 1 ) (a 2 b 2 )) =(a 1 (b 1 b 2 )) (a 2 (b 1 b 2 )) =(a 1 a 2 ) (b 1 b 2 ) =((a 1 a 2 ) b 1 ) ((a 1 a 2 ) b 2 ) =((a 1 b 1 ) (a 2 b 1 )) ((a 1 b 2 ) (a 2 b 2 )) =(a 1 b 1 ) (a 2 b 1 ) (a 1 b 2 ) (a 2 b 2 ) ( ) ( ) ( ) ( ) ( ) ( ) 18.2 : Z[i], + Abel 0 = 0 + 0i a + bi Z[i] ( a) + ( b)i Z[i] a + bi, c + di Z[i] (a + bi)(c + di) = (ac bd) + (ad + bc)i Z[i] Z[i], Z[i], Z[i], +, 18.3 : P(B) 1.7 P(B),, 18.4 : Z a, b, c Z (a b) c = (a + b 1) + c 1 ( ) = a + (b + c 1) 1 ( ) = a (b c) ( ) (a b) c = (a + b ab) + c (a + b ab)c ( ) 153

154 = a + b ab + c ac bc + abc = a + b + c bc ab ac + abc ( ) = a + (b + c bc) a(b + c bc) = a (b c) ( ) a (b c) = a (b + c 1) ( ) = a + (b + c 1) a(b + c 1) ( ) = a + b + c 1 ab ac + a = a + b ab + a + c ac 1 = (a + b ab) (a + c ac) ( ) = (a b) (a c) ( ) (b c) a = (b + c 1) a ( ) = (b + c 1) + a (b + c 1)a ( ) = b + c 1 + a ba ca + a = b + a ba + c + a ca 1 = (b + a ba) (c + a ca) ( ) = (b a) (c a) ( ) 1 0 a Z 2 a Z a (2 a) = 1 Z Z,, 18.5 a R a = a : a R a = ( a) 2 ( ) = a 2 ( 18.1(3)) = a ( ) (1) : a, b R a + b = (a + b) 2 ( ) = a 2 + ab + ba + b 2 ( ) = a + ab + ba + b ( ) ab + ba = 0 ab = ba ba = ba ab = ba = ba (2) : a R a = a a + a = a + ( a) = 0 (3) 154

155 : R > 2 a, b R {0} a b ab + ab = 0 ( (2) ) a 2 b + ab 2 = 0 (a 2 = a, b 2 = b) a(ab + b 2 ) = 0 a(a + b)b = 0 ( ) ( ) a + b = a b 0 a(a + b) = 0 a a + b a(a + b) 0 a(a + b) b R 18.6 (1) 15.6 (2) 15.6 (3) R 1 = R 2 = Z 2,, 2 2 Z 2 Z 2,, 0, 1, 1, 0 Z 2 Z 2 0, 1 1, 0 = 0, (1) n 1 < p, q < n pq = n p q = q p = pq mod n = 0 (2) : k k r = 0 n kr = [r, n] [r, n] [r, n] rn r n k = = r r (r, n) = n (r, n) (r, n) = 1 k = n / Z n r r (r, n) > 1 0 < k < n k Z n r (3) 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, : b R ab = 0 b = 1b = (a 1 a)b = a 1 (ab) = a 1 0 = 0 a a a : R, +, R = R {0} a, b R R a 0 b 0 R ab 0 ab R R R, R R 1 R R a R ϕ a : R R x R ϕ a (x) = ax R x, y R ϕ a (x) = ϕ a (y) x = y ϕ a ϕ a R ϕ a (R ) R ϕ a (R ) ϕ a (R ) R R ϕ a (R ) R ϕ a (R ) = R a R 1 R = ar b R 1 = ba = ab ar a 1 1 R ( ) 155

156 a R R, Abel R, +, : pq R 18.1 R : S, S a, b S, c R abc = 0 bc = 0 c = 0 (a ) (b ) a, b S ab ab ab S S, +, R, +, 18.7 (3) R = Z 18 S = {0, 1, 5, 7, 11, 13, 17} = 2 / S S S, +, R ( Z, +, ) S = R R : 18.3 n (a + b) n = n Cna i n i b i 0 < i < n Cn i n i=0 n n C i n C i n = k i n k i Z i = 1, 2,, n a R, k i Z kna = kn(1 a) = k(n 1)a = k(0 a) = 0 n n 1 (a + b) n = Cna i n i b i = a n + b n + k i na n i b i = a n + b n. i=0 : T S T 2 a T a 0 x T T xa T x = (xa)a 1 S x T S S T S S S i=1 x, y S x y S xy 1 S x, y S a 1, a 2, b 1, b 2 T b 1, b 2 x = a 1 b 1 1, y = a 2b 1 2 x y = a 1 b 1 1 a 2 b 1 2 (x = a 1 b 1 1, y = a 2b 1 2 ) = a 1 b 2 b 1 2 b 1 1 a 2 b 1 b 1 1 b 1 2 (b 2 b 1 2 = b 1 b 1 1 = 1) = (a 1 b 2 a 2 b 1 )(b 1 b 2 ) 1 ( ) T a 1 b 2 a 2 b 1, b 1 b 2 T F b 1 b 2 b 1 b 2 0 x y = (a 1 b 2 a 2 b 1 )(b 1 b 2 ) 1 S x, y S a 1, a 2, b 1, b 2 T b 1, b 2 x = a 1 b 1 1, y = a 2b 1 2 xy 1 = a 1 b 1 1 (a 2b 1 2 ) 1 (x = a 1 b 1 1, y = a 2b 1 2 ) = a 1 b 1 1 b 2a 1 2 ( 17.2) = a 1 b 2 b 1 1 a 1 2 = a 1 b 2 (a 2 b 1 ) 1 ( 17.2) 156

157 a 2, b 1 ( x, y S ) F a 2 b 1 0 a 1 b 2, a 2 b 1 T a 1 b 2 (a 2 b 1 ) 1 S F a 1 b 2 (a 2 b 1 ) 1 0 a 1 b 2 (a 2 b 1 ) 1 S S S 1 T a, b T S 1, b 0 b S1 S1 b 1 S1 S 1 ab 1 S 1 a, b S S : a a b R ab b a + ab b a ( ab b = 0 ab = b ) c = a + ab b x R xc = x(a + ab b) (c = a + ab b) = xa + xab xb ( ) = x + xb xb (xa = x) = x (xb xb = 0) c a a b R ab = b a R : (1) (2) u u a l u a r a r a r = a l ua r (a l u = 1) = a l (ua r = 1) = a l ua r (ua r = 1) = a r (a l u = 1) u (2) (3) u a r u a r u 1 a r u 1 0 u 0 ( ua r = 0a r = 0 1 ) u(a r u 1) = ua r u u ( ) = u u (ua r = 1) = 0 u (3) (1) u b r 0 ub r = 0 a r u u(a r + b r ) = ua r + ub r ( ) = 1 (ua r = 1, ub r = 0) a r + b r u b r 0 a r + b r a r u : a R k a k = 0 a 0 k 2 ( k = 1 a = a 1 = 0 ) k a 1 0, a k 1 0 a k = a 1 a k 1 = 0 a a k 1 R

158 18.2 R D R N(D) = {x x R, n x n D}, N(D) R : x, y N(D) m, n Z + x m, y n D R (x y) m+n 1 = m+n 1 Cm+n 1x i m+n 1 i y i (*) i=0 (*) n x m x m+n 1 i y i = x m x n 1 i y i x m D x n 1 i y i R x m+n 1 i y i D (*) m y n x m+n 1 i y i = x m+n 1 i y i n y n x m+n 1 i y i n R y n D x m+n 1 i y i D D (x y) m+n 1 D x y N(D) x N(D), y R(D) m Z + x m D R (xy) m = x m y m x m D y m R (xy) m D xy = yx N(D) N(D) : N({0}) R {0} R 18.2 N({0}) R R : N({0}) R {0} R : A, B R R x, y A B x y A x y B x y A B r R, x A B rx, xr A rx, xr B rx, xr A B A B R (1) : x, y A + B a 1, a 2 A, b 1, b 2 B x = a 1 + b 1, y = a 2 + b 2 x y = a 1 + b 1 a 2 + b 2 = (a 1 a 2 ) + (b 1 b 2 ) A + B r R, x A + B a A, b B x = a + b rx = r(a + b) = ra + rb A, B ra A, rb B rx = ra + rb A + B r(a + B) A + B (A + B)r A + B A + B (2) R Z[i] = {a + bi a, b Z} C R + Z[i] = {x + ai x R, b Z} 2 + i R + Z[i] ( 2 + i) 2 = i / R + Z[i] R + Z[i] : E ij i j 1 0 xe ij i j x 0 x F D M n (F ) D {(0)} D = M n (F ) D A D A a kt 0(1 k, t n) E ij, i, j = 1, 2, n x F xe ij = (xa 1 kt E ik)ae tj 158

159 D A D E tj M n (F ) AE tj D xa 1 kt E ik M n (F ) xe ij = (xa 1 kt E ik)ae tj D B = (b ij ) M n (F ) B = b ij E ij D D = M n (F ) 1 i,j n Z 5, Z 5 {0} Z {0, 2, 4} {0, 3} Z 6 Z 6 H 1 = {0} H 2 = {0, 3} H 3 = {0, 2, 4} H 4 = Z 6 : 0 D D D x D m Z x = 4m = 2(2m) A D A x, y D m, n Z x = 4m, y = 4n x y = 4(m n) D a A d D m, n Z a = 2m, d = 4n ad = da = 8mn = 4(2mn) D A A/D = { 0, 2}, +, = 2 2 = : A m = {0} (R/A) n = { 0} R mn = {0} r ij R i = 1, 2,, m j = 1, 2,, n r ij = m ij i = 1, 2,, m ā i = r i1 r i2 r in i=1 j=1 n r ij a i = n r ij (a i ) = r i1 r i2 r in ( ) = 0 ((R/A) n = { 0}) = A ( 0 = A + 0 = A) a i A i = 1, 2,, m m n r ij = ij = = 0 i=1 j=1 m i=1 a i r ij R mn = {0} R j=1 ( ) (a i ) (A m = 0) : R/H R D R H D D = R H D x D H x / H x 0 = H R/H ȳ R/H xy + H = x ȳ = 1 = 1 + H xy xy + H = 1 + H 159

160 h H xy = 1 + h x D, y R D xy D H D h D 1 = xy h D r R 1 D r = r 1 D D = R R R R R/H R/H R/H { 0} ā R/H ā 0 a / H A = {h + ax h H x R} h H h = h+a 0 A H A a = 0+a 1 A a / R H A A h 1 +ax 1, h 2 +ax 2 A (h 1 +ax 1 ) (h 2 +ax 2 ) = (h 1 h 2 )+a(x 1 x 2 ) A h + ax H, r R r(h + ax) = (h + ax)r = (hr) + a(xr) A A R A H H 1 A = R h H, b R 1 = h + ab 1 ab = h H 17.25(4) ab = 1 b ā ā R/H { 0} R/H : 0 = 0 x x x m S S r 1 x 1 +r 2 x 2 + +r m x m, r 1 x 1 +r 2 x 2 + +r m x m S (r 1 x 1 +r 2 x r m x m ) (r 1 x 1 +r 2 x 2 + +r m x m ) = (r 1 r 1) x 1 +(r 2 r 2) x 2 + +(r m r m) x m S a R r 1 x 1 + r 2 x r m x m S (r 1 x 1 + r 2 x r m x m )a = a(r 1 x 1 + r 2 x r m x m ) = (ar 1 ) x 1 + (ar 2 ) x (ar m ) x m S S R ϕ : Z 2 Z Z 2 Z 0 ϕ(0) = 0 ϕ(1) + ϕ(1) = ϕ(1 + 1) = ϕ(0) = 0 Z x + x = 0 x = 0 ϕ(1) = ϕ(0) = Z 2 Z ϕ : Z 2 Z, x Z 2, ϕ(x) = 0 : A Abel a 1, a 2, b 1, b 2, c 1, c 2 Z ( ) ( a 1 b 1 0 c 1 a2 ) b 2 0 c 2 = ( a1 ) a 2 a 1 b 2 +b 1 c 2 0 c 1c 2 A A ( ) B B x 1, x 2 Z ( ) ( x ) ( 0 x 2 = 0 0 ) 0 x 1 x 2 B ( 0 0 ) ( 0 x ) ( 0 x 2 = 0 0 ) 0 x 1x 2 B B A ϕ : A B, ( a 0 c b ) A, ϕ(( a b Z} c )) = ( 0 0 a 0 ) ϕ ker ϕ = {( 0 b 0 c ) b, c : f(x) = a 0 + a 1 x + + a n x n, g(x) = b 0 + b 1 x + + b m x m F [x] ϕ(f(x) + g(x)) = a 0 + b 0 = ϕ(f(x)) + ϕ(g(x)) ϕ(f(x) g(x)) = a 0 b 0 = ϕ(f(x)) ϕ(g(x)) ϕ a F f(x) = a F [x] ϕ(f(x)) = a ϕ ker ϕ = {a 1 x + a 2 x a n x n n N, a i F, i = 1, 2,, n} F [x]/ ker ϕ = {ā a F }, +, ā = {a + a 1 x + + a n x n n N, a i F, i = 1, 2,, n} a, b F ā + b = a + b ā b = a b

