18 5 10

1 1 1.1 1.1.1 P Q P Q, P, Q P Q P Q P Q, P, Q

2 1 1.1.2 P.Q T F Z R 0 1 x, y x + y x y x y = y x x (y z) = (x y) z x + y = y + x x + (y + z) = (x + y) + z P.Q V = {T, F } V P.Q P.Q T F T F 1.1.3 1. (a)...

1.1. 3 (b)............ (c)...... 2. P, Q, R etc. 3. (a) ( ), { } x + y z + y z x x + y (x + y) z,,, ( ), { } (b) P, Q, etc. x + y z φ(x, y) φ X Y P(X, Y ) P 1.1.4 x + y + z x + y z

4 1 x + y + x + y z x, y +, X, Y,,, (1) T F (2) (3) A A (4) A, B A B, A B, A B T F A, B X, Y. T, F (1), (2) X, Y X, Y (3) X, Y, T, F, X, Y X X, X X X T, X F, X Y, X Y (4) X, Y, T, F, X, Y X X,, X Y, X Y (4) X Y X Y X Y X Y (4) L 1.1.5 Z + 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 2, V = {T, F },,, V 1. X X X T, F F, T

1.1. 5 X X T F F T 2. () X, Y T ( ) X Y T X Y X Y T T T T F F F T F F F F 3. ( ) X, Y T X Y T X Y X Y T T T T F T F T T F F F 4. ( ) X, Y X Y X, Y X Y X Y X Y X Y Y X Y T T F F T F T T F T F F F F T F

6 1 X Y X Y X Y X Y T T T T F F F T T F F T X Y X Y X Y X Y X Y T, F 1.1.6 1.1.5 V = {T, F },,, (true) T (false) F V = {T, F } V X, Y,..., A, B,... 2 n X, Y X Y V V V = {(T, T ), (T, F ), (F, T ), (F, F )} 1.1.5 V φ : (X, Y ) V V X Y V (1.1) (T, T ) T (T, F ) F (F, T ) F (F, F ) F (X 1, X 2,..., X n ) P(X 1, X 2,..., X n ) φ : (X 1, X 2,..., X n ) V n P(X 1, X 2,..., X n ) V (1.2) (X 1, X 2,..., X n ) 2 n, φ(x 1, X 2,..., X n ) T F φ 2 2n

1.1. 7 1. X φ 1 (X) φ 2 (X) φ 3 (X) φ 4 (X) T T T F F F T F T F T X X F (a) φ 1 (X) X T T (b) φ 2 (X) X (c) φ 3 (X) X X (d) φ 4 (X) X F F 2. X Y φ 1 (X, Y ) φ 2 (X, Y ) φ 3 (X, Y ) φ 4 (X, Y ) T T T T T T T F T T T T F T T T F F F F T F T F T X Y X X Y φ 5 (X, Y ) φ 6 (X, Y ) φ 7 (X, Y ) φ 8 (X, Y ) T T T T T T T F F F F F F T T T F F F F T F T F X Y Y X Y X Y X Y φ 9 (X, Y ) φ 10 (X, Y ) φ 11 (X, Y ) φ 12 (X, Y ) T T F F F F T F T T T T F T T T F F F F T F T F Y X Y φ 13 (X, Y ) φ 14 (X, Y ) φ 15 (X, Y ) φ 16 (X, Y ) T T F F F F T F F F F F F T T T F F F F T F T F X F

8 1 1.1.7 1.1.5,,, 1.1.6 1.1.5 X X, X (Y Z) T ( ) 1.1 (i) T F, F T (ii) X T X, X F X (iii) X F F, X T T φ 10 (iv) X X (v) X X X, X X X (vi) X X F, X X T (vii) X Y Y X, X Y Y X (viii) X (Y Z) (X Y ) Z (ix) X (Y Z) (X Y ) Z (x) X (X Y ) X (xi) X (X Y ) X (xii) X (Y Z) (X Y ) (X Z) (xiii) X (Y Z) (X Y ) (X Z) (xiv) (X Y ) X Y (xv) (X Y ) X Y (xvi) X Y X Y X Y X Y X Y X Y X Y T T T T T T T F F T F F F T F T T F F F F F T T X Y X, Y T (xii) (xv) (xvi)

1.1. 9 X Y, (X Y ), X, Y X Y X Y (X Y ) X (X Y ) X Y X Y X Y X Y T T T F T F F F T T T F F T T F T T F F F T F T F T F T T T F F F T F T T T T T (10) X (X Y ) X (14) (X Y ) X Y (16) X Y X Y 1.1.8 (x + y) 2 = x 2 + 2x y + y 2 x, y = = X 1, X 2,..., X n P(X 1, X 2,..., X n ), Q(X 1, X 2,..., X n ) φ : (Y 1, Y 2,..., Y n ) V n P(Y 1, Y 2,..., Y n ) V (1.3) η : (Y 1, Y 2,..., Y n ) V n Q(Y 1, Y 2,..., Y n ) V (1.4) (Y 1, Y 2,..., Y n ) V n V = {T, F } φ(y 1, Y 2,..., Y n ) = η(y 1, Y 2,..., Y n ) (1.5) = P(X 1, X 2,..., X n ) = Q(X 1, X 2,..., X n ) (1.6) 1.1 = 1.2 (i) T = F, F = T (ii) X T = X, X F = X (iii) X F = F, X T = T

10 1 (iv) X = X (v) X X = X, X X = X (vi) X X = F, X X = T (vii) X Y = Y X, X Y = Y X (viii) X (Y Z) = (X Y ) Z (ix) X (Y Z) = (X Y ) Z (x) X (X Y ) = X (xi) X (X Y ) = X (xii) X (Y Z) = (X Y ) (X Z) (xiii) X (Y Z) = (X Y ) (X Z) (xiv) (X Y ) = X Y (xv) (X Y ) = X Y (xvi) X Y = X Y 1.2 1.2.1 X Y X (X Y ) φ(x) = (φ(t ) X) (φ(f ) X) (1.7) φ(x) = (φ(t ) X) (φ(f ) X) (1.8) (1.7) X T φ(t ) 1.2 (φ(t ) T ) (φ(f ) T ) = (φ(t )) (φ(f ) F ) = φ(t ) (F ) = φ(t ) X F φ(f ) 1.2 (φ(t ) F ) (φ(f ) F ) = (F ) (φ(f ) T ) = φ(f ) (1.8) (1.7) η(x, Y ) = (η(t, Y ) X) (η(f, Y ) X) = ((η(t, T ) Y ) X) ((η(t, F ) Y ) X) ((η(f, T ) Y ) X) ((η(f, F ) Y ) X) = (η(t, T ) X Y ) (η(t, F ) X Y ) (η(f, T ) X Y ) (η(f, F ) X Y ) (1.9)

