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1 8.1 f G(f) G(f) f G(f) f <a 1, b 1 > = <a 2, b 2 > a 1 = a 2 b 1 = b 2 <a, b> <a, b> {a, b} a, b {a} = {a, a}{a} <a, b> = {{a}, {a, b}} a, b <a, b>a b a, b {a}, {a, b}{a} {a, b} <a 1, b 1 > = <a 2, b 2 > a 1 = a 2 b 1 = b 2 a 2 = b 2 {{a 1 }, {a 1, b 1 }} = <a 2, a 2 >= <a 1, b 1 > = <a 2, b 2 > 255 = {{a 2 }, {a 2, a 2 }} = {{a 2 }} {a 1, b 1 } = {a 2 } a 1 = a 2 b 1 = a 2 = b 2 a 2 b 2 {a 1 } {a 2, b 2 }<a 1, b 1 > = <a 2, b 2 > {{a 1 }, {a 1, b 1 }} = {{a 2 }, {a 2, b 2 }} {a 1 } = {a 2 } {a 1, b 1 } = {a 2, b 2 } {a 1 } = {a 2 } a 1 = a 2 a 1 = a 2 {a 1, b 1 } = {a 2, b 2 } b 1 = b A B A B = {<x, y> x A y B} A B A B A B P(P(A B)) <x, y> A B x A y B x, y A B x A B {x} A B x, y A B {x, y} A B {x}, {x, y} A B {x}, {x, y} A B {x}, {x, y} P(A B) {x}, {x, y} P(A B) 256

2 {{x}, {x, y}} P(P(A B)) <x, y> P(P(A B)) A B P( P(A B)) A B P( P(A B)) A B A B A B n 1 n <a 1,, a n > (1) <a 1 > = a 1 (2) <a 1,, a k+1 > = <a 1,, a k >, a k+1 > n n 2 n n 1 n n n A 0 A k+1 = (A 0 A k ) A k+1 n n 2 n n 1 n n R R 1 = {<x, y> <y, x> R} R Q Q R = {<x, y> z <x, z> R <z, y> Q} (R 1 ) 1 = R S (Q R) = (S Q) R R {x y <x, y> R} R dom(r) {y x <x, y> R} R ran(r) R dom(r) ran(r) R B {<x, y> <x, y> R x B} R B R B A B B A (1) dom(r 1 ) = ran(r) ran(r 1 ) = dom(r) (2) dom(q R) dom(r) ran(q R) ran(q) (3) dom(r B) = dom(r) B ran(r B) ran(r) f f <x, y>, <u, v> f x = u y = v f f <x, y>, <x, z> f y = z 258

3 f x dom(f) <x, y> f y y f(x) f <x, f(x)> f {<x, f(x)> x dom(f)} f f <x, z>, <y, z> R x = y f A B f dom(f) = A ran(f) B f A B f A B A B f A B ran(f) = B f A B f A B f A B f A B f dom(f) ran(f) f dom(f) ran(f) (1) dom( ) = ran( ) = (2) (3) (4) B B B C A B A C ( ) A B A C 260

4 f g h = {<<x, y>, <u, v>> <x, u> f <y, v> g} dom(h) = dom(f) dom(g) ran(h) = ran(f) ran(g) <<x, y>, <u, v>>, <<x, y>, <u, v >> h <x, u> f, <y, v> g, <x, u > f, <y, v > g <x, u>, <x, u > f f u = u <y, v>, <y, v > g g v = v <u, v> = <u, v > <x, y> dom(h) <u, v> <x, u> f <y, v> g <x, u> f x dom(f) <y, v> g y dom(g) <x, y> dom(f) dom(g) <x, y> dom(f) dom(g) x dom(f) y dom(g) x dom(f) u <x, u> f y dom(g) v <y, v> g <x, y> dom(h) ran(h) = ran(f) ran(g) f, g B 261 (1) f B f B (2) f g g f (1) <x, y>, <x, z> f B <x, y>, <x, z> f f y = z f B (2) <x, y>, <x, z> g f u, v <x, u>, <x, v> f <u, y>, <v, z> g f u = v <u, y>, <u, z> g g y = z g f f f 1 f f f f 1 f f f 1 <x, y>, <x, z> f 1 <y, x>, <z, x> f f y = z f 1 f 1 f <x, z>, <y, z> f <z, x>, <z, y> f 1 f 1 x = y f 1 f A B f 1 B A 262

