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3章 集合の演算

GAIRON-book : 2018/6/21(19:23). 33. 3 . , ( ) . , . , .. 2 A B . , A B = {x | x A x B}.. , A B . , A B = {x | x A x B} = {x | x A, x B}.. 2 A, B , A B = , A, B . , A B = A B . A B A B. : A B A B. GAIRON-book : 2018/6/21(19:23). 34 3 . , x A B (x A) (x B), ( ). x A B (x A) (x B) ( ).. , 2 A = {x | P (x)}, B =. {x | Q(x)} , A B = {x | P (x) Q(x)}, A B = {x | P (x) Q(x)}.. , A B = {x | P (x), Q(x)} . , .. , , .. ( ) A, B , A B = B A, A B = B A. ( ) A . A A = A, A A = A. ( ) A . A = A = A, A = A = . 3 , . ( ), ( ) .. A, B , A A B, A B A. A B A B , A .. , A A B . A B A B . , A .. , A B A . GAIRON-book : 2018/6/21(19:23). 35. A, B, C , . (1) A C B C A B C. (2) C A C B C A B. (1) a A B , a A b B . a A . , A C a C . , a B , B C . a C . , a C A B C . (2) a C , C A a A , C B a B.. , a A B , C A B . , A B A B , A B A B .. ( ) . , . , . , . A, B A B , A B = B, A B = A. 1 (2 ). , B A B.. A B . B B , A B B.. ( ) A, B.

GAIRON-book : 2018/6/21(19:23) 3.1. 和集合と積集合 35 定理3.5 集合A;B;Cに対して, 次が成り立つ. (1) AˆCかつBˆC)A[BˆC. (2) CˆAかつCˆB)CˆA\B. 証明 (1) a2A[Bとすると, a2Aまたはb2Bである. a2Aであれ ば, 仮定AˆCからa2Cである.また, a2Bであれば, 仮定BˆCから やはりa2Cである.いずれにせよ, a2CであるからA[BˆCが示された.

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Transcription of 3章 集合の演算

1 GAIRON-book : 2018/6/21(19:23). 33. 3 . , ( ) . , . , .. 2 A B . , A B = {x | x A x B}.. , A B . , A B = {x | x A x B} = {x | x A, x B}.. 2 A, B , A B = , A, B . , A B = A B . A B A B. : A B A B. GAIRON-book : 2018/6/21(19:23). 34 3 . , x A B (x A) (x B), ( ). x A B (x A) (x B) ( ).. , 2 A = {x | P (x)}, B =. {x | Q(x)} , A B = {x | P (x) Q(x)}, A B = {x | P (x) Q(x)}.. , A B = {x | P (x), Q(x)} . , .. , , .. ( ) A, B , A B = B A, A B = B A. ( ) A . A A = A, A A = A. ( ) A . A = A = A, A = A = . 3 , . ( ), ( ) .. A, B , A A B, A B A. A B A B , A .. , A A B . A B A B . , A .. , A B A . GAIRON-book : 2018/6/21(19:23). 35. A, B, C , . (1) A C B C A B C. (2) C A C B C A B. (1) a A B , a A b B . a A . , A C a C . , a B , B C . a C . , a C A B C . (2) a C , C A a A , C B a B.. , a A B , C A B . , A B A B , A B A B .. ( ) . , . , . , . A, B A B , A B = B, A B = A. 1 (2 ). , B A B.. A B . B B , A B B.. ( ) A, B.

2 A (A B) = A, A (A B) = A. 1 (2 ). , . A B A . , A (A B) = A .. ( ) A, B, C , . (A B) C = A (B C), (A B) C = A (B C). GAIRON-book : 2018/6/21(19:23). 36 3 . 1 (2 ). , . , x (A B) C (x A B) (x C). ((x A) (x B)) (x C) ( ).. , x A (B C) (x A) (x B C). (x A) ((x B) (x C)) ( ).. ( ) , ( ) ( ) . , (A B) C = A (B C) . , 3 A1 , A2 , A3 . , A1 (A2 A3 ) = (A1 A2 ) A3 ( ).. , ( ) , A1 A2 A3.. , 2 .. , n A1 , A2 , .. , An , .. n A1 A2 An = Ak ( ). k=1.. , A1 , A2 , .. , An .. , . n A1 A2 A3 , A1 A2 An = Ak ( ). k=1.. , ( ) 1 Ak . , ( ) Ak .. GAIRON-book : 2018/6/21(19:23). 37. ( ) A, B, C , . A (B C) = (A B) (A C), ( ). A (B C) = (A B) (A C). ( ). ( ) 2 ( ( ) ). ( ). , . , A (B C) (A B) (A C) . x A (B C) .. , 2 . (i) x A, (ii) x B C. (i) , x A B x A C , x (A B) (A C).. (ii) , x B x C . , x A B. x A C, , x (A B) (A C) .. x (A B) (A C) . , (A B) (A C) A (B C) . x (A B) (A C) .. , x A B x A C . , x A (B C) . x , 2 . (i) x A, (ii) x A.

3 , (i) x A (B C) . (ii) .. x A B x A C , x A , x B. x C . , x B C . , x A (B C) . , x A (B C) . A A. B C B C. : . GAIRON-book : 2018/6/21(19:23). 38 3 . ( ) 2 A, B , 3 .. (i) A B, (ii) A B = B, (iii) A B = A. A, B, C , . (A B) (B C) (C A) = (A B) (B C) (C A). A1 , .. , An , B , . ( . n ) . n Ak B = (Ak B), k=1 k=1. ( . n ) n Ak B = (Ak B). k=1 k=1.. A, B , .. , . A . B A B . , A\B A B . , A\B = {x|x A, x B}.. , x A\B (x A) (x B).. , A\B B\A . , A , A\A = , A\ = A, \A = .. A, B, C , . (A B)\C = (A\C) (B\C), ( ). (A B)\C = (A\C) (B\C). ( ). GAIRON-book : 2018/6/21(19:23). 39. ( ) . ( ) . , x (A B)\C (x A B) (x C). ((x A) (x B)) (x C).. , , ((x A) (x C)) ((x B) (x C)). (x A\C) (x B\C). x (A\C) (B\C). , (A B)\C = (A\C) (B\C) . ( ) A, B, C , . C\(A B) = (C\A) (C\B), ( ). C\(A B) = (C\A) (C\B). ( ). ( ) ( ( ) ). ( ) , .. , x C\(A B) (x C) (x A B). (x C) ((x A) (x B)).. , ( ) . (x C) ( (x A) (x B)).. , , , , ((x C) (x A)) ((x C) (x B)).

4 (x C\A) (x C\B). x (C\A) (C\B).. , C\(A B) = (C\A) (C\B) . A, B, C , . GAIRON-book : 2018/6/21(19:23). 40 3 . A A. B C B C. : . (1) (A\B)\C = A\(B C). (2) A\(B\C) = (A\B) (A B C). 2 A, B , 2 . (i) A\B = A, (ii) A B = . 2 A, B , 3 . (i) A B, (ii) A\B = , (iii) B\(B\A) = A. A , a A . B A\{a} B A . , B = A B = A\{a} . 2 A, B , A B = (A\B) (B\A) = (A B)\(A B) ( ). A B .1) , ( ) 2 .. , A , A A = , A=A =A.. 1) A B . GAIRON-book : 2018/6/21(19:23). 41. A B A B. : A\B A B. A, B, C . (A B) C = A (B C). A, B , A X = B X . X , .. , A X , . Ac = X\A. A (X ) .2) A = {x X | P (x)} , Ac = {x X | P (x)}.. A = {2, 3, .. } . N , Ac = {x N | x A} = {1}.. Z , Ac = {x Z | x A} = {1, 0, 1, 2, .. }.. , .. 2) Ac complement . A , .. GAIRON-book : 2018/6/21(19:23). 42 3 . X X. Ac A. : Ac ( ) X , A, B . , . (1) X c = , c = X. (2) A Ac = , A Ac = X. (3) (Ac )c = A. (4) A B B c Ac (5) [ ] (A B)c = Ac B c , (A B)c = Ac B c . X , A, B.

5 (1) A\B = A B c (2) A B = (A B c ) (Ac B) = (A B) (A B)c A1 , .. , An , . P (A1 , .. , An ) , . P (A1 , .. , An ) . , P (A1 , A2 , A3 ) = A1 (A2 A3 ), P (A1 , A2 , A3 ) = A1 (A2 A3 ). Q(A1 , .. , An ) , , A1 , .. , An . P (A1 , .. , An ) Q(A1 , .. , An ) ( ). , A1 , .. , An . P (A1 , .. , An ) Q (A1 , .. , An ) ( ). GAIRON-book : 2018/6/21(19:23). 43.. , ( ) ( ) .. , A1 , .. , An . P (A1 , .. , An ) = Q(A1 , .. , An ). , P (A1 , .. , An ) = Q (A1 , .. , An ).. , .. , . , ZFC , , .. , . , A, B, .. , a, b, .. , , A = B A B .. , . , , . [ ] A = A. [ ] P (x) , P (A) A = B P (B).. 2 . [ ] A = B B = A. [ ] A = B, B = C A = C.. , 3 . ZF (S1) (S9) ZF .3). (S1) . A B( x(x A x B) A = B). 3) (1908 ) (Adolf Abraham Halevi Fraenkel, 1891 1965. ), (Thoralf Albert Skolem, 1887 1963. ) , 1920 . GAIRON-book : 2018/6/21(19:23). 44 3 . , 2 A = B, A B . , . , A B(A = B x(x A x B)) . , . A B x(x A x B) . , , , .. (S2) . A x(x.

6 / A). , , . (S3) x, y , x y . x y A z(z A (z = x z = y)). , {x, y} . , x = y , {x, x} =. {x} . , x {x, y}, y {x, y} . 2 x, y (x, y) . , , (x, y) = (x , y ) x = x y = y .. , (x, y) = {{x}, {x, y}} . , {x, y} . (S4) X , .. X A t(t A x X(t x)).. X , X . , . X = {x, y} , {x, y} = x y . , x . x = x . (S5) X X . X A x(x A x X)). , X , 2X . GAIRON-book : 2018/6/21(19:23). 45. (S6) , , x x {x} .. A( A x A(x {x} A)). , A . x x {x} .. x+ = x {x} . A , + , ++ , +++ , .. A . A .. , .. , , + , ++ , +++ , .. , 0. 1, 2, .. ( 15 ). (S7) ( ) X , .. , X x P (x) .. X A x(x A (x X P (x))).. {x X | P (x)} . , X Y =. {x X | x Y } . , , . , , .. , . (S8) P (x, y) (x, y) . A . x A P (x, y) y , y . B , x A P (x, y) y B .. A( x A y P (x, y) B x A y B P (x, y)). (S9) A , y A y x . x A .4). A(A = x A y A(y . / x)). 4) , x A . GAIRON-book : 2018/6/21(19:23). 46 3 .. x y y x x, y . , 2 x, y , A = {x, y} .. x y y x , A , x y . x x y.

7 / x , x y y . / y , .. , x y y x . x, y . (1) x x . (2) x y, x = y, y x 1 . (3) {x} x x . , x = {x} .. (1) x = y , x x .. (2) (1) . (3) {x} x x x (1) . x1 , x2 , .. , xn , .. x1 x2 xn . ( ) . A = {xn | n N} .5). ZFC ZF (S10) ZFC.. ZF , .. , , .. , . , . 5) A , . GAIRON-book : 2018/6/21(19:23). 47. (S10) . X . , 2 , x X. x A 1 A . X(( . / X x X y X(x = y x y = )). A x X t(x A = {t})). , A X 1 . , . , .. A .. , . 11 ( ). , P (x) .. {x | P (x)} ( ).. , , K = {x | x . / x} ( ).. , K K , . ( ). , , X P (x) . x , . {x X | P (x)} = {x | x X P (x)} ( ).. ( ) P (x) . , X , ( ) .. , ZF , ( ) , .. , K . , , .. BG ( ) .. , (1937 ) , . BG .. GAIRON-book : 2018/6/21(19:23). 48 3 . 6) , NBG .. , BG BGC .7) . , ( ) , .. , ZF BG , BG ZF .. 6) John von Neumann (1903 1957).. 1930 . , . 1933 .. , , , , . , 20 .. ( ) .. , , (1925). 7) BGC ( , 2014).

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