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. 2. 3. 4. 5. 6. 7
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2.. 2+3=5.. 32.!?! x+y>5.. 9
2.. 2+3=5.. 32. 3; 2; ; ABAB
p: 2; -- q: ; -- P: x+y>5; -- (T); (F). 2
3
p q p q (p)(q):p q,:p q (p),(q):p q (p),(q): ( p) q p 4
p q p q p q p q 5
p q p q p q q p p q 6
p q p q 7
: p q : ( p q) (p q) p q : ( p q) (p q) : p q) 968969 : p q 8
9
. 2. 3. 4. 5. 6. 2
.. p,q,r,...,p i,q i,r i,...,, 2. 3.,B(A B), (A B), (A B), (A B) (p q) r (p q) r (p q) r p q)r p q r p q r 2
A()p,q,r,...,p i,q i,r i,...,,,a An+(n)A A=7BBn A=B CBCij,n=max(i,j); A=B CBCij,n=max(i,j); A=B CBCij,n=max(i,j); A=B CBCij,n=max(i,j); K. 7p q : 2 p q r : 2 7(7p q) (r s) : 7(7p)(q(r s)) : 3 4 22
A, p,p 2,...,p n p,p 2,...,p n AA A A A=( pq) r A = A = 23
2 3 4 5 6 7 p q n(n>),2 n, A,A r r q r p (q r ) 24
p(qr) P7..) (p(pq)) q P7..2) (pq) q P7..3) A A A 25
. 2. 3. 4. 5. 6. 26
A,B,A B, AB,A<=>B <=> = A BAB A <=> B <=> A <=> B A BA BA <=> B A B4. 27
7pvq 7pV7q 7pvq 7p7q 28
p q 7pvq 7pV7q 7p 7q pvq 7(pvq) 7pv7q 29
2 p q 7pvq 7p7q 7p 7q pvq 7(pvq) 7p7q 3
(), 2 3 4 5 6 7 8 9 A77A AAVA A AA AVB BVA ABBA (AVB)VC AV(BVC) (AB)C A(BC) AV(B^C) (AVB)^(AVC) A^(BVC) (A^B)V(A^C) ( ( 3
2 2 3 4 5 6 7 8 9 7(AVB) 7A^7B 7(A^B) 7AV7B AV(A^B) A A^(AVB) A A V A^ AV A A^ A AV7A A^7A.. 32
3 2 2 22 23 24 25 A->B 7AVB A <-> B (A->B)^(B->A) A->B 7B->7A A<->B7A<->7B A->(B->C)(AB)->C (A->B)^(A->7B) 7A () () 33
()(A)A(B) B(A)A AB(A)(B) P7(qr)P(7qV7r) (AB)(A)(B) (A)AB(B)BB AB(A)BB(A)(B) (A)(B)A(A) B(B)AB 34
: p(q r)(p q)rp.9()) p(qr) p(qr) p(qr) (pq)r (pq)r (pq)r (pq)r (qp) r (qp)r (qp)r q(pr) q(pr) q(pr) 35
: p(p q) (p q) P.9(2)) p p p(q q) (pq)(p q) P. p-a; q-b; r-c; s-d 36
. 2. 3. 4. 5. 6. 37
38
p,q pq pq pq pqpq 39
p,q pq pqpq pq pq pq7(p q) (pq) (p q) (pq) (p q) 4
p,q pq pqpq pqpq pq7(p V q) (pq) (p q) (pq) (p q) 4
2 n n(n){} n {} nfn F {} n {} {}<=>{x x(x-)=} n2 n : n 2 2 n 42
2 n 2 n k n! C k = n ( n )! k! p, q 2 4 2 6 3 4 4 pq; pq; 7(pq); 7(pq) p; q; 7p; 7q; p q; 7(p q) pq; pq; (pq); (pq) <=>p7p 43
p q pq (pq p (qp q (p q pq F F 2 F 3 F 4 F 5 F 6 F 7 F 8 p q (pq F 9 (p q F Fq (qp F 2 Fp 3 (pq F 4 pq F 5 F 6 44
{7,,v,,} pq7pvq7(p7q) p q(pq)(qp)(7pvq)(7qvp) 7(p7q)7(q7p) pvq77(pvq)7(7p7q) {v,,} {7, } 45
{7,,v,,{7,,v,, {7,,v{7V{7,{ {7, V{7,{ 46
{}. pq pq (pq) ((pp)(q q )) (p p)q q) 2. pq (pq) (pq) (pq) (pq) (pp)<=>a 3. pqpq (pp)q Aq (A(A ((pp)((pp) 47
. 2. 3. 4. 5. 6. 48
7,,VA,V, V,A, A A* pq pvq 7(pq) 7(pvq) 49
AA*,p, p 2,.,p n AA*,AA*n, ()7A(p, p 2,.,p n )A*(7p,7 p 2,.,7p n ); (2) A(7p,7 p 2,.,7p n ) 7A*(p, p 2,.,p n ). :A(p,q,r)p(7qvr) 7A(p,q,r)7(p(7qVr))7pV(q7r) A*(p,q,r)pv(7qr) A*(7p,7q,7r)7pv(q7r) 7A*(p,q,r)7(pv(7qr))7p(qV7r) A(7p,7q,7r)7p(qv7r) 5
2: A,B,AB,A*B*, A*,B*A,B. : (p^q)v(7pv(7pvq))7pvq (p^q)v(7pv(7pvq))(p^q)v(7pvq) (pv(7pvq))^(qv(7pvq)) ^(7pvq) 7pVq (pvq)^(7p^(7p^q))7p^q (pvq)^(7p^(7p^q))(pvq)^(7p^q) (p^(7p^q))v(q^(7p^q)) v(7p^q)7p^q 5
,,. :pvq,7pvq,7pv7q :p^q, 7p^q,7p^7q,.,. 52
,. AAVA2VA3..,.. AAA2A3..,. 53
. {7,,V},,,V, pq7pvq p q (7pvq)^ (pv7q) 54
n,,,i i(, ), 7p^7q^7r------- -----m; 7p^7q^r ------- -----m; 7p^q^7r ------- -----2m2; 7p^q^r ------- -----3m3; p^7q^7r ------- -----4m4; p^7q^r ------- -----5m5; p^q^7r ------- -----6m6; p^q^r ------- -----7m7; 55
An,A,A.. A BB^<=>(B^p)V(B^7p) 56
An,A A2 n An,A A An,A A 57
((pvq)r) p ((pvq)r) p 7((pVq)->r)vp 7(7(pvq)vr)vp ((pvq)^7r)vp(p^7r)v(q^7r)vp (p^q^7r)v(p^7q^7r)v(p^q^7r)v(7p^q^7r) v(p^q^r)v(p^7q^r)v(p^q^7r)v(p^7q^7r) (p^q^7r)v(p^7q^7r)v(7p^q^7r)v(p^q^r)v(p^7q^r) m6 v m4 v m2 vm7 vm5 (2,4,5,6,7) 58
A<=>((pvq)r) p(2,4,5,6,7) p q r A B B(7p7q 7r) V (7pq 7r) V (7pq r) V (p7q r) V (pq 7r) (,2,3,5,6) 59
n,,,i i (, ), pvqvr------------m; pvqv7r----------m; pv7qvr----------2m2; pv7qv7r--------3m3; 7pVqVr----------4M4; 7pVqV7r--------5M5; 7pV7qVr--------6M6; 7pV7qV7r ------7M7; 6
An,A,A.. A BBV<=>(BVp)^(BV7p) 6
An,A A An,A A2n An,A A 62
(p^q)vr (p^q)vr(pvr)^(qvr) (pvqvr)^(pv7qvr)^(pvqvr)^(7pvqvr) (pvqvr)^(pv7qvr)^(7pvqvr) M^M2^M4 (,2,4) 63
A(p^q)vr (,2,4) (,3,5,6,7) p q r A B B(pVqV7r) (,4,7) (7pVqVr)(7pV7qV7r) (7p7q 7r) V (7pq 7r) V (7pq r) V (p7q r) V (pq 7r) (,2,3,5,6) 64
7mi Mi. AmVmVm5Vm7 (,,5,7) 7MV7MV7M5V7M7 7(M^M^M5^M7) (M2^M3^M4^M6) (2,3,4,6) 65
. 2. 3. 4. 5. 6. 66
,,. 67
(A ^A 2^ ^A K )B, A ^A 2^ ^A K B, BA ^A 2^ ^A K, (A ^A 2^ ^A K )B A ^A 2^ ^A K B. 68
() A<=>B A B A=>B AB (A ^A 2^ ^A K ) B (A^A2^ ^AK)B 69
(),.,. p:; q: p->7q, p 7q. : ((p->7q)^p) -> 7q : () (2) (3) 7
(2) ((p->7q)^p)->7q ((7pv7q)^p)->7q ((p^7p)v(p^7q))->7q (p^7q)->7q 7(p^7q)v7q 7pvqv7q ((p->7q)^p)->7qm vm vm 2 vm 3 (,,2,3) 7
A=>(AVB) AB=>A ((AB)A)=>B ((AB)7B)=>7A ((AVB)7A)=>B ((AB)(BC))=> (AC) ((A B)(B C))=> (A C) (AB)(CD)(AVC)=> (BVD) 72
A =AVB A^B =A A->B,7B =7A AVB,7B =A A->B,7B =7A A->B,C->D,AVB =BVD A,B =A^B 73
: pvq,p->7r, s->t, 7s->r, 7t : q : () s->t,t (2) 7t 7t (3) 7s 7s (4) 7s->r (5) r (6) p->7r (7) 7p (8) pvq (9) q 74
(A ^A 2^ ^A K )(AB) 7(A ^A 2^ ^A K )V(7AVB) (7(A ^A 2^ ^A K ) V7A)VB 7 (A ^A 2^ ^A K^A)VB (A ^A 2^ ^A K^A)B (A ^A 2^ ^A K )C C(AB)(CA)B 75
:p->(q->r),7svp,q :s->r :() 7svp (2) s (3) p (4) p->(q->r) (5) q->r (6) q (7) r 76
A ^A 2^^A K,A,A 2,,A K. (A ^A 2^ ^A K ), A,A 2,,A K. (A ^A 2^A K- )->A K <=>7(A ^A 2^ ^A K ), A,A 2,,7A K A K A,A 2,,A K 7A K 77
A ^A 2^^A K- <=>B B: A K : BA K <=>7(B^7A K => B^7A K <=> BA K => B^7A K <=> <=>7A K^B => 7A K 7B 78
:p->(7(r^s)->7q)p7s :7q ()p->(7(r^s)->7q) (2)p (3)7(r^s)->7q (4)7(7q) (5)(r^s) (6)7s (7)s (8)s^7s 79
End of Chpt.. 8