5 f 1 ran(f) A ran(f) = B f f 1 B A f A B f 1 B A f A B f 1 y = f(x) x = f 1 (y) ( ) Γ Γ Γ Γ f Γ f Γ Γ Γ Γ Γ f, g Γ x dom(f) dom(g) f(x) = g(x) f, g Γ x dom(f) dom(g) f(x) = g(x) (1) Γ(1) Γ <x, y>, <x, z> Γ f, g Γ <x, y> f <x, z> g x dom(f) dom(g) (1) y = f(x) = g(x) = z ΓΓ(1) f, g Γ x dom(f) dom(g) <x, f(x)> f <x, g(x)> g <x, f(x)>, <x, g(x)> Γ Γ f(x) = g(x) 263 A B Γ ΓΓ f Γ A, B f = {<x, y> x A, y B A(x) B(y)} (1) f A(x) B(f(x)) (2) f f dom(f) ran(f) (3) f dom(f) ran(f) (4) dom(f) A ran(f) B (5) dom(f) = A ran(f) = B (1) <x, y>, <x, z> f A(x) B(y) A(x) B(z) B(y) B(z) y = z (2) <x, z>, <y, z> f A(x) B(z) A(y) B(z) A(x) A(y) x = y (3) x, y dom(f) x y A(x) A(y) B(f(x)) B(f(y)) 264

6 f(x) f(y) (4) x dom(f) A(x) B(f(x)) y<x A(x)(y) A(x) B(f(x)) B(f(x))(z) A(x)(y) B(f(x))(z) A(y) = A(x)(y) B(f(x))(z) = B(z) f <y, z> f y dom(f) ran(f) B (5) dom(f) A ran(f) B x A, y B dom(f) = A(x) dom(f) = B(y) A(x) B(y) dom(f) = A(x) x dom(f) A(x) B(y) x dom(f) A, B A B B A f = {<x, y> x A, y B A(x) B(y)} (3), (4) (5) A B B A A f A n ~ f = {<< ~ x 1,, ~ x n >, ~ x > <<x1,, x n >, x> f} ~ f A f A n f x 1 y 1,, x n y n f(x 1,, x n ) f(y 1,, y n ) 265 f A f A n f ~ f A / n << ~ x 1,, ~ x n >, ~ x >, << ~ y 1,, ~ y n >, ~ ~ y > f < ~ x 1,, ~ x n > = < ~ y 1,, ~ y n > ~ x 1 = ~ y 1,, ~ x n = ~ yn x 1 y 1,, x n y n f f(x 1,, x n ) f(y 1,, y n ) x y ~ x = ~ y n 1<a 1,, a n > = <b 1,, b n > 1 i n a i = b i (1) f A B f P(P(A B)) (2) B A P( P(P(A B))) f f 1 y = f(x) x = f 1 (y) f g h = {<x, <y, z>> <x, y> f <x, z> g} (1) h (2) dom(h) = dom(f) dom(g) (3) dom(f) = dom(g) dom(ran(h)) = ran(f) ran(ran(h)) = ran(g) A f A n ~ f A / n f 266

7 N N = {<<n, m>, <s, t>> n+t = s+m} ( 3.2.4) N N N N / Z Z <n, m> [n, m] [n, m] = [s, t] <n, m> <s, t> n+t = s+m Z {[n, m] n, m N} f N Z f(n) = [n, 0] n, m N f(n) = f(m) [n, 0] = [m, 0] n+0 = m+0 n = m f N Z f[n] = {[n, 0] n N} N {[n, 0] n N}N Z <n, 0>[n, 0] n n N<0, n>[0, n] n (1) 0 = 0 (2) n N n = m n = m 267 (3) Z = N { n n N} (1) (2) n = m [0, n] = [0, m] 0+n = 0+m n = m (3) [n, m] Z s N a = s a = s n m s = n m n+0 = s+m a = [n, m] = [s, 0] = s n m n = m n n+s = 0+m a = [n, m] = [0, s] = s Z N N f (N N) 2 N N f(<n, m>, <s, t>) = <n+s, m+t> g (N N) 2 N N g(<n, m>, <s, t>) = <n s+m t, n t+m s> N N h N N N N f(<n, m>) = <m, n> f, g h <n, m> <n, m > <s, t> <s, t > n+m = n +m s+t = s +t (n+s)+(m +t ) = (n +s )+(m+t) <n+s, m+t> <n +s, m +t > f(<n, m>, <s, t>) f(<n, m >, <s, t >) 268