161 : A/B R/B B A/B A/B x, ȳ A/B x, y A x y A x ȳ = x y A/B ( x ȳ = x y y y 0 ȳ + y ȳ + y = 0 y = ȳ x ȳ = x + y = x + ( y) = x y) x A/B ȳ R/B x A, y R xy A yx A x ȳ = xy A/B ȳ x = yx A/B A/B R/B ϕ : R/B R/A x + B R/B ϕ(x + B) = x + A ϕ x, y R/B x + B = y + B x + y B ( 17.25(4)) = x + y A (B A) x + A = y + A ( 17.25(4)) ϕ(x + B) = ϕ(y + B) ϕ (ϕ ) x + A R/A x + B R/B ϕ(x + B) = x + A ϕ ϕ 17.25(4) ϕ(x + B) = / x + A = A x A ker ϕ = {x + B x A} = A/B R/B (A/B) = R/A : x R 1 x ϕ 1 (ϕ(s)) y(y ϕ(s) ϕ(x) = y) y z(z S y = ϕ(z) ϕ(x) = y) (ϕ 1 ) (ϕ(s) ) = z(z S ϕ(x) = ϕ(z)) ( ) z(z S x + ker ϕ = z + ker ϕ) ( 17.36(2)) z(z S x z + ker ϕ) ( 17.25(4)) = x S + ker ϕ (S + ker ϕ ) x ker ϕ + S x ϕ 1 (ϕ(s)) ker ϕ + S x R 1 x ker ϕ + S y s(y ker ϕ s S x = y + s) (ker ϕ ) (ker ϕ + S ) = y s(y ker ϕ s S ϕ(x) = ϕ(y) + ϕ(s)) (ϕ ) = y s(y ker ϕ s S ϕ(x) = ϕ(s)) (y ker ϕ) = s(s S ϕ(x) = ϕ(s)) ( ) = ϕ(x) ϕ(s) (ϕ(s) ) x ϕ 1 (ϕ(s)) (ϕ 1 ) 161

162 ker ϕ + S ϕ 1 (ϕ(s)) ϕ 1 (ϕ(s)) = ker ϕ + S : F 1 = A 1, + 1, 1 F 2 = A 2, + 2, ker ϕ F F 1 {0} F 1 ϕ(f 1 ) {0} x F 1 ϕ(x) 0 x / ker ϕ ker ϕ F 1 ker ϕ = {0} ϕ A 1, + 1 A 2, + 2 ker ϕ = {0} ϕ A 1, + 1 A 2, + 2 ϕ A 1 A 2 F 1 F : f, g End G f + g f g x, y G (f + g)(x + y) = f(x + y) + g(x + y) ( ) = f(x) + f(y) + g(x) + g(y) (f, g ) = f(x) + g(x) + f(y) + g(y) (G Abel ) = (f + g)(x) + (f + g)(y) ( ) (f g)(x + y) = f(g(x + y)) ( ) = f(g(x) + g(y)) (g ) = f(g(x)) + f(g(y)) (f ) = (f g)(x) + (f g)(y) ( ) f + g f g G + End G + ϕ 0 ϕ : G G x G ϕ(x) = x ϕ f End G ϕ f End G f + ϕ f = ϕ 0 End G End G, + Abel 2.5 End G, f, g, h End G x G (f (g + h))(x) = f((g + h)(x)) ( ) = f(g(x) + h(x)) (+ ) = f(g(x)) + f(h(x)) (f ) = (f g)(x) + (f h)(x) ( ) ((g + h) f)(x) = (g + h)(f(x)) ( ) = g(f(x)) + h(f(x)) (+ ) = (g f)(x) + (h f)(x) ( ) + End G, +, ϕ : G G ϕ(a) = ia ϕ(ka) = kϕ(a) = kia k, i Z ϕ i (ka) = kia, ka G ϕ i = ϕ j i j (mod n), 162

163 End G = {ϕ i i = 0, 1,, n 1} ϕ i, ϕ j End G a t G (ϕ i + ϕ j )(ka) = ϕ i (ka) + ϕ j (ka) (+ ) = kia + kja (ϕ i, ϕ j ) = k(i + j)a ( ) = ϕ i+j (ka) (ϕ i+j ) (ϕ i ϕ j )(ka) = ϕ i (ϕ j (ka)) ( ) = kjia (ϕ i, ϕ j ) = ϕ ji (ka) (ϕ ji ) G {ϕ i i = 0, 1,, n 1}, +, ϕ i, ϕ j End G ϕ i + ϕ j = ϕ (i+j mod n) ϕ i ϕ j = ϕ (ji mod n) : ϕ : Q Q ϕ(1) = a x Q ϕ(x) = ax End G = {ϕ a a Q} ϕ a x Q ϕ a (x) = ax σ : End Q Q ϕ a End Q σ(ϕ a ) = a σ σ End Q, +, Q, +, ϕ a, ϕ b End Q x Q (ϕ a + ϕ b )(x) = ϕ a (x) + ϕ b (x) (+ ) = ax + bx (ϕ a, ϕ b ) = (a + b)x ( ) = ϕ a+b (x) (ϕ a+b ) (ϕ a ϕ b )(x) = ϕ a (ϕ b (x)) ( ) = abx (ϕ a, ϕ b ) = ϕ ab (x) (ϕ ab ) σ(ϕ a + ϕ b ) = σ(ϕ a+b ) = a + b = σ(ϕ a ) + σ(ϕ b ) σ(ϕ a ϕ b ) = σ(ϕ ab ) = ab = σ(ϕ a )σ(ϕ b ) σ End Q, +, = Q, +, x x x x + 1 x 0 x x 0 x x x + 1 x, (x + 1) F 2 [x]/(x + x 2 ) x 0 x x (x + 1) = 0 F [2]/(x + x 2 ) : 1 f(x) = a 0 + a 1 x + + a n x n F 2 [x] f(x) a i1 = a i2 = = a ik = 1 i 1 < i 2 < < i k k 163

164 f(x) = x i 1 (x i 2 i 1 + 1) + x i 3 (x i 4 i 3 + 1) + + x i k 1 (x i k i k 1 + 1) ( ) m Z + x m + 1 = (x + 1)(x m 1 + x m x + 1) ( ) (x + 1) (x + 1) f(x) x f(x) 1 f(x) F 2 [x] 1 ( ) x x+1 x 2 +x+1 x 3 +x+1 x 3 +x 2 +1 x 4 +x+1 x 4 +x 3 +1 x 4 +x 3 +x 2 +x F 2 [x] n 2 n 3 f(x) F 2 [x]/f(x) 8 f(x) = x 3 + x + 1 f(x) = x 3 + x f(x) = x 3 + x x x + 1 x 2 x x 2 + x x 2 + x x x + 1 x 2 x x 2 + x x 2 + x x + 1 x x x 2 x 2 + x + 1 x 2 + x x x x x 2 + x x 2 + x + 1 x 2 x x + 1 x + 1 x 1 0 x 2 + x + 1 x 2 + x x x 2 x 2 x 2 x x 2 + x x 2 + x x x + 1 x x x 2 x 2 + x + 1 x 2 + x 1 0 x + 1 x x 2 + x x 2 + x x 2 + x + 1 x x x x x 2 + x + 1 x 2 + x + 1 x 2 + x x x 2 x + 1 x x x + 1 x 2 x x 2 + x x 2 + x x x + 1 x 2 x x 2 + x x 2 + x + 1 x 0 x x 2 x 2 + x x x 2 + x + 1 x x x + 1 x 2 + x x x 2 + x + 1 x 2 1 x x 2 0 x 2 x + 1 x 2 + x + 1 x 2 + x x x x x x 2 x x + 1 x + 1 x 2 + x x 2 + x 0 x 2 + x x 2 + x x x 2 + x + 1 x x 2 x 2 + x x 2 + x + 1 x x 1 x 2 + x x 2 x + 1 f(x) = x 3 + x x x + 1 x 2 x x 2 + x x 2 + x x x + 1 x 2 x x 2 + x x 2 + x + 1 x 0 x x 2 x 2 + x x x 2 + x x + 1 x x + 1 x 2 + x x x x 2 + x + 1 x 2 x 2 0 x 2 x x 2 + x + 1 x + 1 x x 2 + x x x x 2 + x + 1 x x + 1 x 2 + x x 2 1 x 2 + x 0 x 2 + x 1 x 2 + x + 1 x x 2 x + 1 x x 2 + x x 2 + x + 1 x + 1 x 2 x 2 + x 1 x x x 5 1 = (x 1)(x 4 + x 3 + x 2 + x + 1) = (x + 1)(x 4 + x 3 + x 2 + x + 1) x + 1 x 4 + x 3 + x 2 + x

165 19.1 (1) (2) {c, d} (3) {a, b} (4) (5) (6) {d, f} (7) {c, d} (8) 19.2 (1) {4, 6} (2) (3) (4) 19.3 (1) : a b = b ( 19.2) = b c ( 19.2) (2) : (a b) (b c) = a b ( 19.2) = b c ( (1) ) = (a b) (a c) ( 19.2) 19.4 (1) : 19.1(1) a b a 19.1(2) a a c a b a c a b b d 19.1(3) a b (a c) (b d) c d (a c) (b d) 19.1(4) (a b) (c d) (a c) (b d) 165

166 (2) : 19.1 a b a a b a b b b c a b a c a 19.1(3) a b (a b) (b c) (c a) b c (a b) (b c) (c a) c a (a b) (b c) (c a) 19.1(4) (a b) (b c) (c a) (a b) (b c) (c a) 19.5 : 1 i n a 1 a 2 a n a i ( 19.1(1)) a 1 a 2 a n ( 19.1(2)) = a 1 a 2 a n ( ) a i = a 1 a 2 a n i a 1 = a 2 = = a n = a 1 a 2 a n 19.6 : a b a b b a 19.2 a b a a b b a b a a b b a b a a b b a b a a b a 19.2 a b b a a b 19.7 (1) a (b c) = (a b) (a c) (2) (a b) (b c) = (a b) (a c) (3) (a b) (c d) (a c) (b d) (4) (a b) (b c) (c a) (a b) (b c) (c a) (4) ( (2) (3) ) 19.8 L 1 {a, b, c} {a, b, d} {a, b, e} {a, c, e} {a, d, e} {b, c, e} {b, d, e} L 1 {a, b, c, e} {a, b, d, e} {b, c, d, e} L 1 L 1 L 2 {a, b, e} {a, b, g} {a, c, f} {a, c, g} {a, d, e} {a, d, f} {a, d, g} {a, e, g} {a, f, g} {b, e, g} {c, f, g} {d, e, g} {d, f, g} L 2 {a, b, c, g} {a, b, d, e} {a, b, e, g} {a, b, f, g} {a, c, d, f} {a, c, e, g} {a, c, f, g} {a, d, e, g} {a, d, f, g} {d, e, f, g} L 2 {a, b, c, e, g} {a, b, c, f, g} {a, b, d, e, g} {a, c, d, f, g} {a, d, e, f, g} 19.9 : x, y L 1 x, y L x a y a L x y, x y L 19.1(1) x y x a 19.1(4) x y a x y, x y L 1 L 1 L x, y L 2 x, y L a x a y L x y, x y L 19.1(3) a x y 19.1(2) a x x y x y, x y L 2 L 2 L L 3 = L 1 L 2 ( A = {L i i = 1, 2, k} 166

167 L x, y A x, y A x y x y A x y, x y A A L ) L 3 L L 1 {d, e, f, g, h} L 1 L 2 {a, b, c, f, g} L 3 {a, b, d, e, h} L 4 {a, b, d, d, e} ( d) L 5 {a, b, c, d, f} L 1 L 2 L 3 L 1 L 2 L : 19.1(2) a a c L a a c a (b (a c)) = (a b) (a c) a, b, c L a (b (a c)) = (a b) (a c) x, y, z L x y 19.2 x y = x x (z y) = x (z (x y)) (x y = x) = (x z) (x y) ( ) = (x z) y (x y 19.2) L : a b c a b 19.2 (a b) c = (a b) c = c (a c) (b c) (a b) = ((a b) c) (a b) ( ) = c (a b) ((a b) c = c) = c ((a b) = c) 167

168 c = (a c) (b c) (a b) a b ((a b) c) (a b) ( 19.1(2)) = (a c) (b c) (a b) ( ) = c ( ) = (a c) (b c) (a b) ( ) = ((a b) c) (a b) ( ) = (a b) ((a b) c) ( ) = (a b) (c (a b)) ( ) = ((a b) c) (a b) (a b a b L ) a b ( 19.1(1)) (1) : b (a c) = (b (b c)) (a c) ( ) = b ((b c) (a c)) ( ) = b ((a c) (b c)) ( ) = b ((c a) (b c)) ( ) = b (c (a (b c))) (c b c L ) = b (c ((a b) (a c))) ( ) = b ((c (a c)) (a b)) ( ) = b (c (a b)) ( ) = (c (a b)) b ( ) = ((a b) c) b ( ) = (a b) (c b) (a b b L ) = (b a) (b c) ( ) (2) : a (b c) = (a (b a)) (b c) ( ) = a ((b a) (b c)) ( ) = a (b (a c)) ( (1) ) = (a b) (a c) (a a c L ) (1) 168

169 : 0 a 19.1(2) a a b = 0 a = 0 b = 0 (2) : a (1) 1 = a b a a = 1 b = (1) : L a x L x 1 = a a = a a = a a = 0 x x = a x L = {a} L = 1 (2) : L 3 a T a 0 a 1 a b a b = 1 L a b b a b a 19.2 a = a b = 1 a 1 a b a b a b = a 0 b a a L L : a, b L 1 ā, b L ā b, ā b L (a b) (ā b) = (a (ā b)) (b (ā b)) ( ) = ((a ā) b) (b ( b ā)) ( ) = ((a ā) b) ((b b) ā) ( ) = (0 b) (0 ā) (a ā = b b = 0) = 0 0 (0 ā 0 b 19.2) = 0 ( 19.3(3)) (a b) (ā b) = ((a b) ā) ((a b) b) ( ) = (ā (a b)) (a (b b)) ( ) = ((ā a) b) (a (b b)) ( ) = (1 b) (a 1) (ā a = b b = 1) = 1 1 (a, b ) = 1 ( 19.3(3)) a b ā b L a b L 1 a b = ā b L a b L 1 L