1.2. 11 (1.8) η(x, Y ) = {η(t, Y ) X} {η(f, Y ) X} (1.10) 1.2 X (Y Z) = (X Y ) (X Z) η(t, Y ) X = {(η(t, T ) Y ) (η(t, F ) Y )} X = (η(t, T ) X Y ) (η(t, F ) X Y ) (1.11) η(f, Y ) X = {(η(f, T ) Y ) (η(f, F ) Y )} X = (η(f, T ) X Y ) (η(f, F ) X Y ) (1.12) η(x, Y ) = (η(t, T ) X Y ) (η(t, F ) X Y ) (η(f, T ) X Y ) (η(f, F ) X Y ) (1.13) 2.1 φ(x) = (φ(t ) X) (φ(f ) X) φ(x) = (φ(t ) X) (φ(f ) X) η(x, Y ) = (η(t, T ) X Y ) (η(t, F ) X Y ) (η(f, T ) X Y ) (η(f, F ) X Y ) η(x, Y ) = (η(t, T ) X Y ) (η(t, F ) X Y ) (η(f, T ) X Y ) (η(f, F ) X Y ) 1.2.2 2 φ(t ), φ(f ), η(t, T ), η(t, F ), η(f, T ), η(f, F ) V = {T, F } X Y η(x, Y ) X Y η(x, Y ) T T v 1 X Y T F v 2 X Y F T v 3 X Y F F v 4 X Y

12 1 η(x, Y ) v i (i = 1, 2, 3, 4) T X Y η(x, Y ) X Y η(x, Y ) T T v 1 X Y T F v 2 X Y F T v 3 X Y F F v 4 X Y η(x, Y ) v i (i = 1, 2, 3, 4) F X Y F (X, Y ) G(X, Y ) T T F F X Y X Y T F T T X Y X Y F T T T X Y X Y F F T F X Y X Y 2 F, G -: [F -] 1 1 X Y X Y [G -] 1,4 1 X Y 4 X Y (X Y ) ( X Y ) [F -] 2,3,4 2 X Y 4 X Y 3 X Y ( X Y ) ( X Y ) (X Y )

1.2. 13 [G -] 2,3 2 X Y 3 X Y ( X Y ) (X Y ) 1.2.3 n n 2 n X 1 X 2 X n X i (1.14) X 1 X 2 X n i=1 n X i (1.15) 2.2 φ n {(X 1,, X n ) φ(x 1,, X n ) = F } {( σ k1, σ k2,, σ kn ) k = 1, 2,, m} { X i, σ ki = F σ ki (X i ) = (i = 1, 2,, m) X i σ ki = T φ (principal disjunctivenormal form) m n φ(x 1,, X n ) = ( σ ki (X i )) (1.16) k=1 i=1 i=1 φ m n φ(x 1,, X n ) = ( σ ki (X i )) K=1 i=1 2.2 : 2.3 φ n {(X 1,, X n ) φ(x 1,, X n ) = T } {(σ k1, σ k2,, σ kn k = 1, 2,, m} { X i, σ ki = T σ ki (X i ) = (i = 1, 2,, m) X i σ ki = F φ - (principal conjunctive normal form) m n φ(x 1,, X n ) = ( σ ki (X i )) k=1 i=1

14 1 - φ - φ X 1 X n φ(x 1,, X n ) T T v 1.. F F v 2 n (1) φ v j F m (2) (1) i = 1, 2,, n X i T X i F X i K = 1, 2,, m n i=1 σ ki(x i ) (3) (2) m m K=1 ( n i=1 σ ki(x i )) m n φ(x 1,, X n ) = ( σ ki (X i )) K=1 i=1

1.2. 15 3,, 2.2, 2.3 4 (1), (2), (1) 1.2 X Y = (X Y ) = ( X Y ) 4 (1) (2) 2.2 X Y = (X Y ) = ( X Y ) 4 (2)

16 1 1.2.4 X X T F X Y X Y X Y X Y X Y T T T T T T T F F T F F F T F T T F F F F F T T F T (negation) X X (conjunction) X Y X Y (disjunction) X Y X Y (implication) X Y X Y (equivalence) X Y X Y

1.2. 17 3, 4,,, 1.1.5 (1) T F (2) (3) A A (4) A, B A B, A B, A B

19 2 2.1 L g L g (n 2 n ) () L g () ( ) T (tautology)

20 2 2.1.1 ABC 1. BC C D 2. C AB A E CE 3. CAB = ACE ABC = ECD ABC + BCA + CAB = ECD + BCA + ACE ECD + BCA + ACE = 180 ABC + BCA + CAB = 180 1. AB CE 2. AB CE ABC = ECD 3. AB CE CAB = ACE 4. ECD + BCA + ACE = 180 5. ( ABC = ECD CAB = ACE) ( ABC + BCA + CAB = ECD + BCA + ACE) 6. ABC + BCA + CAB = 180 1, 2, 3, 4 ABC 6

2.1. 21 L g α α, A, B, C L g α (1) (15) (1) A A (2) (A B) [(B C) (A C)] (3) (A B) B (4) (A B) A (5) (A C) {(B C) [(A B) C]} (6) B (A B) (7) A (A B) (8) (C A) {(C B) [C (A B)]} (9) [A (A B)] B (10) [(A B) F ] (B A) (11) (A A) F (12) [(A C) B] [C (A B)] (13) A T (14) F A (15) A A 2.1.2 (modus ponens) A A B B L g (provable formula) : (1) (2) A A B ( ) B

22 2 A 1, A 2,, A n A k (1) (2) A k A i A j (i, j < k) A j A i A k B A A, (B A A) (A (B A)), A (B A) 1 = 2 2 > 6 2 > 6 (1 = 2 2 > 6 2 > 6) (2 > 6 (1 = 2 2 > 6)) 2 > 6 (1 = 2 2 > 6) A () B A A B A A (B A) A B A A, B A B C () (C C) A, (C C) B (8) {(C C) A} [{(C C) B} {(C C) (A B)}] (C C) (A B) (15) C C A B