8 <n, m> <n, m > <s, t> <s, t > n+m = n +m s+t = s +t n+m = n +m s, t n s+m s = n s+m s n t+m t = n t+m t s+t = s +t n, m n s+n t = n s +n t m s +m t = m s+m t (n s+m t)+(n t +m s )= (n s +m t )+(n t+m s) <n s+m t, n t+m s> <n s +m t, n t +m s > g(<n, m>, <s, t>) g(<n, m >, <s, t >) <n, m> <n, m > n+m = n +m m+n = m +n <m, n> <m, n > h(<n, m>) h(<n, m >) f, g h f, g h Z ~ f Z 2 ~ Z f ([n, m], [s, t]) = [n+s, m+t] g ~ Z 2 Z g ~ ([n, m], [s, t]) = [n s+m t, n t+m s] Z h ~ Z Z h ~ ([n, m]) = [m, n] ~ f g ~ ~ Z f (a, b) a+b g ~ (a, b) a b h ~ Z h ~ (a) a Z a b = a+( b) n, m N n+m = k [n, 0]+[m, 0] = [k, 0] 269 n, m N n m = k [n, 0] [m, 0] = [k, 0] f, g h (1) a+b = b+a a b = b a (2) a+(b+c) = (a+b)+c a (b c) = (a b) c (3) a (b+c) = a b+a c (4) a+( a) = 0 (5) a+0 = a a 1 = a (1) a = [n, m] b = [s, t] a+b = [n, m]+[s, t] = [n+s, m+t] = [s+n, t+m] = b+a a b = b a (2) a = [n, m] b = [s, t] c = [k, l] a+(b+c) = [n, m]+([s, t]+[k, l]) = [n, m]+[s+k, t+l] = [n+(s+k), m+(t+l)] = [(n+s)+k, (m+t)+l] = [n+s, m+t]+[k, l] = ([n, m]+[s, t])+[k, l] = (a+b)+c a (b c) = (a b) c (3) a = [n, m] b = [s, t] c = [k, l] a (b+c) = [n, m] ([s, t]+[k, l]) = [n, m] [s+k, t+l] = [n (s+k)+m (t+l), n (t+l)+m (s+k)] = [(n s+n k)+(m t+m l), (n t+n l)+(m s+m k)] = [n s+n k, n t+n l]+[m t+m l, m s+m k] = [n, m] [s, t]+[n, m] [k, l] = a b+a c (4) a = [n, m] a+( a) = [n, m]+[m, n] = [0, 0] = 0 270

9 (5) (1) a+b = 0 a = b (2) a ( 1) = a (3) a = ( a) (4) ( a) b = (a b) a ( b) = (a b) ( a) ( b) = a b (1) a = a+0 = a+(b+( b)) = (a+b)+( b) = 0+( b) = b (2) a ( 1)+a = a ( 1)+a 1 = a (( 1)+1) = a 0 = 0 (1) a ( 1) = a (3) a+( a) = 0(1) a = ( a) (4) ( a) b+a b = (( a)+a) b = 0 b = 0 (1) ( a) b = (a b) a ( b) = (a b) ( a) ( b) = (a ( b)) = ( (a b)) = a b Z = {<[n, m], [s, t]> s+m n+t} [n, m] [s, t] s+m n+t s+m n+t [n, m] [s, t] [n, m] [s, t] n, m, s, t [n, m ] = [n, m] [s, t ] = [s, t] s +m n +t [n, m ] = [n, m] [s, t ] = [s, t] n+m = n +m s+t = s +t s+m n+t Z [n, m] Z n+m n+m [n, m] [n, m] [n, m], [s, t] Z [n, m] [s, t] [s, t] [n, m] s+m n+t n+t s+m n+t = s+m [n, m] = [s, t] [n, m], [s, t], [k, l] Z [n, m] [s, t] [s, t] [k, l] s+m n+t k+t s+l s+m+k+t n+t+s+l k+m n+l [n, m] [k, l] [n, m], [s, t] Z s+m n+t n+t s+m [n, m] [s, t] [s, t] [n, m] Z N n, m N n m [n, 0] [m, 0] Z + = {n n Z n>0} Z = {n n Z n<0}