170 L 1 L 2 L 3 L 4 L 5 L 1, L 2, L 3 L 4 L 5 L 4 L : x L T (x) = {y y L y x} i x = T (x) x L 1 i x t L x, y L : x y i x i y. L x y y x x y y x y x T (y) T (x) x y x T (x) x / T (y) T (y) T (x) i y < i x ϕ : L L(G) x L ϕ(x) = a pt i x+1 ϕ x, y L ϕ(x) = ϕ(y) i x = i y ( p t ix+1, p t iy+1 p t ϕ(x) = p i x 1 ϕ(y) = p i y 1 i x i y ϕ(x) ϕ(y) ϕ(x) = ϕ(y) ) 19.1 x y y x x = y ϕ L(G) = { a pi i = 0, 1,, t} 0 i t x L i = t i x + 1 x, y L x y i x = i y i x = i y 19.1 x = y y L(G) x L ϕ(x) = y ϕ x y ϕ(x) ϕ(y) x, y L 0 t i x + 1, t i y + 1 t p a x+1 t pt i = (p t, p t ix+1 ) = p t p t ix+1 = pix 1 a pt iy +1 p t = (p t, p t iy+1 ) = p t p t iy+1 = piy 1 ϕ(x) ϕ(y) a pt i x+1 a pt i y +1 (ϕ ) = a pt i x+1 a pt i y +1 ( 5.7 ) p i x 1 p i y 1 i x i y ( a pt i x+1 = p i x 1, a pt i y +1 = p i y 1 ) ( ) x y ( 19.1) 170

171 x y i x i y ( 19.1) p t iy+1 p t ix+1 (t i y + 1 t i x + 1) = a pt i x+1 a pt i y +1 ( ) ϕ(x) ϕ(y) ϕ : x, y L f(x y) = (x y) a (ϕ ) (f ) = (x a) (y a) ( ) = f(x) f(y) (f ) f(x y) = (x y) a (f ) = (x y) (a a) ( 19.3(3)) = (x a) (y a) ( ) = f(x) f(y) (f ) f g x f(l) y L x = f(y) = y a 19.1(2) a x x a 19.2 f(x) = x a = x x f(l) f(l) = {x x L a x} g(l) = {x x L x a} : 19.9 X Y f X Y f x 1, x 2 X f(x 1 ) = f(x 2 ) x 1 b = x 2 b (f ) = a (x 1 b) = a (x 2 b) (x 1 b = x 2 b) (a x 1 ) (a b) = (a x 2 ) (a b) ( ) x 1 (a b) = x 2 (a b) (x 1, x 2 a 19.2) x 1 = x 2 (a b x 1, x ) f y Y b y a b a y y a b a y a (a b) = a a y X f(a y) = (a y) b (f ) = (a b) (y b) ( ) = (a b) y (b y 19.2) = y (y a b 19.2) 171

172 y Y y = f(a y) f(x) f f L X, Y L f X Y x X g f(x) = g(f(x)) ( ) = (x b) a (f, g ) = (x a) (b a) ( ) = x (b a) (x a 19.2) = x (b a x 19.2) g f f 3.10(4) g = f 1 f Y X ( (2) ) : f, g A, x, y L f g(x y) = f(g(x y)) ( ) = f(g(x) g(y)) (g ) = f(g(x)) f(g(y)) (f ) = f g(x) f g(y) ( ) f g(x y) = f(g(x y)) ( ) = f(g(x) g(y)) (g ) = f(g(x)) f(g(y)) (f ) = f g(x) f g(y) ( ) f, g A f g A A I L A A L L 0 = {0} L a = {0, a} L b = {0, b} L c = {0, c} L 1 = {0, a, b, c, 1} L 1 L a L b L c L : ϕ : L I 0 (L) x L ϕ(x) = {x x L x a} ϕ L I 0 (L) ϕ(l) = I(L) x L ϕ ϕ(x) I 0 (L) = I(L) { } x ϕ(x) x ϕ(x) I(L) x ϕ(l) I(L) L I I(L) I = x a = I x I a I I = ϕ(a) ϕ(l) 172

173 x I x a x ϕ(a) I ϕ(a) x ϕ(a) ϕ(a) x a x L a I I x I ϕ(a) I ϕ(a) = I ϕ(l) = I(L) ϕ L I(L) , 1, 1 1, 1 1, 1, 0 1, 0, 1 a, 1, 1 1, 0 a, 1 1, 0, 0 a, 1, 0 a, 0, 1 0, 1, 1 a, 0 0, 1 a, 0, 0 0, 1, 0 0, 0, 1 0, 0, 0 0, 0 L 1 L 2 L 1 L 2 L : a, b B a (ā b) = (a ā) (a b) ( ) = 1 (a b) ( ) = a b (a b ) a (ā b) = (a ā) (a b) ( ) = 0 (a b) ( ) = a b (a b ) B,,,, 0, 1 a, b B (a b) (ā b) = (ā b) (a b). : (a b) (ā b) = a b ā b ( 19.23(2)) = (ā b) (ā b) ( 19.23(2)) = (ā b) (a b) ( 19.23(1)) = (ā (a b)) (b (a b)) ( ) = (ā a) (ā b) (b a) (b b) ( ) = 0 (ā b) (b a) 0 ( ) = (ā b) (b a) (0 19.2) = (ā b) (a b) ( ) 173

174 : B B a, b, c B (a b) c = (((a b) (ā b)) c) ((a b) (ā b) c) ( ) = (((a b) (ā b)) c) (((ā b) (a b)) c) ( 19.2) = (a b c) (ā b c) (ā b c) (a b c) ( ) = (a b c) (a b c) (ā b c) (ā b c) ( ) = (a (( b c) (b c))) (ā ((b c) ( b c))) ( ) = (a ((b c) ( b c))) (ā ((b c) ( b c))) ( 19.2) = a (b c) ( ) a B 0 a = a 0 ( ) = (a 0) (ā 0) ( ) = (a 1) (ā 0) ( 0 = 1) = a 0 (0 ā a ) = a (0 a 19.2) 0 a B a a = (a ā) (ā a) = 0 0 = 0 B B, Abel : B, Abel B B, B,, 19.3(3) a B a a = a B,, : ϕ : B P(A) x B ϕ(x) = {a a B, a, a x} ϕ B P(A) ϕ(0) = a i A a i ϕ(a i ) x 0 ϕ(x) a A a i ϕ(x) a ϕ(x) ϕ(a) = ϕ(x a i ) ϕ(x a) = ϕ(0) = : n n = 1 n = k n = k + 1 a 1 a 2 a k a k+1 = a 1 a 2 a k ā k+1 ( 19.23(2)) 174

175 = ā 1 ā 2 ā k ā k+1 ( ) a 1 a 2 a n = ā 1 ā 2 ā n (1) (a b) (a b) (ā b) = (a (b b)) (ā b) ( ) = (a 1) (ā b) (b b = 1) = a ā b (a ) = 1 b (a ā = 1) = 1 (b ) (2) (a b) (a b c) c = (a (b b c)) c ( ) = (a (b ( b c))) c ( 19.23(2)) = (a (1 c)) c ( b b = 1) = (a 1) c ( c ) = a c (a ) : a, b B 1 ϕ(a b) = ϕ(a) ϕ(b) ϕ(a b) = ϕ(a b) ( 19.23(1)) = ϕ(a b) ( ) = ϕ(ā b) ( 19.23(2)) = ϕ(ā) ϕ( b) ( ) = ϕ(a) ϕ(b) ( ) = ϕ(a) ϕ(b) ( 19.23(2)) = ϕ(a) ϕ(b) ( 19.23(1)) ϕ : 19.9 [a, b] B,, [a, b] B [a, b] a [a, b] b [a, b] x [a, b] y = ( x a) b a = a b (a b 19.2) ( x b) (a b) ( 19.1(2)) = ( x a) b ( ) = y ( ) = ( x a) b ( ) b ( 19.1(1)) 175

176 y [a, b] x y = x (( x a) b) (y = ( x a) b) = (x ( x a)) b ( ) = (x a) b ( (2) ) = a b (a x 19.2) = a (a b 19.2) x y = x (( x a) b) (y = ( x a) b) = (x ( x a)) (x b) ( ) = ((x x) a) (x b) ( ) = (1 a) (x b) (x x = 1) = 1 (x b) (a ) = x b (x b ) = b (x b 19.2) y [a, b] x [a, b] [a, b] a 0 0 / [a, b] b 1 1 / [a, b] a = 0 b = 1 ( [a, b] = B B ) [a, b] B,,,, 0, 1 B,,,, 0, (1) : y B 2 ϕ x B 1 y = ϕ(x) 0 y ϕ(a) x a ( 0 x 19.24(1) ϕ(0) = 0 ϕ(x) 0 x x 0 x) a x = a y = ϕ(a) y ϕ(a) (2) : (1) B 1 B 2 B 1 = B2 B 1 B 2 n A P(A) n a A {a} P(A) a A B P(A) 0 B {a} B 0 = x B x B {a} x {a} {a} {x x = a} x {a} x = a B = {x} = {a} a A {a} P(A) n P(A) C / {{x} x A} C 0 = x A x C 0 {x} C {x} = C ( C = {x} {{x} x A} ) C P(A) n : (1) 19.24(1) ϕ(0) = 0 0 J (2) 0 x a = ϕ(0) ϕ(x) ϕ(a) = 0 ϕ(x) = 0 176

177 x J (3) ϕ ϕ(a b) = ϕ(a) ϕ(b) = 0 0 = 0 a b J (1) ϕ : B 1 B (1) ϕ(0) = 0 ϕ(1) = 1 ϕ(b 1 ) = B 2 ϕ(a) = ϕ(b) = 0 ( ϕ(a) = ϕ(b) = 1) ϕ(a) ϕ(b) = 0 1 = ϕ(a b) ( ϕ(a) ϕ(b) = 1 0 = ϕ(a b)) ϕ ϕ(a) ϕ(b) ϕ B 1 B 2 ϕ 1 = { 0, 0, a, 0, b, 1, 1, 1 } ϕ 2 = { 0, 0, a, 1, b, 0, 1, 1 } (2) ϕ 1 B 1 / = {{0, a}, {b, 1}},,,, {0, a}, {b, 1} {0, a} {b, 1} {0, a} {0, a} {0, a} {b, 1} {0, a} {b, 1} {0, a} {b, 1} {0, a} {0, a} {b, 1} {b, 1} {b, 1} {b, 1} x x {0, a} {b, 1} {b, 1} {0, a} ϕ 2 a b A, B X 1, X 2 A Y 1, Y 2 B X 1 Y 1 X 2 Y 2 X 1 X 2 Y 1 Y 2. : X 1 Y 1 X 2 Y 2 x X 1 x X 1 Y 1 X 2 Y 2 x A A B = x / Y 2 B x X 2 X 1 X 2 Y 1 Y 2 : P(A B),,,,, A B P(A) P(B),,,,,, A, B ϕ : P(A) P(B) P(A B) X, Y P(A) P(B) ϕ( X, Y ) = X Y ϕ 19.3 ϕ ϕ P(A) P(B),, P(A B),, X 1, Y 1, X 2, Y 2 P(A B) ϕ( X 1, Y 1 X 2, Y 2 ) = ϕ( X 1, Y 1 ) ϕ( X 2, Y 2 ) ϕ( X 1, Y 1 X 2, Y 2 ) = ϕ( X 1, Y 1 ) ϕ( X 2, Y 2 ) X, Y P(A) P(B) ϕ( X, Y ) =ϕ( A X, B Y ) ( ) =(A X) (B Y ) (ϕ ) =(A X) (B Y ) =((A X) B) ((A X) Y ) ( ) =(A B) ( X B) (A Y ) ( X Y ) ( ) =(A B) X Y ( X Y ) (B X A Y 1.21 ) =(A B) X Y ( Y X Y 1.21 ) 177

178 =(A B) (X Y ) ( ) =(A B) (X Y ) = (X Y ) (E = A B) = ϕ( X, Y ) (ϕ ) P(A B),,,,, A B ϕ = P(A) P(B),,,,,, A, B : f : B 1 / B 2 [x] B 1 / f([x]) = ϕ(x) [x], [y] B 1 / f([x]) = f([y]) ϕ(x) = ϕ(y) (f ) x y ( ) [x] = [y] ( 2.27) f y B 2 ϕ x B 1 f([x]) = ϕ(x) = y f ϕ B 1 k i i x 1, x 2,, x k B 1 i (f([x 1 ]), f([x 2 ]),, f([x ki ])) = i (ϕ(x 1 ), ϕ(x 2 ),, ϕ(x ki )) (f ) = ϕ( i (x 1, x 2,, x ki )) (ϕ ) = f([ i (x 1, x 2,, x ki )]) (f ) i B 1 f x B 1 f g(x) = f(g(x)) = f([x]) = ϕ(x) f f g = ϕ f f g = ϕ x B 1 f g(x) = ϕ(x) ( ) f ([x]) = ϕ(x) (g ) f ([x]) = f([x]) (f[x] = ϕ(x)) f = f f {0, 1} 3,,,, 000, 111 {0, 1} {0, 1} ( ) {0, 1} 3 B i 000, 111 B i {0, 1} 3 1 {0, 1} 3 8 {0, 1} 3 2 {000, 111} {0, 1} 3 4 B x B x 000 x 111 x B (1) x x x 000 x 111 ( x = x 000 ) B = {000, x, x, 111} ( ) x {0, 1} 3 {0, x, x, 1} {0, 1} 3 {0, 1} 3 178