2.1. 23 A B () B A A B () B A (2) A A, A A A B, B C () A C (A B) [(B C) (A C)] A B (B C) (A C) B C A C A (B C), A B () A C (8) (9) (A B) [{A (B C)} {A B (B C)}] A (B C), A B A {B (B C)} {B (B C)} C A C H H = α + L g A H α A A H A A B B H A H A B H B H

24 2 2.1.3 5() H α A 1, A 2,, A n B A 1, A 2,, A n H B A 1, A 2,, A n 1 H A n B A 1, A 2,, A n 2 H A n 1 (A n B) H α A 1, A 2,, A n H 1 H α H 2 A 1, A 2,, A n 1 A 1, A 2,, A n H B H 1 B H 2 A n B B 1, B 2,, B n 1, B n (= B) H 1 B k (1) B i A 1, A 2,, A n (2) B i H (3) B k B i B j (i, j < k) B j B i B k A n B 1, A n B 2,, A n B n 1, A n B n H 2 A n B k

2.1. 25 (1) B k A 1, A 2,, A n 1 A B A ( ) A n B k H 2 B k A n (1) H 2 A n A n (2) B k H A n B k H 2 (3) B k B i B j (i, j < k) B j B i B k A n B k A n B i A n (B i B k ) A n B k A (B C), A B () A C (14) (1) A H F H A A H F H A F H A A A A (5) ( A A) [(A A) {(A A) A}] (15) A A H A A F A ( A F ) A H

26 2 A B B A () H A B H B H 1 A A H 2 H 2 A H 2 B H 2 F H 1 B A 2.1.4 G L g G = {A 1, A 2, A 3,, A n } A 1, A 2,, A n H B G H B B G H B G H B G B G 6 G H G C G H C G B G H B G H B (11) C H (B B) C B B C 2.1.5 H : 7() H A A

2.1. 27 H A A H A H A 1, A 2,, A n A A n A i (1) A i (2) A i A j A k (j, k < i) A k A j A i i A j A j A i T A j A i A j A i F F T F T T T F F T T T A i T A i 8( ) H H A H A H 6 C C 7 H X 9( ) A H A (completeness theorem) H A A A H A A X 1, X 2,, X n X 1, X 2,, X n T, F A T δx 1, δx 2, δx n H A A F δx 1, δx 2, δx n H A δx i X i T X i F X i 9 A X 1, X 2,, X n X 1, X 2,, X n T, F A T

28 2 X 1, X 2,, X n T, F 2 n [ ] X 1, X 2,, X n H A X 1, X 2,, X n H A X 1, X 2,, X n 1, X n H A X 1, X 2,, X n 1, X n H A X 1, X 2,, X n H A X 1, X 2,, X n 1 X n A X 1, X 2,, X n 1 X n A (5) (15) X 1, X 2,, X n 1 H A X n X n 2 n 1 2 n 1 X n 1 X n 1 2 n 2 X 1, X 2,, X n H A A X X T X H X F X H X (1) A B, B C, B C B, C A B A T B F δx 1, δx 2,, δx n H B A F B T δx 1, δx 2,, δx n H B B, B A A

2.1. 29 A B C A T B, C T B T δx 1, δx 2,, δx n H B (3) B B C δx 1, δx 2,, δx n H B C B C A A F B, C F δx 1, δx 2,, δx n H B δx 1, δx 2,, δx n H C P, Q P Q δx 1, δx 2,, δx n H B C B C A(= (B C)) A B C A T B, C T δx 1, δx 2,, δx n H B δx 1, δx 2,, δx n H C P, Q P Q δx 1, δx 2,, δx n H B C B C A A F B, C F B F δx 1, δx 2,, δx n H B (3) P P Q δx 1, δx 2,, δx n H B C B C A = (B C)

31 3 3.1 1 = 1 2 > 1 x x = x T V (predicatelogic) P, Q P Q, P, Q x x x Is human(x), Is motal(x) Is human, Is motal(x) x Is human( ), Is motal( )

32 3 T ( ) Is human( ), Is motal( ) F ( ) T ( ) Is human(x) Is motal(x) x Is human(x) Is motal(x) T ( ) Is human(x), Is motal(x) x 3.1.1 n V = {T, F } D Z R D D V = {T, F } P : x D P(x) V (3.1) 1 (propositional function) P(x) (predicate) D D = D 1 D 2 D n (= {(x 1, x 2,, x n ) x i D i }) P n n D P (objectdomain) D P P D P(x) D N prime : x N prime(x) V prime(x) = { T F x x N (1 ) prime(x) N (1 ) x prime(x) x T, F P : x D P(x) V P(x) x( D) P

3.1. 33 3.1.2 P : x D P(x) V) x D P(x) V (3.2) P(x) P, Q : D V x D P(x) Q(x) V (3.3) P(x) Q(x) P(x) Q(x), P(x) Q(x), P(x) Q(x) (3.4),,,, 2.2 n k (k < n) (n k) x, y x < y 2 x < 2π 1 3 < 2π 0

34 3 3.1.3 P P : x D P(x) V (3.5), ( x)(p(x)) P(x) (3.6) x D ( x D)(P(x)) D, D = {a 1, a 2, a 3,, a n } (3.7) P(x) = P(a 1 ) P(a 2 ) P(a n ) (3.8) x D,D D x P(x) { T x D P(x) = T ( x D)(P(x)) = (3.9) F P(x) = F x D D D ( x)(p(x)) () x P(x) P(x) (3.10) x D ( x)(p(x)) ( x D)(P(x)) D, D = {a 1, a 2, a 3,, a n } (3.11) P(x) = P(a 1 ) P(a 2 ) P(a n ) (3.12) x D,D D x P(x) { T P(x) = T x D ( x D)(P(x)) = (3.13) F x D P(x) = F D D ( x)(p(x)) P(x) x P(x) 1 ( x)(p(x)), ( x)(p(x)) 0 2 1 P(x, y)

3.1. 35 y 1 ( x D)(P(x, y)) = P(x, y) (3.14) x D ( x D)(P(x, y)) = P(x, y) (3.15) x D D D = {a 1, a 2, a 3,, a n } P(x, y) = P(a 1, y) P(a 2, y) P(a n, y) (3.16) x D P(x, y) = P(a 1, y) P(a 2, y) P(a n, y) (3.17) x D P(a i, y) i = 1, 2,, n y 1 1 n (n 1) n P(x 1,, x i 1, x i, x i+1,, x n ) (3.18) x i ( x i D)(P(x 1,, x i 1, x i, x i+1,, x n )) = P(x 1,, x i 1, x i, x i+1,, x n ) (3.19) x i D ( x i D)(P(x 1,, x i 1, x i, x i+1,, x n )) = x i D P(x 1,, x i 1, x i, x i+1,, x n ) (3.20) x 1,, x i 1, x i, x i+1,, x n n 1 (universal quantifire) (existential quantifire)