10 8.2.9 (1) b c a+b a+c (2) b c a 0 a b a c (3) a b b a (1) a = [n, m], b = [s, t] c = [k, l] a+b = [n+s, m+t] a+c = [n+k, m+l] b c[s, t] [k, l] k+t s+l n+m (n+m)+(k+t) (n+m)+(s+l) (n+k)+(m+t) (n+s)+(m+l) [n+s, m+t] [n+k, m+l] a+b a+c (2) a = [n, m] b = [s, t] c = [k, l] a b = [n k+m l, n l+m k] a c = [n k+m l, n l+m k] b c a 0[s, t] [k, l] [n, m] 0 k+t s+l n m 0, k+t s+l n m (n m) (k+t) (n m) (s+l) (n k+m l)+(n t+m s) (n s+m t)+(n l+m k) [n k+m l, n l+m k] [n k+m l, n l+m k] a b a c (3) a = [n, m] b = [s, t] a = [m, n] b = [t, s] a b[n, m] [s, t] s+m n+t m+s t+n 273 [t, s] [m, n] b a Z Z + = {<<a, b>, <c, d>> a d = c b} ( 3.2.4) Z Z + Z Z + / Q Q n N<a, b> b a a = c <a, b> <c, d> a d = b c b d Q { a a Z b Z + } b f Z Q f(a) = 1 a 274 a, c Z f(a) = f(c) 1 a = 1 c a 1 = 1 c a = c f Z Q f[z] = { a 1 a Z} Z { 1 a a Z}Z Q <a, 1> 1 a a Q Z Z + f (Z Z*) 2 Z Z* f(<a, b>, <c, d>) = <a d+b c, b d> g (Z Z*) 2 Z Z* g(<a, b>, <c, d>) = <a c, b d>

11 h (Z Z*) 2 Z Z* h(<a, b>) = < a, b> f, g h <a, b> <a, b > <c, d> <c, d> a b = a b c d = c d a b = a b d d a b d d = a b d d c d = c d b b b b c d = b b c d (a d+b c) b d = (a d +b c ) b d <a d+ b c, b d> <a d + b c, b d > f(<a, b>, <c, d>) = f(<a, b >, <c, d >) <a, b> <a, b > <c, d> <c, d> a b = a b c d = c d a c b d = a c b d <a c, b d> <a c, b d > g(<a, b>, <c, d>) = g(<a, b >, <c, d >) <a, b> <a, b > a b = a b a b = a b < a, b> < a, b > h(<a, b>) = h(<a, b >) f, g h f, g h Q ~ f Q 2 Q ~ f ( a, c ) = a d + b c b d b d a c b d ~ g Q 2 Q ~ g ( b a, d c ) = Q ~ h Q Q h ~ ( a ) = a b b ~ f g ~ Q ~ f (p, q) p+q g ~ (p, q) p q h ~ Q h ~ (p) p b a 0 b a ( b a ) 1 ( b a ) 1 = b a b a a>0 a<0 a, b Z a+b = c 1 a + 1 b = 1 c a, b Z a b = c 1 a 1 b = 1 c a Z a = b 1 a = 1 b f, g h (1) p+q = q+p p q = q p (2) p+(q+r) = (p+q)+r p (q r) = (p q) r (3) p (q+r) = p q+p r (4) p+( p) = 0

12 (5) p+0 = p p 1 = p (6) p 0 p p 1 = 1 (1) p = b a q = d c p+q = a d + b c = c b + d a = q+p b d d b p q = q p (2) p = a q = c r = u b d v p+(q+r) = a + c v + d u b d v a ( d v) + b ( c v + d u) = b ( d v) ( a d + b c) v + ( b d) u = ( b d) v = a d + b c + u = (p+q)+r b d v p (q r) = (p q) r (3) p = a q = c r = u b d v p (q+r) = a c v + d u a ( c v + d u) = b d v b ( d v) ( a d) ( b v) + ( b d) ( a u) = ( b d) ( b v) = a d + a u = p q+p r b d b v (4)(5)(6) (1) p+q = 0 p = q (2) p ( 1) = p (3) p = ( p) 277 (4) ( p) q = (p q) p ( q) = (p q) ( p) ( q) = p q (5) p q = 1 p = q 1 (6) p = (p 1 ) 1 Q = {< a, c > a d b c} b d a c a d b c b d a d b c a c b d b a d c a, b, c, d a = a c = c a b = a b c d = c d b b d d a d b c a d b c Q b a Q a b b a b a b a a, c Q a c c a b d b d d b a d b c c b d a a d = b c b a = d c a, c, u Q a c c u b d v b d d v a d b c c v d u a, c, u>0 278

