2 2.1 (set) A, B, C, (element). ( ) a A, a A. a A, a A. A a A a 19 N. 3 N, 2 / N ( ) 20 {1, 2, 3, 4, 5} {5, 2, 1, 3, 4} {1, 1, 1} {{1, 2}, {2, 3

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1 . (set),, C, (element). ( ),.,. 9 N. N, / N.. ( ) 0 {,,, 4, 5} {5,,,, 4} {,, } {{, }, {,, 4}} (empty set) ϕ {} : N = {0,,, } N, Z, Q, R (0 ).. P (x) x {x P (x)} {x R 0 x } [0, ] (0 ) S = {x P (x)} x S x P (x) (x S) (x P (x))

2 : {,, } {,,,, } {x N x }.. ( ) {x P (x)} {x x P (x)} {x P (x) x } x [0, ] {x 0 x x } {e P (x)}, {y x.(y = e P (x))} {k k Z}, {k + k Z} {x P (x)} P (x) x {x x = y y N } {x y(x = y y N )}., =. x (x x ), {,, } = {,, } {,,, } = {,, } {, } = {,, } = {,, } = {,,, } {, } = {x x = x = } ϕ = {x x = x + }. {,, } = {, } x x {x P (x)} Russell ( )

3 .. x (x x ). (suset). = 4 {} {, } N R.4.4. x x 5 Prime(x) x (prime numer) = {x Prime(x) 4 x 5}, = {x x = 4k + k N }. = 4 = 47. = 4 = 4 0 +, 0 N = 47 = 4 +, N,.4. x x x 6 = {k + k N }, = {4k + k N }, =. 7 = {x x x 8}, = {x k {, 4} x = 4k + }, =.

4 . x (x = x = 7) = 4 +, 7 = x, x = 4 + x = 4 4 +, x 8., =.4.4 (Pigeon hole priniple) ( ) n m. m > n n n + = {k N k n} k < n k k = {k (0 ) (k n)} k k n n = n n + k k.5.5. (power set) P() x. (x x ) x 0 = {,, } = {ϕ, {}, {}, {}, {, }, {, }, {, }, }.5. (, union) (, intersetion), 4

5 x.(x (x x )) x.(x (x x )) = {, }, = {,, 4} = {,,, 4} = {} () () () (d) (e) ϕ = = ( ) ( C) = ( ) C ( ) = ( ) = (e) () = () ( ) x x x x, x x () ( ) x x x x () = x x = x () () () (d) (e) ϕ = ϕ = ( ) ( C) = ( ) C ( ) = ( ) = () ( C) = ( ) ( C) () ( C) = ( ) ( C) 5

6 .5. x.(x (x x )) = {,, }, = {, d} = {, } ( ) = ( ) =.5.4 ( ). N U, (omplement),. = U x.(x x ) x x U () = () ϕ = U, U = ϕ () = ϕ, = U (d) (e) =, = ( ) (d) x x x x x x x. x ( x ),, x., x,., x.. 6

7 .5.5 ( tuple),, ,,... ( 0, ) 0,,,, (,,, 0, ) 0. ( pir). x, x,..., x n, x i i. (rtesin produt) z.((z ) x y ((z = x, y ) (x ) (y ))) v w ( v, w (v w )) 4 = {, }, = {0,, }. = {, 0,,,,,, 0,,,, } = =,..., n,. n = {,..., n n n } n n {}}{ n 0 = { }( ) 5 = {0, }. = { } 0 = { } = { 0, } = { 0, 0, 0,,, 0,, } = 7

8 .6 ( ) ( ) (, denumerle) ( ) 6 N Z 0,,,,, C ( C ) SCII it ( ) C ( it C ) (ountle set), (unountle) 7 N 8 {x Q 0 < x} ( ) 9 = 0 n, 0,,, 0 = 00, 0 0 = 0, = 0,. N 8