36 3 3.1.4 ( x D)(P(x, y)) = P(x, y) (3.21) x D ( x D)(P(x, y)) = P(x, y) (3.22) x D D D = {a 1, a 2, a 3,, a n } ( x D)(P(x, y)) = P(a 1, y) P(a 2, y) P(a n, y) (3.23) ( x D)(P(x, y)) = P(a 1, y) P(a 2, y) P(a n, y) (3.24) x (bound variable) y (free variable) x < y 2 ( x)(x < y) ( x)(x < y) 1 ( y)( x)(x < y), ( y)( x)(x < y), ( y)( x)(x < y), ( y)( x)(x < y) 0 3.1.5 D D = {a 1, a 2, a 3,, a n } (3.25) b D a 1, a 2, a 3,, a n b = a k 1.2 ( x D)(P(x)) P(a k ) ( ) = P(a 1 ) P(a 2 ) P(a k ) P(a n ) P(a k ) = P(a 1 ) P(a 2 ) P(a k ) P(a n ) P(a k ) = P(a 1 ) P(a 2 ) P(a k 1 ) P(a k+1 ) P(a n ) P(a k ) P(a k ) = P(a 1 ) P(a 2 ) P(a k 1 ) P(a k+1 ) P(a n ) T = T (3.26) b D ( x)(p(x)) P(b) (3.27)

3.1. 37 i A P(a i ) A ( x D)(P(x)) ( ) = A P(a 1 ) P(a 2 ) P(a n ) = ( A P(a 1 )) ( A P(a 2 )) ( A P(a n )) = T T = T (3.28) b D A P(b) A ( x)(p(x)) 1.2 ( A ) (3.29) P(a k ) ( x D)(P(x)) ( ) = P(a k ) P(a 1 ) P(a 2 ) P(a k ) P(a n ) = P(a k ) P(a 1 ) P(a 2 ) P(a k ) P(a n ) = P(a k ) P(a k ) P(a 1 ) P(a 2 ) P(a k 1 ) P(a k+1 ) P(a n ) = T P(a 1 ) P(a 2 ) P(a k 1 ) P(a k+1 ) P(a n ) = T (3.30) a D P(a) ( x)(p(x)) (3.31) k P(a k ) A 1.2 ( x)(p(x)) A ( ) = P(a 1 ) P(a 2 ) P(a 3 ) P(a n ) A ( ) = P(a 1 ) P(a 2 ) P(a 3 ) P(a n ) A = ( P(a 1 ) A) ( P(a 2 ) A) ( P(a 3 ) A) ( P(a n ) A) (3.32) = (P(a 1 ) A) (P(a 2 ) A) (P(a 3 ) A) (P(a n ) A) (3.33)

38 3, b D P(b) A (3.34) ( x)(p(x)) A ( A ) (3.35) P Q D ( x)[p(x) Q(x)] = (P(a 1 ) Q(a 1 )) (P(a n ) Q(a n )) = (P(a 1 ) P(a n )) (Q(a 1 ) Q(a n )) = ( x)(p(x)) ( x)(q(x)) (3.36) ( x)[p(x) Q(x)] = ( x)(p(x)) ( x)(q(x)) (3.37) ( x)[p(x) Q(x)] = (P(a 1 ) Q(a 1 )) (P(a n ) Q(a n )) = (P(a 1 ) P(a n )) (Q(a 1 ) Q(a n )) = ( x)(p(x)) ( x)(q(x)) (3.38) ( x)[p(x) Q(x)] = ( x)(p(x)) ( x)(q(x)) (3.39) 1 P(x) ( x)(p(x)) = (P(a 1 ) P(a n )) = ( y)(p(y)) (3.40) ( x)(p(x)) = P(a 1 ) P(a n ) = ( y)(p(y)) (3.41) 2 P(x, y), 1.2

3.1. 39 (k=n ) ( x)( y)(p(x, y)) = ( x) (P(x, a k ) = = k=1 i=n(k=n i=1 ) (P(a i, a k ) k=1 k=n(i=n k=1 ) (P(a i, a k ) i=1 = ( y)( x)(p(x, y)) (3.42) (k=n ) ( x)( y)(p(x, y)) = ( x) (P(x, a k ) = = k=1 i=n(k=n i=1 ) (P(a i, a k ) k=1 k=n(i=n k=1 ) (P(a i, a k ) i=1 = ( y)( x)(p(x, y)) (3.43), 1.2 [ ] ( x)(p(x)) ( x)(p(x)) = ( x)(p(x)) ( x)(p(x)) (i=n ) (i=n ) = (P(a i ) (P(a i ) i=1 i=1 (3.44) i=n ( ) = P(a i ) i=1 ) (P(a i ) (i=n i=1 i=n ( ) = P(a i ) P(a i ) i=1 i=n ( ) = T i=1 = T (3.45) ( x)(p(x)) ( x)(p(x)) (3.46) 1.2 ( y)( x)(p(x, y)) ( x)( y)(p(x, y)) = T

40 3 ( y)( x)(p(x, y)) ( x)( y)(p(x, y)) (3.47),P Q D, ( x)[p(x) Q(x)] [( x)(p(x)) ( x)(q(x))] (3.48) ( x)[p(x) Q(x)] [( x)(p(x)) ( x)(q(x))] (3.49) D 7 (i) D P (a) a D ( x)(p(x)) P(a) (b) a D A P(a) A ( x)(p(x)) ( A ) (c) a D (d) a D P(a) ( x)(p(x)) P(a) A ( x)(p(x)) A ( A ) (ii) D P Q (a) ( x)[p(x) Q(x)] ( x)(p(x)) ( x)(q(x)) ( x)[p(x) Q(x)] = ( x)(p(x)) ( x)(q(x))

3.1. 41 (b) ( x)[p(x) Q(x)] ( x)(p(x)) ( x)(q(x)) ( x)[p(x) Q(x)] = ( x)(p(x)) ( x)(q(x)) 1 (iii) (a) Q Q G Q G Q = G ( x)(p(x)) = ( y)(p(y)) ( x)(p(x)) = ( y)(p(y)) (b) Q G Q G Q = G ( x)( y)(p(x, y)) = ( y)( x)(p(x, y)), ( x)( y)(p(x, y)) = ( y)( x)(p(x, y)) (c) (d) ( x)(p(x)) ( x)(p(x)) ( y)( x)(p(x, y)) ( x)( y)(p(x, y)) (iv) D P, Q (a) ( x)[p(x) Q(x)] [( x)(p(x)) ( x)(q(x))] (b) ( x)[p(x) Q(x)] [( x)(p(x)) ( x)(q(x))] 2 1 ( x)[p(x) Q(x)] ( x)(p(x)) ( x)(q(x)), ( x)[p(x) Q(x)] ( x)(p(x)) ( x)(q(x))!! 2 (iii) (c),(d), (iv) (a),(b)!!