179 B 1 = {000, 111} B 2 = {000, 001, 110, 111} B 3 = {000, 010, 101, 111} B 4 = {000, 011, 100, 111} B 5 = {0, 1} : X = {0, x, x, 1} a, b X a = 0 ( b = 0) a b = 0 X a b = b X ( a b = a X) a = 1 ( b = 1) a b = b X ( a b = a X) a b = 1 X a, b {x, x} a = b a b = a b = a = b X a b a = b a b = 0 X a b = 1 X a, b X a b X a b X 0 1 x x X 0, 1 X X 0 1 X B X B 179

180 180

181 (1) 3 2m = 3n 2n 3 = m n = 6, m = 9 (2) G 1 G 2 G 3 G 1 G 2, G 3 G 2 K 3,3 G 3 G 2 G 3 2. (1) T 3 3 C b = bade C c = cde C g = gef (2) {C b, C c, C g } (3) C C C = 0 1 G C b = bade C c = cde C g = gef C b C c = bac C b C g = badgf C c C g = cdgf C b C c C g = bacegf G {, C b, C c, C g, C b C c, C b C g, C c C g, C b C c C g } 181

182 (1) (A C) B = A B A C B ( ) (2) 1 A A = A A x A y A x y : A = A A A 4.10 A A = A x A x A A y A x y A A A x A y A x y x A x A A A = A 1 A = A = A A = A : A = A = A A A = A A A A x 0 A x 1 A x 0 x 1 x 2 A x 1 x 2 x 0, x 1, x i x i+1 x i A(i = 0, 1, ) 2 x i A A 2 A A ( 6.8) : A A A A A A x A x + A A x + A = x + A x + = x {x} A = x A x x A A = x + A x A x + A x + x x + x A x x + A x A A A A = A 3 B A = B {B, {B}, {{B}}, } : A = A A = A 1 2 S x S x S = x 0, x 1,, x n x 0 x 1 x 2 x n x 0 ( )

183 (3) (4) card A card B (5) (6) G 4 H 1 = 0 = {0} H 2 = 4 = {0, 4} H 3 = 2 = {0, 2, 4, 6} H 4 = 1 = G H 1 = {0} (7) a I, a 0 = 0 2. (1) f(r 0 (t)) = f(r) = {x 2 x R} = {x x R x 0} = R R (2) f 1 (R 4 (t)) = R 2 (t) (3) f 1 ({(t 2 + 2t + 1)}) = {(t + 1)} (4) f 1 (f({(t 1), (t 2 1)})) = f 1 ({(t 2 2t + 1), (t 4 2t 2 + 1)}) = {(1 t), (t 1), (1 t 2 ), (t 2 1)} 1. (1) (2) G G ( ) G 2 G 1 G 1 G (3) G 1 = G2 G G a a d d b c e b c e G 1 G 2 (4) (5) K 3,3 8 m = 3n 6 = : T m = n 1 T 1 T k 1 G n k

184 (G) 2m = d(v) (G) + (k 1) + 2(n k) = 2n 1 > 2n 2 v G v T v N(v) = {v 1, v 2,, v k } Γ i vv i v Γ i v i ( d(v i ) 2 v i Γ i T ) 1 i, j k i j v i v j ( ) T k : ϕ : G H/H 1 x G ϕ(x) = H 1 σ(x) ϕ σ H 1 y H/H 1 x G σ(x) = y ϕ(x) = H 1 y ϕ a, b G ϕ(ab) = H 1 σ(ab) (ϕ ) = H 1 σ(a)σ(b) (σ ) = H 1 H 1 σ(a)σ(b) (H 1 H 1 = H 1 ) = H 1 σ(a)h 1 σ(b) (H 1 σ(a) = σ(a)h 1 ) = ϕ(a)ϕ(b) (ϕ ) ϕ ϕ(g) = H x G x ker ϕ ϕ(x) = H 1 (ker ) H 1 σ(x) = H 1 (ϕ(x) = H 1 σ(x)) σ(x) H 1 ( 17.22) x G 1 (G 1 = σ 1 (H 1 )) G 1 = ker ϕ G/G 1 = H/H1 184

185 (1) A A = card A = 2 (2) A (3) A A (4) A 4 16 (5) A 1 A : f x G, x = (x 1 ) 1 x = f(x 1 ) ran f f G x, y G f(xy) = (xy) 1 (f ) = y 1 x 1 ( 17.2) = x 1 y 1 (G ) = f(x)f(y) (f ) f f x, y G xy = ((xy) 1 ) 1 ( 17.2) = f((xy) 1 ) (f ) = f(y 1 x 1 ) ( 17.2) = f(y 1 )f(x 1 ) (f ) = (y 1 ) 1 (x 1 ) 1 (f ) = yx ( 17.2) G 13. (1) a, c, d (2) c, d (3) c ( ) (4) c 5 185

186 (5) d G m n + 1 = = 1 n 1 = G G G : G G ( 11.15) G 1 ( ) G G G ( 8.1) 186

187 (1) K 5 K 6 2 (2) 6 ( 2, 4, ) (3) 3 ( K 5 1 K 3,3 2 ) 2. : G T T v 1, v 2 V (G) v 1 v 2 v 1, v 2 T {v 1, v 2 } G {v 1, v 2 } 3. : G = V 1 V 2, E V 1 = {u i i = 1,, 7} 7 V 2 = {v i i = 1,, 7} 7 (u i, v j ) E(G) v j u i G = V 1, V 2, E V 1 3 V G G 1. (1) ( B A) (2) (P(A) A = {{{ }}, {, { }}}) (3) ( R = I A ) (4) ( 2x + y, 6 = 5, x + y 2x + y = 5 x + y = 6) (5) (M n (R) ) (6) (A ) (7) ( 19.21) 2. (1) 0 N 0 / ran f f f( 0, 1 ) = f( 1, 0 ) = 2 f f f(n {1}) = {x x N} = {x + 2 x N} = N {0, 1} (2) x, y N x, x + 1 = y, y + 1 x = y f 187

188 0, 2 N N 0, 2 / ran f f f {0, 1, 2} = { 0, 0, 1, 1, 1, 2, 2, 2, 3 } 3. : κ = card B, µ = card(a B) B (A B) = κ + µ = card(b (A B)) = card A = λ κ ℵ λ = κ + µ = max{κ, µ} ( 5.24 κ µ κ κ µ > κ µ ) κ < λ κ max{κ, µ} = λ card(a B) = µ = max{κ, µ} = λ 4. x = 3 : y y 1 = y yxy 1 = x 2 yx = x 2 y ( y) x = y 1 x 2 y ( y 1 ) = x 2 = (y 1 x 2 y)(y 1 x 2 y) x 2 = y 1 x 4 y (yy 1 = e) x 2 = yx 4 y 1 (y = y 1 ) yx 4 y 1 = x 2 = yxy 1 x 3 = e x 3 x x = 1 x = 3 188

189 G G v 3 R 1 R 2 R 3 R 4 R 5 v 1 v 2 v 5 G G v 4 G v 1, v 3, v 4, v 5 G v 1 v 2 v 1 C v 1 v 2 C > 2 C G ( V 1 = {v 2} p(g V 1 ) = 2 > V G ) 2. B i = i A k A = A 2 = k= A 3 = A 4 =

190 B 2 = B 3 = B 4 = (1) v 2 v 5 b (4) 25 (2) 3 = 5 4 = i,j 5 a (3) ij (3) B 3 0 D 3. (1) i=1 a (3) ii = 12 : G G n 2 G[V i ] v G[V i ] G d(v) n 2 V i n (2) : G G V 1 V (G) V 1 = k 1 δ(g V 1 ) δ(g) (k 1) = n k + 1 G V 1 = n k (1) G V 1 κ(g) k G k ( A = B, C A B = A C = ) 3. ( E A E 2 A = E A A 2 E A I A ) 4. (f 1 f 1 B A f 1 : B A f 1 : B A ) 5. ( ) ( ) 8. ( ) 1. (1) f( 1, 0 ) = f( 0, 1 ) = 1 (2) 3 N 3 / ran f (3) f 1 (0) = { 0, 0 } ( f 1 (0) dom f f 1 (0) = { 0, 0 } = 0, 0 ) (4) f { 0, 0, 1, 2 } = { 0, 0, 0, 1, 2, 5 } 2. (1) : f B A, x A f(x) f(x) frf R f, g, h B A frg grh x A f(x) g(x) g(x) h(x) f(x) h(x) frh R 190

191 f, g B A frg grf x A f(x) g(x) g(x) f(x) f(x) = g(x) x f = g R R (2) B A, R B, B, m f : A B, x A, f(x) = m B A, R : B, m f : A B x A f(x) = m g B A x A g(x) f(x) = m grf B A, R f B, a A, f B A b B b f(a) ( f(a) B ) g : A B, x, g(x) = b g(a) f(a) g Rf f f B A, R : G Lagrange G G G G a G ( G G = {e} G G = 1 ) a G ( G Abel G ) a a = a > 1 a = {e} a = G G = {a k k Z} a G k G, 1 < k < G 1 < a k = a k = G k < G ak G 191

192 (1) κ = 2 (2) λ = 3 (3) χ = 4 (4) ( 2 G 2 G tight ) ( ( p q r) ( p q r) ( p q r)) 5. 1, 0 6. (1) Pn m n! = (n m)! (2) n! { { m n} m n} Stirling 2. : R 1 = R B B B B R 1 B R 1 b B b, b R b, b B B b, b R 1 R 1 a, b, b, c R 1 a, b, b, c R R a, c R a, b, b, c R 1 B B a, b, c B a, c B B a, c R 1 R 1 a, b A a, b R 1 R b, a R 1 R R a = b R 1 3. R 1 B : a G a 1 a a, b G a = a 1, b = b 1 aabb = aa 1 bb 1 (a 1 = a, b 1 = b) 192

193 = e (aa 1 = bb 1 = e) = (ab)(ab) 1 ((ab)(ab) 1 = e) = abab ((ab) 1 = ab) ab = ba a, b G a G a 1 a c = a, d = a 1 c d cd = dc = e 4. : H = G 1 G 2 v i V (H) v i G 1 G 2 a i = N G1 (v i ) b i = N G2 (v i ) d H (v i ) = N H (v i ) = N G1 (v i ) N G2 (v i ) = a i + b i 2 N G1 (v i ) N G2 (v i ) G 1 G 2 a i, b i d H (v i ) V k H v i V k d H[Vk ](v i ) = d H (v i ) (v i, v j ) E(H) (v i, v j ) / E(H[V k ]) H[V k ] V k v j / V k v j v i v i V k v j V k v j V k v i V k d H[Vk ](v i ) = d H (v i ) H H[V k ] V k H[V k ] 193

194 N/R 1 = {{n} n N} N/R 2 = {{2k k N}, {2k + 1 k N}} N/R 3 = {{3k + j k N} j = 0, 1, 2} N/R 4 = {{6k + j k N} j = 0, 1, 2, 3, 4, 5} 2. N/R 2 N/R 3 N/R 4 N/R 1 3. f 1 (H) = H 1. f 2 (H) = {0} f 3 (H) = {0, 1, 2} f 4 (H) = {0, 2, 4} (1) ker ϕ 1 = G G 2 ϕ 1 (2) ker ϕ 2 = {0} 1 Z 1 / ϕ 2 (Z) ϕ 2 (3) ker ϕ 3 = {0} x R + ln x R x = ϕ 3 (ln x) ϕ 3 2. : e A, e B e = ee AB AB A b B Ab = ba BA = {ba b B} = {Ab b B} = AB x, y AB a 1, a 2 A, b 1, b 2 B x = a 1 b 1, y = a 2 b 2 xy 1 = a 1 b 1 (a 2 b 2 ) 1 = a 1 b 1 b 1 2 a 1 2 a 1b 1 b 1 2 AB = BA a 3 A, b 3 B a 1 b 1 b 1 2 = b 3 a 3 xy 1 = b 3 a 3 a 1 2 BA = AB AB G

195 (1) κ = 2 (2) ξ = 5 (3) α 0 = 4 (4) χ = 3 (5) β 1 = 3 2. : G 3 2m = n d(v i ) 4n m 2n G 3 G 4 m l (n 2) m 2n 4 l 2 i=1 195

196 m = d d = 2m = 34 d = : λ(g ) 2 G G G G = G G e G e e ( e G e e G e ) G = G G λ(g ) 2 3 vi G d G (vi ) = deg(r i) = 3 G 3-1. R = { 0, 4, 3, 3, 6, 2, 9, 1, 12, 0 } R 2 = { 3, 3, 12, 4 } 2. A B B {x x P (A) x = 1} A n = A B {x x P (A) x = n 1} {x x P (A) x = 1} B 3. G(1) = {1, 2}, G(2) = {3}, G(3) = G G ( {1} P(A) {1} / ran G) ran G = {, {1, 2}, {3}} 4. : I a, b I a b A a b a a b I I I 5. (1) : x, y G xax 1 = yay 1 ax 1 y = x 1 ya ( x 1 y) x 1 y N(a) (N(a) ) 196

