Элементы теории множеств: Учебно-методическое пособие для вузов

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  • - 20 2007 , 4

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    , 1 100; ; ; ; , [a, b]; ; . .

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    , , , , , - , .

    A B, , : A = B.

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  • a a = {b1, b2, . . . } , , {a} = a {a} {b1, b2, . . . } ( !).

    ( - ) , , . - .

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  • , 5, , 5.

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    A ( )

    A, I , A.

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    , , , B A. 4. Z \ 2Z

    . A\B B A -

    B ( A) CAB CB, B, , .

    A4B A B, A \B B \ A (. 5), . .

    A4B = (A \B) (B \ A).

    A B. 4 . 5

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    5. . - ( ) AB A B - (a, b),

    8

  • A, B, . .

    AB = {(a, b) : a A, b B}. 5. A B ,

    (a, b), a A, b B, - a b. AB A = [, ], B = [, ] , . 6, R R .

    . 6x

    y

    A1 Ak - A1, . . . , Ak k (a1, . . . , ak), a1 A1, . . . , ak Ak, . .

    A1 Ak = {(a1, . . . , ak) : ai Ai, i = 1, . . . , k}.

    6. - . A . , - A. ,. . A, 2A

    ( , ). , 2A. .

    S = {B}I A A,

    9

  • A B S, . .I

    B = A. S = {B}I A A, A - B S. , - S = {B}I A A, , A, . .

    1) B1 B2 = ,1, 2 I, 1 6= 2;2)I

    B = A.

    B1

    B2

    . 7

    A

    B S -, S, . . , , - A. , - . . .

    A . R - A A A, . . - A. (a, b) R (a, b) R a

    Rb , a

    b R.

    10

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    2) a Rb, b

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    3) a Rb b

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    R A - a A [a] x A, a : x

    Ra,

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    A. {[a]}aA - A.

    . a A - [a],

    aA

    [a] = A.

    , a, b A [a] [b] , . [a] [b] 6= c [a] [b]. x - [a], . . x

    Ra (. 8).

    x

    a

    bc

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    A

    . 8

    11

  • c Ra, , a

    Rc.

    x R

    a, a R

    c, c R

    b, , x R

    b, . .x [b], [a] [b]. [b] [a]. , [a] = [b].

    , -. , {B}I A , , a

    Rb

    , a - {B}I , b. {B}I - {[a]}aA. - A, R, -- A R A/R.

    7. P l P R : a

    Rb, , l, b,

    a. - P/R - P , l.

    8. , 6, , - .

    3. . , , \,4, - .

    , :

    A B = B A,

    A B = B A

    12

  • :(A B) C = A (B C),(A B) C = A (B C).

    , , . . - :

    (A B) C = (A C) (B C) (1)

    :

    (A B) C = (A C) (B C). (2)

    , , (1). x , (1), . . x (AB)C. , x C x A B. x A C B C, , , x (A C) (B C). , x (A C) (B C). x A C B C. , x C, , , x A B, . . x AB. , x (AB)C. (1) . (2).

    , -, .

    . 1. ,

    , -.

    , , \, 4, , , :

    (A4B) C = (A C)4(B C). (3)

    2. (3).

    13

  • , :

    1)

    X \

    A =

    (X \ A); (4)

    2)

    X \

    A =

    (X \ A) (5)

    ( (4), (5) {A} - X).

    3. (4), (5). : -

    , , ( - ), - , -, .

    . - ( ) , ., (AB)\(DF ) CX

    ((A B)\

    \(D F )),

    ((CXA) (CXB)

    )\((CXD) (CXF )

    ).

    9. A,B X

    (A \B) (B \ A) = (A B) \ (A B)

    (!). -, (

    CX(A \B))(CX(B \ A)

    )= CX

    ((A B) \ (A B)

    ).

    14

  • 1. 10 : 3 7?

    2. - ()?

    3. : A B,A B = B, A B = B.

    4. , A B = A \ (A \B).5. , -

    :A (B C) = (A B) (A C), (B C) A = (B A) (C A);A (B C) = (AB) (A C), (B C) A = (B A) (C A).

    6. , A,B,C -

    (A B) \ C = (A \ C) (B \ C),(A B) \ C = (A \ C) (B \ C),

    C \ (A B) = (C \ A) (C \B),C \ (A B) = (C \ A) (C \B)

    (. . - -).

    7. R - R : a

    Rb, x y Z. , R

    , - R/R.8. n 1 . Z

    R : p Rq, pq

    n. , R ,

    15

  • - Z/R (- Z/R Zn, n).

    9. P R1, R2:

    1) (a1, a2) R1

    (b1, b2), a1 = b1, a2 b2 Z;2) (a1, a2)

    R2

    (b1, b2), a1 b1, a2 b2 Z., R1, R2 , - P/R1, P/R2.

    10. , , 2, 5 15 -.

    16

  • 2.

    1. . , , .

    X, Y . X Y , X Y .

    . f , - X Y, f : X Y ; - X , Y f . a f7 b , a X f : X Y b Y . - a f7 b a 7 b, , .

    a X b Y (a, b) XY. - : f - XY X Y , a X b Y, (a, b) f.

    a X, b - f : X Y a ( f) f(a). X, b Y , ( -) b f1(b).

    A X; {f(a) : a A} A f(A). f(X) X f f Imf . B - Y , {f1(b) : b B}

    17

  • ( ) B f1(B)., f(A) = , A = . , f1(B) , B 6= , f1(B) = , B f(X) = (!).

    f : X Y , g : X Y f = g, X = X , Y = Y f(x) = g(x) x X.

    1. , f : X Y A X, B Y f(f1(B)) = B f(X), f1(f(A)) A. , f1(f(A)) 6= A.

    1. X . X X, x X , - ( ) I

    X I (

    ). , IX(x) = x, x X .

    2. X, Y y0 Y . f : X Y , f(x) == y0, x X, .

    3. c X . f : N X X. f(n) xn, {x1, x2, . . . } {xn}.

    4. f : X1 Xn Y , n 2, , () n .

    . f : X Y ( ), x1, x2 X f(x1), f(x2) -, . . f(x1) = f(x2) x1 = x2; , f1(y) y Y . f : X Y

    18

  • ( ), y Y x X , f(x) = y, . . f(X) = Y ; , f1(y) y Y . f : X Y - ( ), , . . f1(y) y Y .

    2. , X -, f : X X .

    . 1. f : X Y

    A B X , ,. .

    f(A B) = f(A) f(B), (1)f(A B) f(A) f(B). (2)

    . (1). y f(A B). y = f(x) x AB. x - A B, y = f(x) f(A) f(B), ,, y f(A) f(B). , y f(A) f(B), y f(A) f(B),, A B x , y = f(x). x A B, y = f(x) f(A B).

    (2). y f(A B). y = f(x) x A B. , x A x B, ,f(x) f(A) f(x) f(B), y = f(x) f(A) f(B).

    19

  • 1. , , . , - f : X Y , f(x) = y0, x X . x1, x2 X, x1 6= x2. A = {x1}, B = {x2}. AB = , , f(AB) = , f(A) f(B) = {y0}.

    2. f : X Y A B Y () - () , . .

    f1(A B) = f1(A) f1(B), (3)

    f1(A B) = f1(A) f1(B). (4) . (3). x

    f1(AB). , f(x) AB, f(x) - A B. x f1(A) f1(B), x f1(A) f1(B). , x f1(A) f1(B), x f1(A) f1(B). f(x) A B, , f(x) A B, xf1(A B).

    (4). x f1(A B), f(x) A B, . .f(x) A f(x) B, , x f1(A) x f1(B), x f1(A)f1(B). , x f1(A)f1(B), x f1(A) x f1(B), , f(x) A f(x) B, f(x) AB, x f1(A B).

    3. , . ?

    2. - . , ,

    20

  • . -, f : X Y , {f1(y): yf(X)},, X. , X - {A}I . Y {A}I . pi : X Y , x X (. . Y ), x . {pi1(y) : y Y } - {A}I .

    2. . - . , .

    1. ( ). f : X Y A X. - f |A : A Y , f |A(x) = f(x), x A, ( ) f - A.

    5. R+ - , R . f : R R, f(x) = |x| : f |R+(x) = x,f |R(x) = x.

    2. . X, Y, Z f : X Y , g : Y Z . ( -) f g g f : X Z, -

    (g f)(x) = g(f(x)),x X.

    6. f, g : R R, f(x) = x2, g(x) == 2x. g f : R R (g f)(x) = 2x2, f g : R R (f g)(x) = 22x. (g f 6= f g !)

    3. . f -

    21

  • X Y . g : Y X f ,

    g f = IX, f g = I

    Y. (5)

    , , - .

    4. f : X Y,g : Y X , g f = I

    X, f g 6= I

    Y.

    3. , .

    . . f : X Y, g : Y X . :1) g f , f ;2) g f , g . . 1) x1, x2 X f(x1) = f(x2),

    (g f)(x1) = g(f(x1)) = g(f(x2)) = (g f)(x2). g f , x1 = x2, . . f .

    2) x - X. g f x X , (g f)(x) = x,. . g(y) = x, y = f(x), , g .

    . f : X Y, g : Y X

    g f = IX, (6)

    f , g . 3. .

    f : X Y g : Y X f. , g f = I

    X

    f, f g = IY

    f, . . f .

    22

  • . f : X Y , y Y f1(y) X. g : Y X g(y)=f1(y) y Y . (f g)(y) = f(g(y)) = f(f1(y)) = y, . .f g = I

    Y; (g f)(x) = g(f(x)) = f1(f(x)) = x, . . g f = I

    X.

    , g f .

    3. . , g : Y X , f : X Y, , f, y Y f1(y) ,, f(g(y)) = y, g(y). , - f , - f1.

    7. f : R R, f(x) = x2k1,k N, f1(x) = x 12k1 . g : R R, g(x) = x2k, k N, , g .

    8. X I1X

    = IX.

    f : X Y X, , , , , f1 .

    , f f1 (f1)1 = f .

    4. f : X Y y Y f1(y). f, f1 : Y X, f1(y) y f1. , - f1(y)

    23

  • y f1.3. .

    , . 4. ,

    . . f : X Y , g : Y Z, h : Z H ,

    (h g) f = h (g f). (7)

    . (7)

    ((hg)f)(x) = (hg)(f(x)) = h(g(f(x))) = h((gf)(x)) = (h(gf))(x).

    5. f : X Y , g : Y Z h = g f . :

    1) f g , h ;2) f g , h ;3) f g , h . . 1) h(x1) = h(x2).

    h(x1) = (g f)(x1) = g(f(x1))||

    h(x2) = (g f)(x2) = g(f(x2)), g(f(x1)) = g(f(x2)). , - g, f(x1) = f(x2), , f , - x1 = x2.

    2) z Z. g, y Y , g(y) = z, f , x X , f(x) = y. h(x) == (g f)(x) = g(f(x)) = g(y) = z, . . h(x) = z.

    3) 1), 2).

    24

  • 5. f : X Y , g : Y Z . , (g f)1 = f1 g1.

    1. , -; , ; .

    2. f g , f -, g , g f .

    3. , f : X Y A X f1(f(A)) = A.

    4. , f : XY B Y f(f1(B)) = B.

    5. , f : X X

    f(X) f(f(X)) f(f(f(X))) . . .

    6. , X Y |X| = n,|Y | = k, X Y kn.

    7. , f : X Y - A X,B Y f |1A (B)=f1(B) A.

    25

  • 3.

    1. . .

    . - , , - - , , - , . . , , . , Z , 10, 10 , -, . - : - ( -) - , .

    -, . . . . - , , - , : -, , . . 5 , , - , ( ). , .

    26

  • - , , .

    , , . , - ( ) , . - . ? (), , , . A B, B A, . . A B , B . A - A B, , B , . -. ; , . - : , , . , , - .

    A -

    27

  • B, f : A B. , A B, B A, A B . A B , , , |A| = |B|.

    1. , - - .

    , . - . , . , . , B - A B 6= A, |B| 6= |A|. : N 6= 2N, |N| = |2N| (. 1 ).

    .2. . ,

    N . 0 ( -, ).

    f : A N A: a n = f(a)( an). , A - N A N. , .

    1. 2N . f : N 2N , - f(n) = 2n, n N.

    2. Z

    28

  • 2N+ 1 . 1. Q . .

    p/q, q > 0. |p| + q p/q. , n N , n, . - , . . 0/1 1, 1/1, 1/1, 0/2 2 . . , . . N Q.

    . 2.

    . . A B

    . B = , . B 6= . A : a1, a2, . . . . an1, an2, . . . , B. n1, n2, . . . , B , , B , an1, an2, . . . 1, 2, . . . .

    3. -.

    . A . A - a1. A \ {a1} a2, A\{a1, a2} a3 . . {a1, a2, a3, . . . }, A.

    29

  • , - .

    4. .

    .

    A1, A2, . . . (1)

    . , , (1) , - A1, A2 \A1, A3 \ (A1 A2), . . . ( , 2), (1). (1) , i- Ai. - (1) , , (1). -, . . , , a11:

    A1 : a11 a12 a13 a14 . . .

    A2 : a21 a22 a23 a24 . . .

    A3 : a31 a32 a33 a34 . . .

    A4 : a41 a42 a43 a44 . . .

    . . . . . . . . . . . . . . . . . . . . . . . . . . .

    , , . . (1) - N.

    30

  • 3. , .

    . 5.

    . . A -

    , B - . 3 A C. - 4, C CB . g:C CB . A = C, , A 6= C, - f : A A B

    f(x) =

    {x, x A \ C,g(x), x C.

    , f .

    3. . , .

    6 (). - (0, 1) .

    . (0, 1) - , . , p10k , p, k N, p < 10k, , : , (, 0, 200 = 0, 199 . . . ). , - , , 9 . , (0, 1). :

    1 = 0, a11a12a13 . . .

    2 = 0, a21a22a23 . . .

    3 = 0, a31a32a33 . . .

    . . . . . . . . . . . . . . . . . . .

    (2)

    31

  • = 0, b1b2b3 (0, 1), b1 1 8, a11, b2 1 8, a22 . . bn 6= ann n N, (2), . . / (0, 1). .

    , - (0, 1). c ( ).

    2. (a, b), R - R+ - . :f : (0, 1) (a, b), f(x) = a + x(b a); g : R+ (0, 1), g(x) = 1x+1 ,h : R (0, 1), h(x) = 12x+1 .

    4. , [a, b] .

    4. . , - A B, |A| |B|, A B. , , |A| 6= |B|, , A B, |A| < |B|.

    -.

    7 ( ). |A| |B| |B| |A|, |A| = |B|.

    . f : A B1, g : B A1 A,B - B1, A1 B,A . h : A B. , ,

    32

  • A B = = . x - A. - {xn} AB . x0 = x. xn , n xn+1 = g1(xn) ( g1(xn) B ), n xn+1 = f1(xn) ( f1(xn) A). .

    1. n xn , xn+1 -, . . {xn} . n - x.

    2. xn+1 n N, . . - {xn} . x .

    A : A , A , A . - B B, B, B. f(A) = B,f(A) = B, g1(A) = B. , {xn} -, x A, y = f(x) B {yn},

    yn =

    {y, n = 0,xn1, n 1,

    f(x) {

    B, x A,B, x A,

    f(A) B, f(A) B. , {yn} , y B, y1 = f1(y), x = y1 A {xn}, xn = yn+1,

    f1(y) {

    A, y B,A, y B,

    33

  • , f1(B) A, f1(B) A, . . f(A) B, f(A) B. , f(A) = B, f(A) = B. g : B A1 - g(B) = A, . . g1(A) = B. - h : A B

    h(x) =

    {f(x), x A A;g1(x), x A.

    - . , -- , - , , , .

    5. , (a, b), - [c, d) [g, h] .

    : - . , - . , .

    8 (). A 2A .

    . A = , . A 6= C =

    {{a} : a A

    }

    A. f : A C, - f(a) = {a}, a A, , A C 2A. , |A| 6= |2A|. . g : A 2A .

    34

  • A:

    A1 = {a A : a g(a)}, A2 = {a A : a / g(a)}. A1, A2 : A1 a : g(a) == A, A2 a : g(a) = . , A1 A2 = A A1 A2 = . g , a A , g(a) = A2. a A1, a g(a) = A2.. a A2, a / g(a) = A2. .

    8 , , . ., .

    2M - M , M . .

    6. , 2M - M , n , 2n ,. ....

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