13 a d c v b c d u b c a v b u b a v u a, c, u<0 a d c v b c d u b c a v b u b a v u a 0 u 0 a v 0 b u 0 a v b u b a v u a, c Q b d a d b c c b d a b a d c d c b a Q Z a, b Z a b 1 a 1 b Q + = { p p Q p>0} Q = {p p Q p<0} b a <0 a<0 b a >0 a> (1) q r p+q p+r 279 (2) q r p 0 p q p r (3) p q q p (4) 0<p q q 1 p 1 (1) p = b a q = d c r = v u p+q = a d + b c p+r = a v + b u b d b v q r c u d v c v d u b b a d b v b b c v+a d b v b b d u+a d b v (a d+b c) b v b d (a v+b u) a d + b c a v + b u b d b v p+q p+r (2) p = b a q = d c r = v u p q = a c p r = a u a 0 b d b v q r p 0 c u a 0 c v d u d v a c a c b v b d a u a c a u b d b v p q p r (3) p = a q = c p = a b d b 280 q = c d

14 p q b a d c a d b c 1 ( c) b d ( a) a c b d q p (4) p = a b q = c d p 1 = a b q 1 = c d p q b a d c a d b c d a c b b d a c q 1 p p<q r p<r r<q p = b a q = d c p = 2 a d q = 2 b c a d<b c 2 b d 2 b d 2 a d<2 a d+1 2 a d+1<2 b c r = 2 a d + 1 p<r r<q 2 b d Archimed p, q>0 n N n p>q p = b a q = d c n = b c+1 ( b c + 1) a > c b d n p>q a, b, c Z (1) a b = b a (2) a (b c) = (a b) c (3) a+0 = a a 1 = a a Z b Z b<a a Z b Z a<b<a p, q, r Q (1) p q = q p (2) p (q r) = (p q) r (3) p+( p) = 0 (4) p+0 = p p 1 = p (5) p 0 p p 1 = (1) p Q q Q q<p (2) p Q q Q q>p ( b c + 1) a n p = b (b c+1) a d = b c a d+a d>b c

15 8.3 Cantor Dedekind Dedekind a Q a Γ(Q) = {Q a a R} R Q a a Q a a Q a a Q a Q a (1) Q a Q a Q (2) p Q a q Q q<p q Q a (3) Q a (2) A B A p B q A q p q B B A A A B A B A B A B A ( 5.1.2) p B q A B p<q (1) B A C B C A 283 (2) B C A B C C B (3) i I A i A i I A i A Q a Q a Q r Q Q r = {p p Q p<r} Q p Q r q Q q<p p Q r p<r q<p p<r q<r q Q r Q r Q p Q r p<r ( ) q p<q<r Q r ( 8.2.6) Q r r Q r Q r A = {p p Q (p<0 p 2 <2)} Q p A q Q q<p q<0 q A q 0 q<p p 0 p 2 <2 q<p p 2 <2 q 2 <2 q A 284

16 A Q A A Q A A Q r r Q A = Q r p 0 p 2 <2 p A p Q r p<r p 2 <2 p 2 <r 2 r 2 = Q R R R P(Q) f Q R f(p) = Q p p, q Q p q Q p Q q f(p) f(q) f Q R f [Q] = {Q p p Q} Q {Q p p Q}Q R r r Q r 0 Q 0 = Q ( ) Q R R R R Q p, q Q p q Q p Q q R + = {x x R x>0} R = {x x R x<0} Q R r x r x r<x r x r<x Q r x Q r x p Q r p<r r x p x Q r x r Q r Q r x r x x r p x r x x p<r x Q r x r (1) x, y x<y r x<r r<y (2) x p, q p<x<q (1) x<y p p x p y y r p<r r y p x x p<r r y r<y r x<r r<y (2) x p p x p<x x Q r r x Q q r<q