197 N(a)x = N(a)y ( 17.22) (2) H G/N(a) H = [G : N(a)] : N(a) C C N(a) Lagrange N(a) = [N(a) : C] C H = [G : N(a)] = G N(a) G = [N(a) : C] C n = [N(a) : C]m n m ( (1) ) (Lagrange ) ( N(a) = [N(a) : C] C ) ( G = n, C = m) ([N(a) : C] ) 197

198 A =, A 2 =, A 3 = (1) G 1 i,j 4 a (3) ij = 35 3 a (3) ii = i (2) v 1 v 3 a (1) 13 + a(2) 13 + a(3) 13 = 6 3 (3) v 1 a (1) 11 + a(2) 11 + a(3) 11 = : G G λ 2 G G G 2 G G G n(n 2) ( G v i, v j deg(r i ) = d(v i ) = d(v j ) = deg(r j ) ) G n(n 2) G[V ] G ( G V = V (G)) V = 1 G n 2 v i, v j V (G) v i v j d(v i ) = d(v j ) = 0 V = k 2 G[V ] v i V 1 d(v i ) k 1 V k k 1 v i, v j V V (G) v i v j d(v i ) = d(v j ) 1. (A C) B = A B A C B : (A C) B = A B (A C) B = A B (A B) ( C B) = A B A B C B ( ) ( ) 198

199 x(x A B x C B) ( ) x((x A x B) ( x C x B)) ( ) x( (x A x B) ( x C x B)) x(( x A x B) ( x C x B)) ( ) x(( x A x C x B) ( x B x C x B)) ( ) x(( x A x C x B) 1) ( ) x( x A x C x B) ( ) x( (x A x C) x B) ( ) x((x A x C) x B) x((x A C) x B) x A C B ( ) A, B A B = A A B A B = A x, x A x A B (A B = A) x A x B = x B ( ) A B = A A B A B x, x A = x B ( ) 1 x B ( ) x A x B ( ) x A B A B A B = A A B = A A B 2. (1) f(n {1}) = {n 1 n N} = N (2) f 1 ({0}) = { m, n m, n N mn = 0} = { 0, n, n, 0 n N} (3) f ( f( 1, 4 ) = f( 2, 2 ) = 4 1, 4 = 2, 2 ) (4) f n N n, 1 N N, f( n, 1 ) = n 3. (1) Aut G = {ϕ i i = 1, 2, 3, 4} ϕ i : Z 5 Z 5 (i = 1, 2, 3, 4) x Z 5, ϕ i (x) = ix mod 5 199

200 ϕ 1 ϕ 2 ϕ 3 ϕ 4 ϕ 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 ϕ 2 ϕ 2 ϕ 4 ϕ 1 ϕ 3 ϕ 3 ϕ 3 ϕ 1 ϕ 4 ϕ 2 ϕ 4 ϕ 4 ϕ 3 ϕ 2 ϕ 1 (2) Aut G 3 H 1 = {ϕ 1 } H 2 = {ϕ 1, ϕ 4 } H 3 = Aut G H 3 H 2 H 1 (3) S < 5 S, R H 2 ( S, R S, R 2 k (k N) S = 3 S, R ) 4. : G a G a = {e, a} G Lagrange 2 = a G G A = {{x, y} x, y G xy = e} A G ( A = G (x 1 ) 1 = x A, B A, A B A = B) G A A A {e} A = 2 x = e A G G Abel x = x = e = e. x G A A x A A A x A 200

201 % a % 55% % 25% b % c d e 0 1 9% f 0 1 g h a 00 b 01 c 100 d 101 e 110 f 1110 g h = n n 1 4. : δ + (D) 1 D D Γ Γ = v 1 v 2 v k Γ v k Γ s = min{i v k, v i E(D)} C = v k v s v s+1 v k v k Γ v s v k v k v i s < i < k C G δ + (D) v k v k C C δ + (D)

202 5. ( Appel Haken 1976 ) 1. (1) P(A) { } = {, { }, {{ }}, {, { }}} { } = {{ }, {{ }}, {, { }}} (2) P(A) A = {, { }, {{ }}, {, { }}} {, { }} = {{{ }}, {, { }}} 2. (1) R 1 = I A R 2 = I A { 1, 2, 2, 1 } R 3 = I A { 1, 3, 3, 1 } R 4 = I A { 2, 3, 3, 2 } R 5 = E A (2) R 1 R 2 R 3 R 4 R 5 R 1 R 1 R 1 R 1 R 1 R 1 R 2 R 1 R 2 R 1 R 1 R 2 R 3 R 1 R 1 R 3 R 1 R 3 R 4 R 1 R 1 R 1 R 4 R 4 R 5 R 1 R 2 R 3 R 4 R 5 (3) R 5 R 1 R 5 (4) V V V 3. 1, x = 0; 2, x = 2; (1) f g(x) = 0, x = 4; k, x = 2k, k N, k 3; 3, x = 2k + 1, k N. f g(4) = 0, f g(0) = 1, f g(2) = 2, f g(2k) = k(k 3) f g f g(6) = f g(3) = 3 f g (2) f g(a) = {1, 2, 3} f g 1 (B) = {0, 4, 8} 4. : A P(A) x(x A x P (A)) ( ) 202

203 x(x A x A) x(x A y(y x y A)) x y(x A (y x y A)) x y( x A ( y x y A)) x y( (x A y x) y A) y( x (x A y x) y A) y( x(x A y x) y A) y( x(x A y x) y A) y(y A y A) A A ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5. : g Ag (e A) g Bh (Ag = Bh) Bg = Bh ( 17.22) Bg = Ag (Ag = Bh) = b(b B a(a A bg = ag)) ( ) = b(b B a(a A b = a)) ( ) b(b B b A) (b = a) B A ( ) A B A = B 203

204 (1) 3! {4 3} = 3! C 2 4 = 36 (2) card(a B) = card( ) = 0 (3) 2n 1 ( tight ) (4) min{r, s} (5) 4 2. (1) (2) Π = {{1}, {2, 3}, {6, 9}, {18, 27}, {54}} (3) B {2, 9} : Γ = v 0, v 1,..., v l Γ v 0 Γ δ(g) 3 v 1 v i, v j (2 i < j l) v 0 v 0, v 1,..., v i, v 0 i + 1 v 0, v 1,..., v j, v 0 j + 1 v 0, v i,..., v j, v 0 j i + 2 i + 1 j + 1 i + 1 j + 1 (j + 1) (i + 1) = j i j i

205 (1) (b c) c = d c = d b = b a = b (c c) (2) a d (3) x 1 = x 2 = y 1 = b, y 2 = c x 1 Ry 1, x 2 Ry 2 x 1 y 1 = b b = a b c = x 2 y 2 = d a, d / R R A, 2. : A y 1 xy A x 1 A x 1 y 1 xy A B x 1 y 1 x B y B x 1 y 1 xy B A B = {e} x 1 y 1 xy = e x y xy = yx 3. (1) A&A A&A = {{x, y} x, y A} A C = 6 (C ) 2 3 = 8 (2) A A 3 x, x 2 3 A A 3 x, y, y, x x, y A, x y ( 3 ) = 6 3 (3) A =

206 (1) A = { } {{ }} = (2) A = { } {{ }} = {, { }} (3) A = {, { }} = { } = { } (4) P(A) = A = {{ }, {{ }}} (5) P( A) = P({, { }}) = {, { }, {{ }}, {, { }}} 2. d(v) = 2m = n = (1) a v V (G) b c d (2) C = D = {d} d 4. a b g f c h i e d (1) κ = 0 (2) χ = 2 V 1 = {a, c, e, h}, V 2 = {b, d, f, g, i} V 1, V 2, E (3) β 0 = 5 V 2 = {b, d, f, g, i} β 0 5 {a, b, c, d, e, f} {h, i} g 5 β 0 = 5 n (4) β 1 = 4 4 β 1 4 = 2 (5) α 0 = 4 V 1 = {a, c, e, h} 4 α 0 β 1 = 4 206

207 5. : G 3 G 4 ( G G G n 3 2m 2(n 1) 2(4 1) = 6) m δ(g) 4 m 2n > 2n 4 l (n 2) 2(n 2) = 2n 4, l 2 (1) A, x, y, z A x y = x + y xy = y + x yx = y x (x y) z = x + y + z xy xz yz + xyz = x (y z) x x = x + x x 2 x 0, 1 x x x x 0 = 0 x = x 0 Z x y = x + y xy = 0 x = 1 x 1 y = x x 1 x = 0, 2 y = x 1 Z y 0 2 0, 2 (2) A A, x A x x = x x = x x x x (3) B 2 A B 2 a, b B, a b {a}, {b} A {a} {b} = / A B 2 A, 2. x = 3 : y y 1 = y yxy 1 = x 2 yx = x 2 y ( y) x = y 1 x 2 y ( y 1 ) = x 2 = (y 1 x 2 y)(y 1 x 2 y) x 2 = y 1 x 4 y (yy 1 = e) x 2 = yx 4 y 1 (y = y 1 ) yx 4 y 1 = x 2 = yxy 1 x 3 = e x 3 x x = 1 x = 3 207

208 (1) A/R 1 = {{a, b, e}, {c}, {d}} (2) {c} {d} {a, b, e} 2. (1) G (2) G G ( ) Γ a 2 Γ g a Γ g b, c Γ g Γ (3) G G a d d a b e g f c e b g c f 3. (1) : n 2 4m n 4mn 2n = 2m (2) : V 4m n v 1 V V (G) V 4m n V v i v i V G V n/2 = k k 1 + 4m/n N g (V ) = {v v V (G) u(u V (u, v) E(G))} V 208

209 4m n N g (V ) V N g (V ) + V ( ) 4m n k + k ( V = k) k(1 + 4m/n) n/2 k < 1 + 4m/n N g(v ) V < n 2 (1) V (G) V 4m n V V 1. : a, b G 2. aba 1 b 1 Naba 1 b 1 =Na Nb Na 1 Nb 1 =Na Na 1 Nb Nb 1 =N(aa 1 bb 1 ) ( ) (G/N Abel ) ( ) =Ne (aa 1 bb 1 = e) =N ( ) : 1 i, j 8 i S i 7 j S j 7 i j S i + S j a ij = a c = c a a c = c a = a c a b c a a c a b c b b c a b c ab = ba = cc = c A, 3 ( ) ( a, b 2 ) ( (a a) b = a b = c a (a b) = a c = a) S k = {3n + k n N 3n + k 20}(k = 1, 2, 3) S 1 = S 2 = 7, S 3 = 6 3 S 1, S 2, S 3 S i (1 i 3) C C

210 I II : A X, B Y R = A B x 1, x 2 X, y 1, y 2 Y x 1, y 1 R x 2, y 2 R = x 1 A y 2 B (R = A B) = x 1, y 2 R (R = A B) A = dom R X, B = ran R Y x A, y B x, y R R = A B x X, y Y x A y B w( x, w R) z( z, y R) w z( x, w R z, y R) (dom, ran ) w z( x, y R) ( I ) 2. = x, y R ( ) (1) N 0 B N B N N- N B N (2) N 2 B N ( ) B 0, B 1 B 2 N 2 B N B N+1 B N+1 ( B N ) B N+1 B N+1 (3) N 3 B N B 0, B 1, B 2, B 3 B N ( (4) ) 3 m l l 2 (n 2) B N m 2n 4 < 2n = 4 2 N 1 2m = N2 N m = N2 N 1 N2 N 1 = m < 4 2 N 1 N < 4 (4) N 1 B N 210

211 B N B N V V B N = V 1, V 2, E 3. : G T ( ) v v v G ( ) n + 2(n 3) D V (D) = {v i i = 0, 1, 2,, n, n + 1} E(D) = { v 0, v i, v i, v n+1 i = 1, 2,, n} { v n+1, v 0 } D v n+1 n D v n+1 n v n+1 v n+1, v 0 D v n+1, v 0 n 1. : Aut G ( I G ) Inn G Aut G = {I G } Inn G = {I G } Inn G g, h G ghg 1 = ϕ g (h) = I G (h) = h g gh = hg g, h G Abel 2. (1) A A A = 3 9 (2) A&A A A&A = {{x, y} x, y A} A C = 6 (C ) A A A&A = 3 6 (3) A A { a, a, b, b, c, c } A = 3 6 (4) E A A A A B A B = (A B) = E A B A A B = A + B A B A B A&A A 3 3 A = 3 3 (3 3 1) 211

212 (1) A 2 n2 n n ( ) (2) A 5 19 (3) {1, 2} = (1 2) = 2 = 1 {2, 3} = (2 3) = 2 = (4) card(z Z) = ℵ 0 card A = ℵ = 2 ℵ 0 2. (1) : x x (A B) S(S A B x S) S((S A x B) x S) S( (S A x B) x S) S(( S A x B) x S) S(( S A x S) ( x B x S)) S((S A x S) (x B x S)) S(S A x S) S(S B x S) x A x B x ( A) ( B) ( ) ( ) ( ) 3. (1) r s G (2) r = s 2 G (3) r 2 s 2 G (4) χ = χ = {0, 0, 3}, {0, 1, 2}, {1, 1, 1} 2 212

213 5. (1) : G 3 2m = d(v) v G 4 = 4n m 2n G 3 4 G v G 5 m 5 (n 2) ( 11.9) 5 2 = 2n n < 2n (2) 6 1. V/ = N/, N/ = {2N, {1}, {3}, {5}, } 2N = {2n n N} 2N, x = 2N y = 2N; x, y N/ x y = {mn}, x = {m}, y = {n}, m, n N. 2. : xh G e xh y H xy = e y = x 1 H x 1 H H x = (x 1 ) 1 H 213

