.
1.
, .
f , Df f, Rf .
1.
1. {(1,1); (2,3); (3,2)}, {1,2,3} ;
2. {(1, 2); (2, 2); (, ); (, )} {1, 2, , } {2, , };
3. {(1,2); (1,1); (2,3)} , ;
4. {((x, y), x + y): (x, y) } - .
2.
f g , :
.
, f Df. - Rf. , Y, , Rf Y. .
3.
Df X Rf Y, , f X Y.
4.
Df = X Rf Y, , f X Y.
5.
, , Df = X, Rf = Y, () f X "" Y, . , f Df Rf.
6.
f , , , f (a) = f (b), , a = b.
7.
X Y (), X Y.
1. f = {(x, y) : y = x 2 } , . f R . f , . f , f (1) = f (-1), , 1 = -1. , , f . , f .
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2. g, x g (x) = 2 x +1. , g - Z . , , 2 x 1+1 = 2 x 2+1, , x 1 = x 2.
- Z . , .
1.
M , - M .
!!!
f g , f g , fg x : , f: , g: , fg: .
8.
f f -1 , f f -1.
2.
f f -1 , f .
.
. , f f -1 . , f .
. f . , , , (a, c) f, (b, c) f, a . , f -1 (c, a), (c, b) , f -1 . . , , f , . . .
2.
, . . - . ? , , : , , -, "".
9.
, A B, f: (.: .3, .4).
" " - "". A ~ B.
3.
1. N . f (n)=2 n, .
2. (0,1) R . , , (0,1) . (. 1). (-1,1) . . (mi) R. , , (0,1) , R.
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10.
3.
, .
.
. | A |=| B |= n, .. A = { a 1, a 2,..., an }, B = { b 1, b 2,..., bn }. f: A B, f (ak) = bk, k = 1,..., n, , , A ~ B.
. A~B f: A B , . A = { a 1, a 2,..., a n }, B = { f (a 1), f (a 2),..., f (an)} f (ak) . , | A |=| B |= n. .
11.
A , : A ~ N. , A .
4.
A , , .
.
. A . , A ~ N. f: . , an = f (n), . A ={ a 1, a 2,... }, A.
. A , .. A ={ a 1, a 2,...}. f: f (n)= an, . , f N ~ A. .
5.
.
6.
Q .
.
, . , . , ?
7.
M = (0,1) .
, M - , , , , .. M = {x1,x2,... }. ( 9 ):
...
; , . : , , 0, . , , xi. , (0,1) , , 1 , , . , (0,1) .
A , (0,1). c.
4.
a < b . , (a, b) . , (a, b) (0,1), f (x) = (b-a) x + a: (0,1) (a, b).
8.
. , .
.
, , , .
, . . , , , . . :
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- , ;
- , ;
- ;
- ;
- .
1. . . . .: , 2001. 384 .
2. .. . .: , 2002. 240 .
3. .., .. . .: , 2001.
4. .. . . --: , 2002. 512 .
5