42 3 3.1.6 prenex normal form D D = {a 1, a 2, a 3,, a n } D 8.1 1.2 ( x)(p(x)) = (P(a 1 ) P(a n )) = P(a 1 ) P(a 1 ) P(a n ) = ( x)( P(x)) (3.50) ( x)(p(x)) = (P(a 1 ) P(a n )) = P(a 1 ) P(a 1 ) P(a n ) = ( x)( P(x)) (3.51), P(x) x B 1.2 8.2 8.2 B ( x)(p(x)) = B (P(a 1 ) P(a n )) = (B P(a 1 )) (B P(a n )) = ( x)(b P(x)) (3.52) B ( x)(p(x)) = B (P(a 1 ) P(a n )) = (B P(a 1 )) (B P(a n )) = ( x)(b P(x)) (3.53) B ( x)(p(x)) = B (P(a 1 ) P(a n )) = (B P(a 1 )) (B P(a n )) = ( x)(b P(x)) (3.54)

3.1. 43 B ( x)(p(x)) = B (P(a 1 ) P(a n )) = (B P(a 1 )) (B P(a n )) = ( x)(b P(x)) (3.55) B ( x)(p(x)) B ( x)(p(x)) = B ( x)(p(x)) = B (P(a 1 ) P(a n )) = ( B P(a 1 )) ( B P(a n )) = ( x)( B P(x)) = ( x)(b P(x)) (3.56) = B ( x)(p(x)) = B (P(a 1 ) P(a n )) = ( B P(a 1 )) ( B P(a n )) = ( x)( B P(x)) = ( x)(b P(x)) (3.57) ( x)(p(x)) B = ( x)(p(x)) B = (P(a 1 ) P(a n )) B = P(a 1 ) P(a 1 ) P(a n ) B = ( P(a 1 ) B) ( P(a 1 ) B) ( P(a n ) B) = ( x)( P(x) B) = ( x)(p(x) B) (3.58) ( x)(p(x)) B = ( x)(p(x)) B = (P(a 1 ) P(a n )) B = ( P(a 1 ) P(a 1 ) P(a n )) B = ( P(a 1 ) B) ( P(a 1 ) B) ( P(a n ) B) = ( x)( P(x) B) = ( x)(p(x) B) (3.59) 7 8.1, 8.2 1. ( x)(p(x, y)) ( y)(q(y, z)) = ( x)(p(x, y)) ( w)(q(w, z)) 2. 8.1( ) ( ) = ( ) ( ) = ( )

44 3 3., 8.2 ( ) = ( ) ( ) = ( ) ( )

45 4 4.1 V = {T, F } D D D = Z D = C x > y D sin(x) = y cos(sin(x)) = y D D D 1. c 1, c 2, 0 φ 0 1, φ 0 2, 2. D x 1, x 2, 3. D D φ 1, φ 2, 1. 2. 3. φ n x 1, x 2,, x n φ(x 1, x 2,, x n ) 4.1.1, 1. (a)... (b)............ (c)...... (d)............ (e)...... 2.

46 4 (a) ( ), { } (b) P, Q, etc. 4.1.2 1. P n s 1, s 2., s n P(s 1, s 2., s n ) 2. A A 3. A, B A B, A B, A B 4. C(a) a a x C(a) ( x)(c(x)), ( x)(c(x)) T F A, B, C 4.1.3 L g D A, B, C L g β L g (1) (17) 1. A A 2. (A B) [(B C) (A C)] 3. (A B) B 4. (A B) A 5. (A C) {(B C) [(A B) C]} 6. B (A B) 7. A (A B) 8. (C A) {(C B) [C (A B)]}

4.1. 47 9. [A (A B)] B 10. [(A B) ]F (B A) 11. (A A) F 12. [(A C) B] [C (A B)] 13. A T 14. F A 15. A A 16. ( x)(a(x)) A(s) ( s ) 17. A(s) ( x)(a(x)) ( s ) 4.1.4 (modus ponens), 1. A A B B 2. A B(a) A ( a)(b(a)) a A 3. A(a) B ( a)(a(a)) B a B 4.1.5 L g (provable formula) : 1. 2. A A B ( ) B 3. A B(a) ( ) A ( a)(b(a)) a A 4. A(a) B ( ) ( a)(a(a)) B a B

48 4 A 1, A 2,, A n A k 1. 2. A k A i A j (i, j < k) A j A i A k 3. A k A j (j < k) A j A B(a) (4.1) A k A ( a)(b(a)) (4.2) a A 4. A k A j (j < k) A j A(a) B (4.3) A k ( a)(a(a)) B (4.4) a B 4.1.6 A () B A A B A ( ) A, B A B A B B A A B B A () () () A A, A A ( ) A B, B C () A C

4.1. 49 A (B C), A B () A C H H = β +L g A H β A A H A H A A B B H A B H B H 4.1.7 10( 1) H β A 1, A 2,, A n B A 1, A 2,, A n H B (4.5) A 1, A 2,, A n 1 H ( a 1 )( a 2 ) ( a l )A n B (4.6) a 1, a 2 a l A n H A 1, A 2,, A n H 1 H H 2 A 1, A 2,, A n 1 A 1, A 2,, A n H B (4.7) H 1 B (4.8) a 1, a 2 a l A n H 2 ( a 1 )( a 2 ) ( a l )A n B (4.9) B 1, B 2,, B n 1, B n (= B) (4.10) H 1 B k

50 4 (1) B k A 1, A 2,, A n (2) B k H (3) B k B i B j (i, j < k) B j B i B k (4) B k B j (j < k) B j C D(b) (4.11) B k C ( b)(d(b)) (4.12) b C (5) B k B j (j < k) B j C(b) D (4.13) B k ( b)(c(b)) D (4.14) b D ( a 1 )( a 2 ) ( a l )A n (4.15) Ā n (4.16) Ā n B 1, Ān B 2,, Ān B n 1, Ān B n (4.17) H 2 Ān B (1) B k A 1, A 2,, A n 1 A 1, A 2,, A n 1 H 2 C D C ( ) Ān B k H 2 B k A n (16) l ( a 1 )( a 2 ) ( a l )A n ( a 2 ) ( a l )A n (a 1, a 2,, a l ) ( a 2 ) ( a l )A n ( a 3 ) ( a l )A n (a 1, a 2, a 3,, a l ) ( a 3 ) ( a l )A n ( a 4 ) ( a l )A n (a 1, a 2, a 3, a 4,, a l ) ( a l )A n A n (a 1, a 2,, a l ) (4.18) Ā n A n (4.19) H 2

4.1. 51 (2) B k H Ān B k H 2 (3) B k B i B j (i, j < k) B j B i B k Ā n B k Ān B i Ān (B i B k ) A (B C), A B () A C Ān B k (4) B k B j (j < k) B j ] C D(b) (4.20) B k C ( b)(d(b)) (4.21) b C Ān B k Ān (C D(b)) Ān B k Ā n C ( b)(d(b)) (4.22) Ān (C D(b)) E F(c) E ( c)(f(c)) Ā n ( b)(c D(b)) (4.23) (16) ( b)(c D(b)) (C D(b)) (4.24) (C D(b)) (C ( b)(d(b))) (4.25) Ā n B k (4.26) (5) B k B j (j < k) B j C(b) D (4.27) B k ( b)(c(b)) D (4.28) b D Ān B k Ān (C(b) D) Ān B k Ā n (( b)(c(b)) D) (4.29)

52 4 (17) (C(b) D) (( b)(c(b)) D) (4.30) Ān (C(b) D) Ā n (( b)(c(b)) D) (4.31) 10 ( 2) a 1, a 2 a l A n a 1, a 2 a l A 1, A 2,, A n H B (4.32) A 1, A 2,, A n 1 H A n B (4.33) (10) H A 1, A 2,, A n H 1 H H 2 A 1, A 2,, A n 1 A 1, A 2,, A n H B (4.34) H 1 B (4.35) H 2 A n B (4.36) B 1, B 2,, B n 1, B n (= B) H 1 a 1, a 2 a l B k (1) B k A 1, A 2,, A n (2) B k H

4.1. 53 (3) B k B i B j (i, j < k) B j B i B k (4) B k B j (j < k) B j C(b) D (4.37) B k ( b)(c(b)) D (4.38) b D A n B 1, A 2 B 2,, A n B n 1, A n B n (4.39) H 2 A n B (1) B k A 1, A 2,, A n 1 A 1, A 2,, A n 1 H 2 C D C ( ) A n B k H 2 B k A n (1) A n A n H 2 (2) B k H A n B k H 2 (3) B k B i B j (i, j < k) B j B i B k A n B k A n B i A n (B i B k ) A (B C), A B () A C A n B k (4) B k B j (j < k) B j C(b) D (4.40) B k ( b)(c(b)) D (4.41) b D A n B k A n (C(b) D) A n B k A n (( b)(c(b)) D) (4.42) (17) (C(b) D) (( b)(c(b)) D) (4.43) A n (C(b) D) A n (( b)(c(b)) D) (4.44)

54 4 4.1.8 D D D D D 3.1 1. D α : x α(x) D (4.45) 2. D 0 D 3. D F n n F(D n, D) = {f f : (x 1, x 2,, x n ) D n φ(x 1, x 2,, x n ) D} (4.46) D φ n F n ρ(φ n ) F(D n, D) ρ(φ n ) : (x 1, x 2,, x n ) D ρ(φ n )(x 1, x 2,, x n ) (4.47) 4. D V = {T, F } P n n, P(D n ; V) = {P P : (y 1, y 2,, y n ) D n P (y 1, y 2,, y n ) V} (4.48) D n P n P n π(p n ) P(D n ; V) π(p n ) : (y 1, y 2,, y n ) D π(p n )(x 1, x 2,, x n ) V (4.49) M = (D, ρ, π) (4.50) M = (D, ρ, π) D α D τ(m, α)[ ]

4.1. 55 1. (a) x i τ(m, α)[x i ] = α(x i ) (4.51) (b) c i 0 τ(m, α)[c i ] = ρ(c i ) (4.52) (c) s 1, s 2,, s n,,d, α,n φ n τ(m, α)[s i ], i = 1,, n (4.53) τ(m, α)[φ n (s 1, s 2,, s n )] = ρ(φ n )(τ(m, α)[s 1 ], τ(m, α)[s 2 ],, τ(m, α)[s n ]) (4.54) 2. (a) P n n,s 1, s 2,, s n, τ(m, α)[p n (s 1, s 2,, s n )] = π(p n )(τ(m, α)[s 1 ], τ(m, α)[s 2 ],, τ(m, α)[s n ]) (4.55) (b) 0 P 0 V = {T, F } τ(m, α)[p 0 ] = π(p 0 ) V (4.56) (c) A τ(m, α)[ A] = τ[m, α](a) (4.57) (d) A, B τ(m, α)[a B] = τ(m, α)[a] τ(m, α)[b] (4.58) τ(m, α)(a B) = τ(m, α)(a) τ(m, α)(b) (4.59) (e) C(a) a a x C(a) τ(m, α)[( x)(c(x))] = τ(m, α)[c](b) (4.60) b D τ(m, α)[( x)(c(x))] = τ(m, α)[c](b) (4.61) D {a 1, a 2,, a n } C(x, y, z) b D

56 4 x, c τ(m, α)[x] = α(x) = a 1 τ(m, α)[c] = ρ(c) = a 3 (4.62) τ(m, α)[( z)(c(x, c, z))] = τ(m, α)[c(x, c, b))] b D = π(c)(τ(m, α)[x], τ(m, α)[c], b) b D = π(c)(a 1, a 3, a 1 ) π(c)(a 1, a 3, a 2 ) π(c)(a 1, a 3, a 3 ) π(c)(a 1, a 3, a n ) (4.63) τ(m, α)[( z)(c(x, c, z))] = τ(m, α)[c(x, c, b))] b D = π(c)(τ(m, α)[x], τ(m, α)[c], b) b D = π(c)(a 1, a 3, a 1 ) π(c)(a 1, a 3, a 2 ) π(c)(a 1, a 3, a 3 ) π(c)(a 1, a 3, a n ) (4.64) 4.1.9 G L G = {A 1, A 2, A 3,, } (4.65) 1. M = (D, ρ, π) D α τ(m, α)[a i ] = T, i = 1, 2,, (4.66) M = (D, ρ, π) α G = {A 1, A 2, A 3,, } τ(m, α)[g] = T (4.67) 2. M = (D, ρ, π) D α τ(m, α)[a i ] = T, i = 1, 2,, (4.68) M = (D, ρ, π) G = {A 1, A 2, A 3,, } τ(m)[g] = T (4.69)

4.1. 57 4.1.10 H : 10() A H ( H A) A M(D, ρ, π) τ(m))[a] = T (4.70) H A H A 1, A 2,, A n A A n A k k A i (1) (1) (15) (16),(17) M = (D, ρ, π) D α (16) τ(m, α)(t) = d D (4.71) τ(m, α)[( x)(c(x)) C(t)] = τ(m, α)[ ( x)(c(x)) C(t)] = τ(m, α)[( x)(c(x))] τ(m, α)[c(t)] = τ(m, α)[c](b) (τ(m, α)[c](d) b D (4.72) = { τ(m, α)[c](b)} τ(m, α)[c](d) b D = { τ(m, α)[c](b)} b D,b d τ(m, α)[c(d)] τ(m, α)[c](d) (4.73) = { τ(m, α)[c](b)} T b D,b d = T (4.74)

58 4 (17) τ(m, α)[c(t)] ( x)(c(x))] = τ(m, α)[c](d) τ(m, α)[( x)(c(x))] (4.75) = τ(m, α)[c](d) τ(m, α)[c](b) b D = τ(m, α)[c](d) τ(m, α)[c](d) τ(m, α)[c](b) b D,b d = T τ(m, α)[c](b) b D,b d = T (4.76) (2) A k A i A j (i, j < k) A j A i A k k A i A i A k T A i A k A i A k F F T F T T T F F T T T A k T A k (3) A k A j (j < k) A j C D(t) (4.77) A k C ( t)(d(t)) (4.78) t C,A k M, α τ(m, α)[c ( t)(d(t))] = τ(m, α)[c] τ(m, α)[c](b) b D = { τ(m, α)[c] τ(m, α)[c](b)} (4.79) b D, A j C D(t) (4.80) M, α b D τ(m, α)[(c D(t)] = T (4.81) τ(m, α)[( C D(b)] = T (4.82)

4.1. 59, { τ(m, α)[c] τ(m, α)[c](b)} = T (4.83) b D τ(m, α)[c ( t)(d(t))] = T (4.84) (4) A k A j (j < k) A j C(t) D (4.85) A k ( t)(c(t)) D (4.86) t D, τ(m, α)[( t)(c(t)) D] = τ(m, α)[c](b) D b D = τ(m, α)[c](b) D (4.87) b D = { τ(m, α)[c](b) D} b D = {τ(m, α)[c](b) D} (4.88) b D A j C(t) D (4.89) M, α b D τ(m, α)[c(t) D] = T (4.90) τ(m, α)[c(b) D] = T (4.91) {τ(m, α)[c](b) D} = T (4.92) b D τ(m, α)[( t)(c(t)) D] = T (4.93) 11( ) H H A H A (4.94)

60 4 H 6 C C M H 0 P 0 12( ) A H A (completeness theorem) H A M τ(m)[a] = T (4.95) A A H A Henkin Henkin G G M = (D, ρ, π), D α, G A τ(m, α)[a] = T (4.96) 12 M τ(m)[a] = T (4.97) τ(m)[ A] = T (4.98) M G = { A} Henkin, G = { A} 5, B A H B (4.99) 8 A H B (4.100) H A B (4.101) H A B (4.102) (5) H ( A) (B B) (4.103)

4.1. 61, (15) H ( B B) A (4.104) H B B (4.105) H A (4.106) Henkin G G 1. L C L {C} C L 2. C L L {A 1, A 2,, A n } L L H B, B (4.107) B B B P 1, P 2,, B (4.108) Q 1, Q 2,, B (4.109) L L B B L 3. L L L L 4. () 5. K H K 0 K K 0 L ( ) K K K 0 K n

62 4 (a) K n + 1 ( a)a(a) K n A v A(v) ( a)a(a) K n K n+1 K n+1 = K n {A(v) ( a)a(a)} (4.110) (b) K n + 1 ( a)a(a) K n+1 = K n (4.111) K 0 K 1 K n K n+1 (4.112) K n n n = 0 K 0 K n K n+1 B K n, A(v) ( a)a(a) H B, B (4.113) K n A w A(a) a w A(w) A(v) A(w) (4.114) [A(w) ( a)a(a)] [A(v) ( a)a(a)] (4.115) K n, A(w) ( a)a(a) H B, B (4.116) K n H ( w)(a(w) ( a)a(a)) B (4.117) K n H ( w)(a(w) ( a)a(a)) B (4.118) K n H ( w)(a(w) ( a)a(a)) (4.119) A(w) a ( w)(a(w) ( a)a(a)) ( w)( a)(a(w) A(a)) (4.120) K n H ( w)( a)(a(w) A(a)) (4.121) K n H ( a)(a(a) A(a)) (4.122) (1) K n H ( a)(a(a) A(a)) (4.123)

4.1. 63 K n K n+1 K 0 K 1 K n K n+1 (4.124) L = n= n=0 K n (4.125) L B B B L H B, B (4.126) P 1, P 2,, B (4.127) Q 1, Q 2,, B (4.128) L L B B L n K n K n 6. L (a) K A L A L A L L (b) K A L A L A L L L {A} B 8 L, A H B, B (4.129) L H A B B (4.130), (15) L H ( B B) A (4.131) L H B B (4.132) L H A (4.133) A L

64 4 (c) K A, B L A B L A, B L L H A B (4.134) A B L (d) K A, B L A B L A B L (6), (7) A, B L A, B L (e) K A, B L A B L (f) K A, B L A B L (g) a c c K A(c) L ( a)(a(a)) L ( a)a(a) K n + 1 K n+1 K n+1, A(w) ( a)a(a) (4.135) w K n A A(w) L ( a)a(a) (4.136) L L ( a)a(a) L (h) a c c K (A(c)) L ( a)(a(a)) L ( a)(a(a)) L (16) A(w) L H A(w) L (i) a c c K A(c) L ( a)(a(a)) L (j) a c c K A(c) L ( a)(a(a)) L 7. L H M = (D, ρ, π) (a) D (b) H D α (c) D ρ n F n D F(D n, D) = {f f : (x 1, x 2,, x n ) D n f(x 1, x 2,, x n ) D} (4.137)