9 00, 0, 0, 0,,., 4 Q N N. T = N T, T, S 0, S,, S n, V. V = {n N n S n } V V N V T V S 0, S, V = S k k V k S k k V k V T T R R N R N 0.d 0 0 d 0 d 0 5 d i (0 ) d i = { i 0 i d 0 d... 0 S 9

10 , N, R N R R.7.7. ( ), (funtion) (mp mpping). 40..: ( {,, } {,, } ).: ( : : ) 0

11 .7. f f : f (,, domin), f (,, odomin) 6. f x y ( ) f(x) = y x f ( rgument) y f x ( vlue).7. (, inry funtion): f : ( ) C C x, y f( x, y ) f(x, y) 4. plus : N N N. plus( x, y ) = x y., plus(x, y). ( ) f : n 4 n mx mx : N n N n ( ).7.4 (imge) f : C f C f(c) f(c) = {f(x) x C}.,. y f(c) x C y = f(x).7.5 (inverse imge) f : D f D f (D),. f (D) = {x f(x) D} x f (D) f(x) D 4.5 f = {,, }, f = {,, }, f({, }) = {, }, f() = {, }, f({}) = {}, f({, }) = {}, f ({, }) = {,, }, f ({, }) = {, }, f ({}) = ϕ, f () = {,, } =. 6 (rnge) f f(x) = 0 f f {0}

12 C f f(c).: f C f(c). - f (D) f D.4: f (D)..8 f :, g : x f(x) = g(x) f = g. (extensionl equlity) 44 f(x) = x, g(x) = x + x f = g.9 (omposition) f :, g : C f g g f : C (g f)(x) = g(f(x)) g f f, g f g g f (g f)(x) = g(f(x)) R T R T

13 .5: f. ( ) : (h (g f))(x) = h((g f)(x)) = h(g(f(x))) = (h g)(f(x)) = ((h g) f)(x) f g = g f 45 f(x) = x, g(x) = x +. (f g)(x) = f(x + ) = (x + ). (g f)(x) = g(x ) = x (identity funtion) id : id (x) = x f :, f id = f = id f..0. ( ) f : (injetion) (one to one mp) x y (f(x) = f(y) x = y) x y (x y f(x) f(y)). 46 (.6) f g 4 4.6: f : g :.

14 47 f(x) = x + ( 0 ) f : R R. g(x) = x + x + ( 0 ) g. 48 = {6k + 4 k N }, = {k + 4 k N }, f :, f(x) = x +. f. x y x + y ( ) f : (surjetion) (onto mp) y x f(x) = y f() = f 49 (.7) d f d g.7: f : g :. 50 f :, f(x, y) = x f. 5 g :, g(x) = x, x g..0.4 (ijetion) f : g : g f (inverse funtion) x y (f(x) = y x = g(y)) g f = id, f g = id 5 E, O f : O E, f(x) = x g : E O, g(x) = x + f, g g f f g 4

15 . 7 (prtil funtion) f f; 5 div div R R R div; R R R div(x, 0) ( x R) (totl funtion) ( ) 7 0 5

( [1]) (1) ( ) 1: ( ) 2 2.1,,, X Y f X Y (a mapping, a map) X ( ) x Y f(x) X Y, f X Y f : X Y, X f Y f : X Y X Y f f 1 : X 1 Y 1 f 2 : X 2 Y 2 2 (X 1

( [1]) (1) ( ) 1: ( ) 2 2.1,,, X Y f X Y (a mapping, a map) X ( ) x Y f(x) X Y, f X Y f : X Y, X f Y f : X Y X Y f f 1 : X 1 Y 1 f 2 : X 2 Y 2 2 (X 1 2013 5 11, 2014 11 29 WWW ( ) ( ) (2014/7/6) 1 (a mapping, a map) (function) ( ) ( ) 1.1 ( ) X = {,, }, Y = {, } f( ) =, f( ) =, f( ) = f : X Y 1.1 ( ) (1) ( ) ( 1 ) (2) 1 function 1 ( [1]) (1) ( ) 1: