1. A, B - , ય - . AÇB, AÈB, A\B, B\A, A¢, B¢, A¢DB.
1) A = {2, 1, 8}, B = {9, 7, 2, 5, 1}, ય = {1, 2, 3, 4, 5, 6, 7, 8, 9};
2) A = (-10; 1], B = (-5; 20), ય = R.
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1) A = {2, 1, 8}, B = {9, 7, 2, 5, 1}, ય = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
AÇB = {x | xÎA xÎB} = {1, 2};
AÈB = {x | xÎA xÎB} = {1, 2, 5, 7, 8, 9};
A\B = {x | xÎA xÏB} = {8};
B\A = {x | xÎB xÏA} = {9, 7, 5};
A¢ = ય\A = {x | xÏA} = {3, 4, 5, 6, 7, 9};
B¢ = ય\B = {x | xÏB} = {3, 4, 6, 8};
A¢DB, D - .
XDY = (X\Y)È(Y\X) = (XÈY)\(XÇY).
A¢ÈB = {1, 2, 3, 4, 5, 6, 7, 9}; A¢ÇB = {5, 7, 9};
A¢DB = (A¢ÈB)\(A¢ÇB) = {1, 2, 3, 4, 6}.
2) A = (-10; 1], B = (-5; 20), ય = R.
AÇB = {x | xÎA xÎB} = (-5; 1];
AÈB = {x | xÎA xÎB} = (-10; 20);
A\B = {x | xÎA xÏB} = (-10; -5];
B\A = {x | xÎB xÏA} = (1; 20);
A¢ = ય\A = {x | xÏA} = (-¥; -10] È (1; +¥);
B¢ = ય\B = {x | xÏB} = (-¥; -5] È [20; +¥);
A¢DB, D - .
A¢ÈB = (-¥; -10] È (-5; +¥); A¢ÇB = (1; 20);
A¢DB = (A¢ÈB)\(A¢ÇB) = (-¥; -10] È (-5; 1] È [20; +¥).
2. , X: a) X = A\B¢Ç(B\A); b) X = A¢ \(BÈC\(AÇB)).
:
1) X = A\B¢Ç(B\A).
( 1):
:
B¢ 0, 1.
A\B¢ 3.
B\A 2.
X = A\B¢Ç(B\A) ( 3 2 ).
, X=Æ, , .
2) X = A¢ \(BÈC\(AÇB)).
( 2):
:
A¢ 0, 2, 3, 5.
BÈC 2, 3, 4, 5, 6, 7.
AÇB 4, 7.
BÈC\(AÇB) 2, 3, 5, 6.
X = A¢ \(BÈC\(AÇB)) 0. 0 ( 3).
3. - , 2.
:
, 2. - , .
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1) X = A\B¢Ç(B\A).
A\B¢Ç(B\A) = AÇB¢Ç(BÇA¢) = AÇB¢ÇBÇA¢ = Æ
2) X = A¢ \(BÈC\(AÇB)).
A¢ \(BÈC\(AÇB)) = A¢ \(BÈCÇ(AÇB)¢) = A¢Ç(BÈCÇ(AÇB)¢)¢ =
= A¢Ç(B¢ÇC¢È(AÇB)) = (A¢ÇB¢ÇC¢)È(A¢ÇAÇB) = (A¢ÇB¢ÇC¢)ÈÆ =
= (A¢ÇB¢ÇC¢) = (AÈBÈC)¢
4. F, X, Y, Z - . , , . .
F º (((X & (ù Z)) ((ù Y) V X)) Y)
:
, F º (X & ù Z ù Y V X) Y.
:
F1 º X & ù Z; F2 º ù Y V X; F3 º F1 F2; F º F3 Y.
F:
X | Y | Z | ù Y | ù Z | F1 | F2 | F3 | F |
, F(1,0,0) = 0 F(X,Y,Z) = 1 X, Y, Z.
5. F .
F º (ù X Y) & (X Ú ù Y).
:
(ù X Y) & (X Ú ù Y) º (X Ú Y) & (X Ú ù Y) º X Ú (Y & ù Y) º X Ú 0 º X
6. P(x), T(x,y) - , A. .
A = {0, 10, -3, 14, -1}; P(x) = x 30; T(x,y) = = x+y - .
: IP, IT - P(x) T(x,y) . P(x) - , T(x,y) - , IP Í A, IT Í A´A.
P(x) xÎA, P(x) , .. x, 30. , IP = {10, -3, -1}.
T(x,y) (x,y), T(x,y) . , 0 ( ) 2 (). , IT = {(0; 10), (10; 0), (0; 14), (14; 0), (10; 10), (10; 14), (14; 10), (14; 14)}.
7. P(x) = , R .
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: P(x) - , IP Í R.
P(x), , x.
1) , Û
Û , xÎ[-5; 2,5).
2) ,
:
; ; ; .
:
, xÎ[2,5; 4].
, 1 2:
xÎ[2,5; 4] È [-5; 2,5), .. xÎ[-5; 4].
8. F - A B. ( , , , ). , .
1) A = {1, 5, 3, 7, 2}, B = {10, 2, 4}, F = {(1,10), (5,2), (2,10), (7,2), (3,2)};
2) A = B = N ( ), F = {(x,y) ç y - x}.
:
1) .
: xÎA yÎB, , (x,y)ÎF, .. F - .
: xÎA yÎB, , (x,y)ÎF, .. F - .
: 4ÎB A, .. F .
: (1,10)ÎF (2,10)ÎF, 10 B A, .. F .
F - , F - .
2) .
: xÎ N (, x - ), .. xÎ N . , F .
: xÎ N , .. F - .
: F. , 10 . , F .
: , 5 - 256 3578, .. (256, 5)ÎF (3578, 5)ÎF, 5 B A, .. F .
F , F - .
9. F - A B. (, , ).
A = B = N ( ), "xÎ N F(x)=(x-3)2+1.
:
.
: F. , 3 N, .. (x-3)2+1 = 3, .. (x-3)2 = 2, x N. , F .
: (2-3)2+1=2 (4-3)2+1=2, .. F(2)=2 F(4)=2, 2 B A, .. F .
10. r - , N. (, , , , , ) r.
r: "A,BÎ N A r B Û A B.
:
) : "AÎ N (A r A)
, A A.
, r - , .., , 12 (2¹3).
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) : "A,BÎ N (A r B Þ B r A)
, A B, , B A, r .
r , .. . A=23, B=33, A 6 B, B A (9¹5).
) : "A,B,CÎ N (A r B & B r C Þ A r C)
, A B B C, , A C, r .
r , .. . A=23, B=33, C=45, A 6 B, B 9 C, A C (6¹9).
) : "AÎ N ù (A r A)
, A .
(, A=123), , r .
) : "A,BÎ N (A r B & B r A Þ A=B)
, A B B A, , A=B, r .
r , .. . A=123, B=321, A 6 B, B A, A¹B.
) : "A,BÎ N (A r B Ú B r A Ú A=B)
r , A B : A B; B A; A=B.
r , .. . A=12, B=24, A B (2¹6), B A (8¹3) A¹B.
, r , , , , , .