4.1. 65 ρ φ n F n ρ(φ n )(x 1, x 2,, x n ) = φ n (x 1, x 2,, x n ), D n (4.138) ( ρ(φ n ) φ n D ) (d) H D V = {T, F } H n P n, D n P(D n ; V) = {P P : (y 1, y 2,, y n ) D n P (y 1, y 2,, y n ) V} (4.139) P n P n π(p n ) P(D n, V) (4.140) n P n (y 1, y 2,, y n ) D n Q(y 1, y 2,, y n ) { T P(y 1, y 2,, y n ) L = F P(y 1, y 2,, y n ) L Q H A A L τ(m, α)[a] = F (4.141) A L τ(m, α)[a] = T (4.142) A (a) A n P c 1, c 2,, c n P(c 1, c 2,, c n ) (4.143) L π P(c 1, c 2,, c n ) (4.144) τ(m, α)[a] = π(p)(c 1, c 2,, c n ) = T (4.145) L π τ(m, α)[a] = π(p)(c 1, c 2,, c n ) = F (4.146) (b) A B i. B L B L A τ(m, α)[b] = F (4.147) τ(m, α)[a] = τ(m, α)[b] = T (4.148)

66 4 ii. B L B L A τ(m, α)[b] = T (4.149) τ(m, α)[a] = τ(m, α)[b] = F (4.150) (c) A B C i. B C L B, C L A τ(m, α)[b] = T (4.151) τ(m, α)[c] = T (4.152) τ(m, α)[a] = τ(m, α)[b] τ(m, α)[c] = T (4.153) ii. B C L B, C L B L A τ(m, α)[b] = F (4.154) τ(m, α)[c] = F (4.155) τ(m, α)[a] = τ(m, α)[a] = τ(m, α)[b] τ(m, α)[c] = F (4.156) (d) A B C, B C = B C (4.157) (e) C(a) a a x C(a) A ( x)(c(x)) i. ( x)(c(x)) L L v C(v) L ρ(v) = v τ(m, α)[( x)(c(x))] = τ(m, α)[c(v)] = τ(m, α)[c](v) = T (4.158) b D τ(m, α)[c](b) = T (4.159) ii. ( x)(c(x)) L L v C(v) L τ(m, α)[c(v)] = τ(m, α)[c](v) = F (4.160) τ(m, α)[( x)(c(x))] = τ(m, α)[c](b) = F (4.161) b D

4.1. 67 (f) C(a) a a x C(a) A ( x)(c(x)) ( x)(c(x)) 8. K 0 L K 0 K, K 0 L (4.162) K 0 C τ(m, α)[c] = T (4.163) K 0 9. G K 0 (a) G (b) G K 0 G K 0 ( x)( y)p(x, y, z) (4.164) z c ( x)( y)p(x, y, c) (4.165) (c) K 0 K 0 G G

69 5 5.1 A 1, A 2,, A n B H A 1, A 2,, A n B A 1, A 2,, A n H B (5.1) A 1, A 2,, A n 1 H ( a 1 )( a 2 ) ( a l )A n B (5.2) a 1, a 2 a l A n A 1, A 2,, A n (5.3) A 1, A 2,, A n 1 H A n B (5.4) (6), (7) (A B) A (5.5) (A B) B (5.6) H A 1 A 2 A n 1 A n B (5.7) A 1, A 2,, A n H B (5.8) H A 1 A 2 A n 1 A n B (5.9) A 1 A 2 A n 1 A n B (5.10) A 1 A 2 A n 1 A n B (5.11)

70 5 5.1.1 A 1 A 2 A n 1 A n B (5.12) 1. 7 8.1, 8.2 (a) ( x)(p(x, y)) ( y)(q(y, z)) = ( x)(p(x, y)) ( w)(q(w, z)) (5.13) (b) 8.1( ) ( ) = ( ) ( ) = ( ) (c), 8.2 ( ) = ( ) ( ) = ( ) (d) ( ) 2. ( ) (a) D 2 P(x, y) ( x D)( y D)P(x, y) (5.14) D x P(x, y) D y x D y D f(x) ( x D)(P(x, f(x))) (5.15) f : x D f(x) D (5.16)

5.1. 71 (b) D 2 P(x, y) ( y D)( x D)P(x, y) (5.17) ( x D)P(x, c) (5.18) D c c c 0 f ( x D)P(x, f) (5.19) f 5.1.2 A 1, A 2,, A n A 1, A 2,, A n H B (5.20) A 1 A 2 A n 1 A n B (5.21) ( x 1 )( x 2 ) ( x n )(R 1 R 2 R m ) (5.22) x 1, x 2,, x n R 1 R 2 R m R i C1 i C2 i Ck(i) i (5.23) Cj i Pi j Pi j Pi j (literal) {(A B) (( A) C)} (B C) (5.24) A (A A) A B, A C B C

72 5 R 1 R 2 R m (5.25) (11) A, A (5.26) A A F (5.27) A, A R 1 R 2 R m (5.28) ( x 1 )( x 2 ) ( x n )(R 1 R 2 R m ) (5.29) M R 1, R 2,, R m (5.30) M D H R 1, R 2,, R m (5.31) Ω, F H { Ω, Ω φ H 1 = {a}, Ω = φ H i+1 = H i {g(s 1, s 2,, s n ) g F, s 1, s 2,, s n H i } (5.32)

73 A FAQ FAQ Q1, A1 Q2 A2 Q3 A3 A Q4 A E A4 for all x A ( x) there exists x E ( x) Q5 A5 Q6 ( ) (=) A6 R(X, Y ) P (X, Y ) R(X, Y ), P (X, Y ) R(X, Y ) = P (X, Y ) R = P = X,Y Q7 A7

74 A FAQ Q8 PQP Q Q P Q P A8 Q9 1.2.4 4, 5,,, 1.1.5 (X Y ) = X Y,,,, A9 Q10 2.1.1 n D D A10 Q11 A11 x > 2 x D = (a, b) ( x)(x > 2) = (a > 2 b > 2) Q12 Aj Aj Ai T A12 A j A j A i A i A i A j A j A i T A i T Q13 3.1.6 ()

75 A13 ( x)( y)((x > y) (y = 1) (= 2)) Q14 4.1.1 A A n A14 A A 1, A 2,..., A n,... A Q15 4.1.7 a 1, a 2,..., a l A n A15 A n, (a 1 = a 2 ) (a 3 > a 2