214 A, B A (A B) = A A (A B) = A : x x A = x A x A B ( ) x A (A B) A A (A B) x x A (A B) x A x A B x A (x A x B) (x A x A) (x A x B) ( ) = x A x A ( ) = x A ( ) A (A B) A A (A B) = A = 16 A = {a, b} A I A E A H 1 = I A H 2 = I A { a, b } H 3 = I A { b, a } b a a b a b H 1 H 2 H 3 4 f 1 = { a, a, b, a } f 2 = { a, a, b, b } f 3 = { a, b, b, a } f 4 = { a, b, b, b } 214

215 3. : G V (G) {V 1, V 2 } u V 1, v V 2 (u, v) E(G) {V 1, V 2 } V 1, V 2 V 1 V 2 G N (0) G {N (k) G v V (G) N (k) G (v) v k (k) (k) (k) (v) = {v} k N v N (v) N (v) N (v) V (G) G G G (k) (k) (k) (v), V (G) N (v)} V (G) N (v) V (G) N G G V (G) N (k) N (k+1) G (v) N (0) (1) G (v) N G (v) N (k) (k+1) G (v) N G (v) G G ( N (k) G (v) V (G) N(k+1) G (v) N (k) (k) G (v) N G (v) k k n N (k) (v) = V (G)) v G v G G 4. : 5 G 5 K 5 K 5 G G 5. : V (G) G 27 V V 2 G ( 12 ) G = V 1, V 2, E (1) A, x, y, z A x y = x + y xy = y + x yx = y x (x y) z = x + y + z xy xz yz + xyz = x (y z) x x = x + x x 2 x 0, 1 x x x x 0 = 0 x = x 0 Z x y = x + y xy = 0 x = 1 x 1 y = x x 1 x = 0, 2 y = x 1 Z y 0 2 0, 2 (2) A, x, y, z A x y = x y = y x = y x x = 2, y = z = 1 (x y) z = x y z = 0 2 = x y z = x (y z) 215

216 x x = x x = 0 x 0 x x x x, y A x < 0 0 x y x A, (3) n k A = nz A, n = 1 A n 1 (4) A A, x A x x = x x = x x x x (5) R 1, R 2 R 1 R 2 ( x B x, x R 1, x, x R 2 x, x R 1 R 2 ) x, y, z B x, y, y, z R 1 R 1 R 1, R 2 x, z R 1, x, z R 2 x, z R 1 R 2 R 1 R 2 x, y B x, y R 1 R 2 R 1, R 2 y, x R 1, y, x R 2 y, x R 1 R 2 R 1 R 2 B R 1, R 2 A R 1 R 2 A A, B E B E B 7. Z 12, a n G H 1 = 0 = {0} H 2 = 6 = {0, 6} H 3 = 4 = {0, 4, 8} H 4 = 3 = {0, 3, 6, 9} H 5 = 2 = {0, 2, 4, 6, 8, 10} H 6 = 1 = G H 6 H 5 H 4 H 3 H 2 H 1 L(G) L(G) H 2 H 5 L(G) 8. : g G, h N f(ghg 1 ) = f(g)f(h)f(g) 1 (f ) = f(g)f(g) 1 f(h) (G 2 ) = f(h) (f(g)f(g) 1 = e 2 ) 216

217 f(ghg 1 ) = f(h) ghg 1 ker f = h ker f h ker f hn = N e 1 ker f ghg 1 = ghg 1 e 1 ghg 1 ker f N g G, h N, ghg 1 N N G 217

218 : X, Y Y = (X Y ) (X Y ) Y = Y ( ) = (X X) Y ( ) = (X Y ) ( X Y ) ( ) = (X Y ) (X Y ) ( ) = (X Y ) (X Y ) B = (A B) (A B) = (A C) (A C) = C 2. (1) X = P(X) X P(X) (2) A = 1 3. R 7 = R (2) R 2006 = R = R 7+7 = R t k : k k C 1, C 2,, C k k S k = {G G k } G S k f(g) G ( f S k ) t max = max f(s k ) k G S k f(g) = k ( t max k) G S k C 1, C 2,, C k C i, C j (i j) G k k k G E G = G E H = V, E E (1) H G ( G i e j E e j G i ) G i E G V (G i ) V (G) V (G i ) G H k H k V k (2) G H = G E = G G H G 1 A = A (2) 218

219 H v V (G) d G H (v) = d G (v) + d H (v) d G (v) d H (v) v V ( H v V v E d H (v) 1) d H (v) 2 2 E = v V d(v) 2 V 2k f(g) = E k t max k ( D D ) t max k G S k G p k v 1, v 2,, v p v 1, v 2,, v p G p v 1, v 2, v 2, v 3,, v p 1, v p, v p, v 1 ( G p ) G C k+1 = v 1 v 2 v p v 1 G = G C k+1 = C 1 C 2 C k C k+1 v 1, v 2,, v p G ( ) C k+1 G C k+1 C 1, C 2,, C k G k + 1 G G S k f(g) p k G t max k 5. t max = k (1) 2 : 1-2 K 3,3 χ(k 3,3 ) = 2 2 (2) 5 : G n e E(G) G H = G e H E(H) 3n 6 E(G) = E(H) + 1 3n 5 d(v i ) = 2 E(G) 6n 10 G 5 v v i V (G) G v Heawood G v 5- Heawood G K 5 χ(k 5 ) =

220 : x Z x = 2k(k Z) f( x) = f(2k + 1) = (2k + 1) mod 2 = 1 f(x) = (2k mod 2) = 0 = 1 mod 2 = 1 f( x) = f(x) x = 2k + 1(k Z) f( x) = f(2k + 2) = (2k + 2) mod 2 = 0 f(x) = (2k + 1 mod 1) = 1 = 2 mod 2 = 0 f( x) = f(x) x Z f( x) = f(x) f f(0) = 0, f(1) = 1 f f(2) = f(0) = 0 f 2. (1) : x, y L f a (x y) = (x y) a (f a ) = (x a) (y a) (L ) = f a (x) f a (y) (f a ) f a (x y) = (x y) a (f a ) = (x y) (a a) (a a = a) = (x a) (y a) ( ) = f a (x) f a (y) (f a ) f a g a (2) x L x f {1} (L) x {1} ( x {1} x = x {1} = f {1} (x) f {1} (L) x f {1} (L) x = f {1} (y) = y {1} {1}) f {1} (L) = {, {1}} f {1} {, {1}},, x L x g {1} (L) {1} x g {1} (L) = {{1}, {0, 1}, {1, 2}, {0, 1, 2}} g {1} {{1}, {0, 1}, {1, 2}, {0, 1, 2}},, 3. : G Lagrange G ( ) G G G a G ( G G = {e} G G = 1 ) a G ( G Abel G ) a a = a > 1 a = {e} a = G G = {a k k Z} a G G G a a 2 e, a 2 a a 2 G ( a / a 2 ) a 2 = {e} ( a 2 a 2 ) a 2 G G G G k G, 1 < k < G 1 < a k = a k = G k < G ak G 220

221 221

222 A A = n ( n ) P(A) = 2 n. 1.3 ( ) A 1, A 2,, A n n n n A i = i i=1 i=1 A A i A j + A i A j A k + ( 1) n 1 A 1 A 2 A n. i<j 1.4 {A k } i<j<k (1) lim A k lim A k; (2) lim A k = k k k 1.5 {A k } B n=1 k=n A k ; (3) lim A k = k n=1 k=n A k. (1) B lim A k = lim (B A k ); (2) B lim A k = lim (B A k). k k k k 1.6 {A k } E = A k B k = A k, k = 1, 2, {B k } k N + E = lim A k lim B k = lim A k lim B k. k k k k 222

223 2.1 a, b = c, d a = c b = d. 2.2 a 1, a 2,, a n = b 1, b 2,, b n a i = b i, i = 1, 2,, n. 2.3 F, G (1) dom(f G) = dom F dom G; (2) ran(f G) = ran F ran G; (3) dom(f G) dom F dom G; (4) ran(f G) ran F ran G; (5) dom F dom G dom(f G); (6) ran F ran G ran(f G). 2.4 F (1) dom F 1 = ran F ; (2) ran F 1 = dom F ; (3) (F 1 ) 1 dom F F. 2.5 R 1, R 2, R 3 (R 1 R 2 ) R 3 = R 1 (R 2 R 3 ). 2.6 R 1, R 2, R 3 (1) R 1 (R 2 R 3 ) = R 1 R 2 R 1 R 3 ; (2) (R 1 R 2 ) R 3 = R 1 R 3 R 2 R 3 ; (3) R 1 (R 2 R 3 ) = R 1 R 2 R 1 R 3 ; (4) (R 1 R 2 ) R 3 = R 1 R 3 R 2 R F, G (F G) 1 = G 1 F R, S, A, B, A A (1) R (A B) = (R A) (R B); (2) R A = {R A A A }; (3) R (A B) = (R A) (R B); (4) R A = {R A A A }; (5) (R S) A = R (S A). 2.9 R, S, A, B, A A (1) R[A B] = R[A] R[B]; (2) R[ A ] = {R[A] A A }; (3) R[A B] = R[A] R[B]; (4) R[ A ] = {R[A] A A }; (5) R[A] R[B] R[A B]; (6) (R S)[A] = R[S[A]] R A A (1) R ; (2) I A R; (3) R 1 ; (4) M(R) 1 223

224 (5) G(R) R A A (1) R ; (2) I A R = ; (3) R 1 n ; (4) M(R) 0 (5) G(R) R A A (1) R ; (2) R 1 = R; (3) M(R) ; (4) G(R) R A A (1) R ; (2) R R 1 I A ; (3) M(R) r ij = 1(i j) r ji = 0 (4) G(R) x i, x j (i j) x i, x j x j, x i R A A (1) R ; (2) R R R; (3) M(R R) r ij = 1 M(R) r ij = 1 (4) G(R) x i, x j, x k x i, x j, x j, x k x i, x k ( x i x k 2 x i x k 1 ) R 1, R 2 A A. (1) R 1, R 2 R1 1, R 1 2, R 1 R 2, R 1 R 2, R 1 R 2, R 2 R 1 (2) R 1, R 2 R1 1, R 1 2, R 1 R 2, R 1 R 2, R 1 R 2, R 2 R 1 (3) R 1, R 2 R1 1, R 1 2, R 1 R 2, R 1 R 2, R 1 R 2, R 2 R 1, R 1 (= E A R 1 ), R 2 (4) R 1, R 2 R1 1, R 1 2, R 1 R 2, R 1 R 2, R 2 R 1 (5) R 1, R 2 R1 1, R 1 2, R 1 R A n R A A s, t 0 s < t 2 n2 R s = R t R A A m, n (1) R m R n = R m+n ; (2) (R m ) n = R mn R A A s, t(s < t) R s = R t (1) R s+k = R s+t, k N; (2) R s+kp+i = R s+i k, i N, p = t s; (3) S = {R 0, R 1,, R t 1 } q N R q S R A A A (1) R r(r) = R (2) R s(r) = R (3) R t(r) = R A R 1, R 2 A A R 1 R 2 (1) r(r 1 ) r(r 2 ); (2) s(r 1 ) s(r 2 ); (3) t(r 1 ) t(r 2 ). 224

225 2.21 A R 1, R 2 A A (1) r(r 1 R 2 ) = r(r 1 ) r(r 2 ); (2) s(r 1 R 2 ) = s(r 1 ) s(r 2 ); (3) t(r 1 ) t(r 2 ) t(r 1 R 2 ) R A A A 2.23 R A A A r(r) = R I A. s(r) = R R R A A A t(r) = R R 2. A R A A l t(r) = R R 2 R l R A A A (1) R s(r) t(r) (2) R r(r) t(r) (3) R r(r) R A A A (1) rs(r) = sr(r); (2) rt(r) = tr(r); (3) st(r) ts(r) R A x, y A (1) [x] R [x] R A; (2) x, y R [x] R = [y] R ; (3) x, y / R [x] R [y] R = ; (4) {[x] R x A} = A A. (1) R A A R A/R A (2) A A R A = { x, y x, y A x, y A } R A A A A. (1) (2) I A A (3) I A A A x, y A (1) x y, x = y, y x (2) (x y x = y) (y x x = y) x = y A, A n (1) A (2) A n. 225

226 3.1 f : C D f C C C 1, C 2 C (1) f( C ) = {f(a) A C }; (2) f( C ) = {f(a) A C }; (3) f(c 1 C 2 ) = f(c 1 ) f(c 2 ). 3.2 f : C D D 1, D 2 D D D (1) f 1 ( D) = {f 1 (D) D D}; (2) f 1 ( D) = {f 1 (D) D D}; (3) f 1 (D 1 D 2 ) = f 1 (D 1 ) f 1 (D 2 ). 3.3 g : A B f : B C f g : A C x A f g(x) = f(g(x)). 3.4 g : A B f : B C. (1) f g f g (2) f g f g (3) f g f g. 3.5 g : A B f : B C. (1) f g f (2) f g g (3) f g g f. 3.6 f : A B I A, I B A B f I A = I B f. 3.7 f : R R g : R R f g f g. 3.8 A A 1 A. R R R f : A B f f 1 : B A f : A B A. (1) f f (2) f f (3) f f (4) f f. 226

227 4.1 N. 4.2 N σ : N N σ(n) = n + ( σ ) N, σ, Peano m, n m + n + m n m, n m n, m = n, n m. 4.8 (N ) A a A F : A A h : N A h(0) = a n N h(n + ) = F (h(n)). 4.9 M, F, e Peano N, σ, 0 M, F, e A (1) A ; (2) A A; (3) y A y A; (4) A P(A) A A P(A) A (A + ) = A N m, n N 4.16 m, n N m + 0 = m, ( 1) m + n + = (m + n) +. ( 2) m 0 = 0, ( 1) m n + = m n + m. ( 2) 4.17 m, n m 0 = 1, ( 1) m n+ = m n m. ( 2) 227

228 4.18 m, n, k N (1) m + (n + k) = (m + n) + k; (2) m + n = n + m; (3) m (n + k) = m n + m k; (4) m (n k) = (m n) k; (5) m n = n m N ( N ) N N (< N ) N m, n, k N (1) m n (m + k) (n + k) (m < n m + k < n + k); (2) m n m k n k (m < n m k < n k), k n, m, k (1) m + k = n + k m = n; (2) k 0 m k = n k m = n (N ) A N m A n A m n ( m A ) (N ) A N n N n A n A A = N. 228

229 ( ) 5.1 (1) Z N; (2) N N N; (3) N Q; (4) (0, 1) R; (5) [0, 1] (0, 1). 5.2 A P(A) (A 2) (A 2) 2 A A 2 = {0, 1}. 5.3 A, B, C (1) A A; (2) A B B A; (3) A B B C A C. 5.4 ( ) (1) N R; (2) A A P(A) (1). (2) N A, B A B C B A C. A, B. (1) A B A B; (2) A B A B B A. 5.8 A, B, C. (1) A A; (2) A B B C A C. 5.9 A, B, C, D 4 A B C D (1) B D = A C B D; (2) A C B D A, B, C, D 4 card A = card C = κ, card B = card D = λ A B C D A card A < card P(A). 229

230 5.12 (Schröder-Bernstein ) (1) A, B A B B A A B; (2) κ, λ κ λ λ κ κ = λ R (N 2) N 2 = 2 N (1) A N A; (2) κ ℵ 0 κ. 1 κ κ < ℵ 0 κ A N card A = ℵ A A {a 1, a 2,, a n, } A P(A) K 1, K 2, L 1, L 2 4 K 1 K 2, L 1 L 2 (1) K 1 L 1 = K 2 L 2 = K 1 L 1 K 2 L 2 ; (2) K 1 L 1 K 2 L 2 ; (3) L 1 K 1 L 2 K (1) A 2 card A = card P(A); (2) κ κ < 2 κ. (1) card P(N) = 2 ℵ0 ; (2) card P(R) = 2 ℵ ; (3) ℵ = 2 ℵ κ, λ, µ (1) κ + λ = λ + κ, κ λ = λ κ; (2) κ + (λ + µ) = (κ + λ) + µ, κ (λ µ) = (κ λ) µ (3) κ (λ + µ) = κ λ + κ µ; (4) κ λ+µ = κ λ κ µ ; (5) (κ λ) µ = κ µ λ µ ; (6) (κ λ ) µ = κ λ µ. κ, λ (1) κ + (λ + 1) = (κ + λ) + 1; (2) κ (λ + 1) = κ λ + κ; (3) κ λ+1 = κ λ κ 5.22 κ, λ, µ κ λ (1) κ + µ λ + µ; (2) κ µ λ µ; (3) κ µ λ µ ; (4) µ κ µ λ κ, µ κ κ κ = κ κ, λ 0 κ + λ = κ λ = max{κ, λ}. κ κ + κ = κ κ = κ κ κ κ = 2 κ. 230

231 6.1 A, 1 A f : N A n N f(n + ) f(n). 6.2 A, 1, B, 2, C, 3 (1) A, 1 = A, 1 ; (2) A, 1 = B, 2 B, 2 = A, 1 ; (3) A, 1 = B, 2 B, 2 = C, 3 A, 1 = C, f : A B B B A A x, y A x A y f(x) B f(y) (1) A A ; (2) B B ( ) A A ; (3) B B A A. 6.4 A, B B A. (1) A A A B B ; (2) A A A B B ; (3) A A A B B. 6.5 ( ) A B A B = A. 6.6 A A A A. γ(x, y) A f!yγ(f, y) A F t A, γ(f seg t, F (t)). 6.7 A, A, B, B (1) A, A = B, B ; (2) A, A = seg b, 0 B, b B; 1 A, A, ( = ) nontrivial A = N, = { 0, x x N + } N, dom( ) = {0} ran( ) = N + dom( ) ran( ) = x, y, z N y x ( y dom( ) ) z y ( y / ran( ) ) f : N N ( f(1) f(1) f(0) f(2) f(1) ) N, N N + N, ( b 0 B b 0 B b 1 B b 1 b 0 b 0 b 1 B b 0 b 1 b 0 b 1 b 1 b 0 ) 231

232 (3) seg a, 0 A = B, B, a A. 0 A 0 B A seg a B seg b. 6.8 A A E t A E(t) = ran(e seg t) = {E(x) x t}. 6.9 A, E α A, - (1) t A, E(t) / E(t); (2) E A α ; (3) s, t A, s t E(s) E(t); (4) α = ran E α α α ( α α, α - ) α, β, γ (1) α ( ); (2) α / α ( ); (3) α β β γ α γ ( ); (4) α β, α = β, β α ( ); (5) α, β α < β, α = β, α > β (1) ; (2) 0 ; (3) α α + = α {α} ; (4) A A (1). (2) N ( N ω ) ω, ω +, ω ++, ω +++,. (3) A A A. (4) α α + α. (5) α α = {x x x < α} (Hartogs ) A α A α ( ) A A ( ) A α A α (1) A B card A = card B A B; (2) A card A A. 232

233 6.21 α α α. 233

234 7.1 ( ) G = V, E V = {v 1, v 2,, v n } E = m n d(v i ) = 2m. i=1 7.2 ( ) D = V, E V = {v 1, v 2,, v n } E = m n n n d(v i ) = 2m d + (v i ) = d (v i ) = m. i=1 i=1 i=1 G ( ). 7.3 d = (d 1, d 2,, d n ) ( d i 0 i = 1, 2,, n ) n d i = 0 (mod 2). i=1 i=1 7.4 d = (d 1, d 2,, d n ), (n 1) d 1 d 2 d n 0 d r 1 r (n 1) r n n d i r(r 1) + min{r, d i } d i = 0 (mod 2). i=r d = (d 1, d 2,, d n ) n d i = 0 (mod 2) (n 1) d 1 d 2 i=1 d n 0 d d = (d 2 1, d 3 1,, d d1 +1 1, d d1 +2,, d n ). 7.6 n G v i v j (v i v j ) v i v j n 1. n G v i v j (v i v j ) v i v j n n G v i v i n. n G v i v i n ( ). 7.8 G G. 7.9 G n G G m n (Whitney) G κ λ δ, κ, λ, δ G. G k- G k i=1

235 7.11 G n(n 6) λ(g) < δ(g) K n1, K n n1 n λ(g) G G λ(g) + 2 n 1. 2 (1) δ(g) δ(g ) n 1 1 n 2 1; (2) G u, v d G (u) + d G (v) n 2; (3) d(g) d(g ) G n(n 6). n (1) δ(g) λ(g) = δ(g); 2 (2) G u, v d(u) + d(v) n 1 λ(g) = δ(g); (3) d(g) 2 λ(g) = δ(g) G n G K n κ(g) 2δ(G) n n, δ, κ, λ n G δ(g) = δ κ(g) = κ λ(g) = λ n (1) 0 κ λ δ < ; 2 (2) 1 2δ n + 2 κ λ = δ < n 1; (3) κ = λ = δ = n (Whitney) G n(n 3) G 2- G G n(n 3) G 2 - G v G v G V (G) v V (G) v = V 1 V 2 u V 1 w V 2 v u w. v G v v u w v u v e G e G e G e G e V (G) V (G) = V 1 V 2 u V 1, v V 2 e u v G n(n 3) (1) G (2) G (3) G (4) G (5) G u, v e u v e (6) G 3 3 (7) G

236 7.21 D n D D D D n D D. 236

237 8.1 G (1) G (2) G (3) G. 8.2 G G G. 8.3 D (1) D (2) v V (D) d + (v) = d (v) (3) D. 8.4 D D D G Fleury. 8.6 G = V, E V V 1 p(g V 1 ) G V 1. p(g V 1 ) V 1 G = V, E V V 1 p(g V 1 ) V G n(n 2) G v i, v j G. d(v i ) + d(v j ) n 1 1 G n(n 3) G v i, v j G G. d(v i ) + d(v j ) n 2 G n(n 3) v V (G) d(v) n G u, v n G d(u) + d(v) n G G (u, v). 8.9 D n(n 2) D. D n D n D

238 D n D n D K 2k+1 (k 1) k k K 2k+1. K 2k (k 2) k 1 K 2k k 1 k. 238

239 9.1 G = V, E n m (1) G (2) G (3) G m = n 1 (4) G m = n 1 (5) G G (6) G G u, v (u, v). 9.2 T n T. 9.3 G G. 1 G n m m n 1. 2 T n m G T T m n T G T T C G E(T ) E(C). 9.4 T G e T T e G. 9.5 T G e T G e. e 1, e 2 T. 9.6 G = V, E n ( V = {v 1, v 2,, v n } ) G e τ(g) = τ(g e) + τ(g\e). 9.7 τ(k n ) = n n 2 (n 2) K n n. 9.8 Ω 0 G i = 1 G i = G i i = 1, 2,, 2 m F = {0, 1} m M. 9.9 T n m G C k e k k = 1, 2,, m n + 1 r(1 r m n + 1) e i 1, e i 2,, e i r. C i1 C i2 C ir 9.10 C 1 C 2 G ( ) C 1 C 2 G. C 1, C 2 G C 1 C 2 G ( ). 239

240 9.11 G T G G ( ) T. 1 G. 2 G n m G s ( ) m n + 1 s 2 m n G n m s G ( ) S = 2 m n G n m C G ( ) C Ω m n + 1 Ω G G G G n m T G S T S i1, S i2,, S ik S e i 1, e i 2,, e i k. S i1 S i2 S ik 9.15 S 1, S 2 G S 1 S 2 G G T G G T G n m S = { } {S S G } S Ω n 1 Ω G. 240

241 10.1 n G r(m(g)) = n n G r(m f (G)) = n 1. 1 n G p r(m(g)) = r(m f (G)) = n p M f (G) M(G). 2 G r(m(g)) = r(m f (G)) = n M f (G) n G. M f M f (G) n 1 M f {e i 1, e i2,, e in 1 } G[{e i1, e i2,, e in 1 }] G M f M f = A n D A l(l 2) A l = A l 1 A a (l) ij v i v j l a (l) ij D l a (l) ii D l i j i. A n D B r b (r) ij v i v j r b (r) ij D r b (r) ii D r. i j i ( 10.5 ) G n V = {v 1, v 2,, v n } A G A k a (k) ij = a (k) ji (i j) G v i v j ( v j v i ) k. a (k) ii v i v j k. 1 A 2 a (2) ii = d(v i ). 2 G i j, v i, v j d(v i, v j ) A k a (k) ij k

242 11.1 G G. G G G G = G G m 2 n deg(r i ) = 2m. i= R G G G G 1 R G n(n 3) G G n(n 4) G δ(g) G n, m, r G. n m + r = p(p 2) G n m + r = p + 1 n, m, r G G G l(l 3) G m n m l (n 2). l G p(p 2) l(l 3) m n m l (n p 1). l G n(n 3) m m 3n G n ( n 3 ) m m = 3n G G G G K 5 K 3,3. 242

243 11.14 G G K 5 K 3, G G n, m, r n, m, r G G (1) n = r; (2) m = m; (3) r = n; (4) G v i G R i d G (v i ) = deg(r i) G p(p 2) (1) n = r; (2) m = m; (3) r = n p + 1; (4) v i G R i d G (v i ) = deg(r i). n, m, r, n, m, r G G G G = G G n(n 4) W n n(n 3) G 3 n. n G n(n 3) G n G n(n 3) (1) m = 2n 3 m G (2) G 3 3 (3) G 2 2 (4) G κ = G G K 4 K 2, G n C G. r i, r i C C i n (i 2)(r i r i ) = 0. i=

244 12.1 χ(g) = 1 G χ(k n ) = n G 2- G. 1 χ(g) = 2 G. 2 G 2- G G χ(x) (G) (Brooks) K n (n 3) χ(g) (G) G χ(g)- V i = {v v V (G) v i }, i = 1, 2,, χ(g), Π = {V 1, V 2,, V χ(g) } V (G) G χ(g)- R = { u, v u, v V (G) u, v }, R V (G) f(k n, k) = k(k 1) (k n + 1), f(n n, k) = k n K n, N n n n. f(k n, k) = f(k n 1, k)(k n + 1), n G V (G) = {v 1, v 2,, v n }. (1) e = (v i, v j ) / E(G) (G, k) = f(g (v i, v j ), k) + f(g\(v i, v j ), k). (2) e = (v i, v j ) E(G) (G, k) = f(g e, k) f(g\e, k). G\(v i, v j ) v i, v j w ij v i, v j. f(g, k) = f(k n1, k) + f(k n2, k) + + f(k nr, k). χ(g) = min{n 1, n 2,, n r } V 1 G G[V 1 ] G V 1 G V 1 p(p 2) G 1, G 2,, G p p (f(h i, k)) i=1 f(g, k) = f(g[v 1 ], k) p

245 H i = G[V 1 V (G i )], i = 1, 2,, p T n f(t, k) = k(k 1) n G n f(g, k) = (k 1) n + ( 1) n (k 1) G k- G k G G G G k- G k (Heawood) (Vizing) G (G) χ (G) (G)

246 13.1 G V1 G G V2 V1 V2 = G V G V G G = V, E V V V G V = V V G. G n. V G ( ) V = V (G) V G ( ). α 0 + β 0 = n G n V G V G. G n V G ( ) V G ( ) ν 0 (G) = β 0 (G) G n. (1) M G M v v N W = M N G. (2) W 1 G W 1 N 1 M 1 = W 1 N 1 G. (3) α 1 + β 1 = n. G n M G W G M W. M G W G G n M G N G Y G W G (1) M N (2) Y W M, N, Y, W G. G n β 1 α 0, β 0 α M 1, M 2 G G[M 1 M 2 ] M 1, M

247 13.8 M G Γ G M M = M E(Γ) M = M M G G M n G V V (G) p (G V ) V, p (G V ) G V (Hall ) G = V 1, V 2, E V 1 V 2. G V 1 V 2 S V 1 S N(S) N(S) S N(S) = N(v i ). v i S G = V 1, V 2, E V 1 t(t 1) V 2 t G V 1 V G = V 1, V 2, E k- G k. K k,k k G = V 1, V 2, E α 0 = β

248 14.1 P E = {v T E(v) } T E = V P E T E u T E Γ (u) P E P L = {v T L(v)) } T L = V P L T L u T L Γ + (u) P L T S(v i ) = 0 v i C G = V, E, W G (1) G G (2) G G G = V, E, W V G V = 2k(k 0) F = {e e E G } F G[F ] V k T G = V, E, W (1) T G (2) e E(T ) e S e e S e (3) e E(T ) ( T T ) C e e e C e G = V, E, W C G e C G e G G = V, E, W. S = (V 1, V 1 ) G e S W (e ) = min e S {W (e)} T e T G G = V, E, W e G. G e T G = V, E, W e G G G e T G G T = G[E(T ) {e}] T G w 1 w 2 w t w 1, w 2 v 1, v 2 v 1, v 2 h (Huffman ) T w 1 + w 2, w 3,, w t w 1 w 2 w t T w 1 + w 2 w 1 248

249 w 2 T T w 1, w 2,, w t r(r 2) T i t (r 1)i = t G = V, E, W n v i, v j, v k V (v i, v j ), (v j, v k, ), (v i, v k ) w ij, w jk, w ik w ij + w jk w ik, d d ( log 2 n + 1), d 0 G d G = V, E, W n(n 3) 0 v i, v j, v k V (v i, v j ), (v j, v k ), (v i, v k ) w ij + w jk w ik d 0 G H G d d d 0 < d d 0 < 3 2. d 0 G d. 249

250 15.1 A A e l A e r A x A e l x = x x e r = x e l = e r = e e A A θ l A θ r A x A θ l x = θ l x θ r = θ r θ l = θ r = θ θ A A e θ A e θ A e. x A y l, y r A y l x = e x y r = e y l = y r = y y x V 1 = A, 11, 12,, 1r, V 2 = B, 21, 22,, 2r V V 1 V 2. 1i, 1j, 2i, 2j (1) 1i, 2i V 1 V 2 ( ) i V ( ). (2) 1i 1j V 1 2i 2j V 2 i j V. (3) 1i, 1j V 1 2i, 2j V 2 i, j V. (4) e 1 ( θ 1 ) V 1 1i ( ) e 2 ( θ 2 ) V 2 12 ( ) e 1, e 2 ( θ 1, θ 2 ) V i ( ). (5) 1i, 2i a A, b B 1i 2i a 1, b 1 a 1, b 1 a, b V i V 1 = A, 1, 2,, r, V 2 = B, 1, 2,, r i = 1, 2,, r i, i k i. ϕ : A B V 1 V 2 ϕ(a) V 2 V 2 V 1 ϕ V 1 = A, 1, 2,, r, V 2 = B, 1, 2,, r ϕ : A B V 1 V 2 i, j V 1. (1) i ( ) i ( ). (2) i j i j. (3) i, j i, j. (4) e ( θ ) V 1 i ( ) ϕ(e) ( ϕ(θ) ) V 2 i ( ). 250

251 (5) i x 1 A x i ϕ(x 1 ) ϕ(x) i V = A, 1, 2,, r i = 1, 2,, r i k i. V V V/ = A/, 1, 2,, r. i, j V. (1) i ( ) i V/ ( ). (2) i j i j V/. (3) i, j i, j V/. (4) e ( θ ) V i ( ) [e] ( [θ] ) V/ i ( ). (5) i V x A i x 1 V/ [x] i [x 1 ] V 1 = A, 1, 2,, r, V 2 = B, 1, 2,, r i = 1, 2,, r i, i k i. ϕ : A B V 1 V 2 ϕ A V V = A, 1, 2,, r i k i i = 1, 2,, r. V g : A A/, g(a) = [a], a A V V/ ( ) V 1 = A, 1, 2,, r, V 2 = B, 1, 2,, r i = 1, 2,, r i, i k i. ϕ : A B V 1 V 2 ϕ V 1 V 1 V 1 ϕ V 1 / = ϕ(a), 1, 2,, r. 251

252 16.1 V = S, x, y S (1) x n x m = x n+m ; (2) (x n ) m = x nm S, e S V S S V V S B S. n Z + B n = {b 1 b 2 b n b i B, i = 1, 2,, n}, B = n Z + B n V = S, V = {S S, } V V V ( ) V = S,, e T S S T,, I S S,, e M = Q, Σ, Γ, δ, λ w 1, w 2 Σ (1) δ (q, w 1 w 2 ) = δ (δ (q, w 1 ), w 2 ), (2) λ (q, w 1 w 2 ) = λ (q, w 1 )λ (δ (q, w 1 ), w 2 ), w 1 w 2 w 1 w M = Q, Σ, δ M = Q, Σ, δ M. w Σ f w : Q Q f w (q) = δ (q, w). S = {f w w Σ } T M = S,, f Q Q,, I Q T = S,, e M M T M T M 1 = Q 1, Σ 1, Γ 1, δ 1, λ 1 M 2 = Q 2, Σ 2, Γ 2, δ 2, λ 2. T M1 T M2. M 1 M 2 T M1 T M M 1 = Q 1, Σ, Γ, δ 1, λ 1 M 2 = Q 1 /, Σ, Γ, δ 2, λ 2 M 1 M 1 M

253 17.1 G, e G a G a e = a a G a G a a = e G 17.2 G a, b G (1) (a 1 ) 1 = a; (2) (ab) 1 = b 1 a 1 ; (3) a n a m = a n+m, m, n Z; (4) (a n ) m = a mn, m, n Z; (5) G Abel (ab) n = a n b n, n Z G a, b G ax = b ya = b G G a, b G ax = b ya = b G G G G G = {a 1, a 2,, a n } G G G a G a = r (1) a k = e r k k Z (2) a = a 1 (3) G = n r n ( ) G H G H G (1) a, b H ab H (2) a H a 1 H ( ) G H G H G a, b H ab 1 H ( ) a, b H G H G H G ab H G = a. (1) G G a a

254 (2) G n G φ(n). n = 1 G = e e n > 1 n r a r G (n, r) = G = a (1) G (2) G G {e} (3) G n G n n d G d E(A) A E(A) σ, τ S n σ τ στ = τσ n σ = (i 1 i 2 i k ) A = {1, 2,, n} k k > 1 σ = (i 1 i k )(i 1 i k 1 ) (i 1 i 2 ) σ S n σ(j) = i j j = 1, 2,, n σ π = i 1 i 2 i n G n. (1) σ G σ = (i 1 i 2 i k ) σ = k. (2) τ G τ = τ 1 τ 2 τ l τ i k i i = 1, 2,, l τ k 1, k 2,, k l τ = [k 1, k 2,, k l ] G H G (1) He = H; (2) a G, a Ha G H G a G, Ha H G H G a, b G a Hb Ha = Hb ab 1 H G H G G R a, b G R G [a] R = Ha G H G arb ab 1 H, a, b G, Ha Hb = Ha = Hb Ha = G G H G (1) eh = H (2) a G, a ah (3) a G, ah H (4) a, b G, a bh ah = bh a 1 b H (5) G R a, b G arb a 1 b H R G [a] R = ah (6) a, b G, ah bh = ah = bh a G ah = G (Lagrange ) G H G G = [G : H] H. 1 G n G n a G a n = e. 254 a G

255 G G G C G a G a C ā = {a} G a G N(a) G G a G ā = [G : N(a)] ( ) G C G. G k a 1, a 2,, a k k G = C + [G : N(a 1 )] + [G : N(a 2 )] + + [G : N(a k )] N G. (1) N G (2) g G gng 1 = N (3) g G n N gng 1 N ϕ G 1 G 2 ϕ ker ϕ = {e 1 } G 1 = a ϕ G 1 G 2 G ϕ G 1 G 2. (1) H G 1 ϕ(h) G 2. (2) H G 1 ϕ ϕ(h) G ϕ G 1 G 2 (1) ker ϕ G 1 (2) a, b G 1, ϕ(a) = ϕ(b) a ker ϕ = b ker ϕ ( ) G H G G G/H G. G G G ϕ G G/ ker ϕ = G G End G Aut G G Inn G Aut G G K L G G = K L (1) K G, L G; (2) K L = {e}; (3) G = KL G G 1, G 2,, G n G G = G 1 G 2 G n (1) G i G, i = 1, 2,, n (2) G i G 1 G 2 G i 1 G i+1 G n = {e}, i = 1, 2,, n (3) G = G 1 G 2 G n r- m log r m Z n, a Z n a 0 a Z n n T T 2 log 2 n. 255

256 1 Z n r- Z n log r (2 log 2 n ). 2 Z n H Z n H r- Z n log r (2 log 2 H ) (1) n = p i p i r- Z n log r (2 log 2 n ). (2) n = p i1 1 pi2 2 pi k k n r- Z n log r (2 log 2 t(n) ) t(n) = max{p i1 1, pi2 2,, pi k k }. 256

257 18.1 R (1) a R, a0 = 0a = 0 (2) a, b R, ( a)b = a( b) = (ab) (3) a, b R, ( a)( b) = ab (4) a, b, c R a(b c) = ab ac, (b c)a = ba ca; (5) a 1, a 2,, a n, b 1, b 2,, b m R ( n )( m ) a i b j = i=1 j=1 (6) a, b R, n Z, (na)b = a(nb) = n(ab). n i=1 j=1 m a i b j ; 18.2 R. R R a, b, c R, a 0 ab = ab ( ba = ca ) b = c F F F p F = p n n Z R S R a, b S (1) a b S; (2) ab S ϕ : R 1 R 2 ker ϕ R ϕ : R 1 R 2 (1) S R 1 ϕ(s) R 2 (2) T R 2 ϕ 1 (T ) R 1 (3) D R 1 ϕ(d) R 2 (4) I R 2 ϕ 1 (I) R D R g : R R/D r R g(r) = D + r g R R/D ker g = D ( ) R R/D R. R R R = R/ ker ϕ F [x] F f(x) F [x]. F [x] R g(x), h(x) F [x] g(x)rh(x) f(x) (g(x) h(x)), R F [x]. 257

258 18.11 F F [x]/f(x) f(x) F [x]. 258

259 P L P S, a, b, c S (1) a b a, a b b; (2) a a b, b a b; (3) a b a c a b c; (4) a b a c a b c S, a, b S a b a b = a a b = b L,, L (1) a, b L a b = b a, a b = b a; (2) a, b, c L (3) a L (4) a, b L (a b) c = a (b c), (a b) c = a (b c); a a = a, a a = a; a (a b) = a, a (a b) = a S,,. S S, S, S,, S,, L (1) a, b, c L (2) a, b, c, d L a b a c b c a c b c; a b c d a c b d a c b d L (1) a, b, c L a (b c) (a b) (a c), a (b c) (a b) (a c); (2) a, b, c L a b a (c b) (a c) b. 259

260 19.7 ϕ L 1,, L 2,, a, b L 1 a b ϕ(a) ϕ(b) L 1, L 2 ϕ : L 1 L 2 ϕ L 1 L 2 a, b L 1, a b ϕ(a) ϕ(b) L. S L S ( S ) L L I(L) = {x x L }, I(L) L L I(L) L. I 0 (L) = I(L) { } I 0 (L) L I 0 (L) L L L L a, b, c a b a c = b c a c = b c a = b L L a, b i L, i = 1, 2,, n (1) a ( n ) n b i = (a b i ); (2) a ( n ) n b i = (a b i ). i=1 i=1 i= L a, b, c L a c = b c a c = b c a = b L a, b, c L (a b) (b c) (c a) = (a b) (b c) (c a) L L L a, b, c L a c = b c a c = b c a = b L a L. a a B,,,, a, b a, b B. (1) x, y B x y = y x, x y = y x ( ) (2) x, y, z B x (y z) = (x y) (x z), x (y z) = (x y) (x z) ( ) (3) x B x b = x, x a = x ( ) (4) x B x x = a, x x = b ( ) B,,,, a, b. B a 0 b B,,,, 0, 1 (1) a B, ā = a 260 i=1

261 (2) a, b B, a b = ā b, a b = ā b (3) a, b B, a b a b = 0 ā b = 1 a b = a a b = b (4) a, b B, a b b ā B 1, B 2 ϕ : B 1 B 2. ϕ (1) ϕ(0) = 0, ϕ(1) = 1; (2) ϕ(b 1 ) B ( ) B A B B A P(A) n n N B B 0 n B = {0, 1} n B F n (B) = {f f : B n B} B n. f, g F n (B) f g f g f f 0 f 1 x B n (f g)(x) = f(x) g(x), (f g)(x) = f(x) g(x), f(x) = f(x), f 0 (x) = 0, f 1 (x) = 1. F n (B),,,, f 0, f