17 r x x r<q p, q p<x<q () A x = A (1) x Q (2) x (3) A x (1) x x (2) p x y A p y y q y p<q q x p<q x (3) (1) (2) x Q A z y A y z x = A z z Q x Q A x = A x x A x A ( 3.4.6) () A A A 287 A X X A X R A R x = A x x A x A x A x A A X = {y y A } x A X A = {p p Q (p<0 p 2 <2)} ( 8.3.4) A A X x, y z = {p+q p x q y} p+q z r Q r<p+q r q<p x p x r q x r = (r q)+q z z Q p+q z x p x p<p x q x q<q p +q z p+q<p +q z x Q y Q p, q Q p x q y 288

18 p+q Q p+q z z Q x y p, q Q p x q y p+q Q p+q z z z f(x, y) = {p+q p x q y} f f(x, y) x+y (1) x+y = y+x (2) x+(y+z) = (x+y)+z (3) x+0 = x (4) y z x+y x+z (1) x+y = {p+q p x q y} = {q+p q y p x} = y+x (2) (1) (3) x+0 x x x+0 r x+0 p x, q 0 r = p+q q q<0 r = p+q<p r x r x p x r<p r p 0 r = p+(r p) x+0 (4) y z x+y x+z r x+y p x, q y r = p+q q y y z q z p x q z p+q x+z r x+z x 0 x + = {p p x p 0} x 0, y 0 z = {p q p x + q y + } 0 z f(x, y) = {p q p x + q y + } 0 f f(x, y) x y x, y, z 0 (1) x y = y x (2) x (y z) = (x y) z (3) x (y+z) = x y+x z (4) x 1 = x (5) y z x y x z (1) x y = {p q p x + q y + } 0 = {q p q y + p x + } 0 = y x (2)(3) (1)

19 (4) x 1 x x x 1 x 1 x r x 1 p x +, q 1 + r = p q q q<1 r = p q<p r x r x p x r<p r p 1 r = p r p x 1 (5) y z x y x z {p q p x + q y + } {p q p x + q z + } r {p q p x + q y + } p x +, q y + r = p q q y y z q z + p x + q z + p q {q p q y + p z + } r {q p q y + p z + } X X X* = X X p X* = X {p} x (Q x)* Q x (Q x)* = Q x Q x p q (Q x)* p<q r Q p<r<q 291 x p Q x p<r r Q x p r r (Q x)* q (Q x)* r (Q x)* r<q (Q x)* x z = {p p (Q x)*} p z q Q q<p p< q x p (Q x)* p< q q (Q x)* q z z Q p z p (Q x)*(q x)* q (Q x)* q< p q z p<q z x Q (Q x)* z x (Q x)* Q z Q z f(x) = {p p (Q x)*} f f(x) x 292

20 (1) x+( x) = 0 (2) x+y = 0 x = y (3) x = ( x) (4) x y y x x 0 y 0 x = y () x, y 0 ( x) y = (x y) x ( y) = (x y) ( x) ( y) = x y (1) x y = y x (2) x (y z) = (x y) z (3) x (y+z) = x y+x z (4) x 1 = x (5) y z x 0 x y x z x>0 z = {p 1 (Q x)*} p z> x>0 x x 1 = {p 1 (Q x)*} p 293 x>0 ( x) 1 = (x 1 ) x, y 0 (1) x x 1 = 1 (2) x y = 1 x = y 1 (3) x = (x 1 ) 1 (4) 0<x y y 1 x x, y, z R x+(y+z) = (x+y)+z x, y, z 0 (1) x (y z) = (x y) z (2) x (y+z) = x y+x z r Q, x R r 0 r x = {r p p x} Archimed x, y>0 n N n x>y 294

Solutions to Exercises in "Discrete Mathematics Tutorial"

Solutions to Exercises in Discrete Mathematics Tutorial 1 2 (beta 10 ) 3 SOLVED AND TEXIFIED BY 4 HONORED REVIEWER BBS (lilybbs.us) 1 2002 6 1 2003 1 2 2 ( ) (E-mail: xiaoxinpan@163.com) 3 beta 2005 11 9 ( / ) 40.97% 4 02CS chouxiaoya tedy akaru yitianxing