( [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

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1 , 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

2 ( [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 = X 2 ) (Y 1 = Y 2 ) (( x X 1 )f 1 (x) = f 2 (x)) 2.1 ( ) f X Y X, Y, f, ( ) X = {1, 2, 3}, Y = {4, 5} f : X Y x = 1, 2, 3 f(x) f 8 f j (j = 1, 2,, 8) 2 Y 1 = Y 2 [2] 2

3 f(1) f(2) f(3) f f f f f f f f f 2 (1) = 4, f 2 (2) = 4, f 2 (3) = 5. g(x) := max{x + 2, 4} g : X Y g(1) = max{1 + 2, 4} = 4, g(2) = max{2 + 2, 4} = 4, g(3) = max{3 + 2, 4} = 5. h(x) := [ ] x+7 2 h: X Y h(1) = [ ] 8 = 4, h(2) = 2 [ ] 9 = 4, h(3) = 2 [ ] 10 = 5. 2 f 2, g, h : f 2 = g = h. X, Y, 3 f(x) ( f 2 ) 2.1 ( ) X x Y X Y ( ) ( ) ( ) ( ) f (x) = sin f(x), f(0) = 1, f (0) = 0 ( ) f : R R ( ) f(x) x ( x f(x) ) ( ) f(x) f f(x) 3

4 f(x) f x (the image of x under f), f x (the mapping value at x) f x y f : x y f : x y y = f(x) X f (the domain of f, the domain of definition of f),, source set ( [3] X ) Y f Y f X A f(a) := {y ( x A) y = f(x)} ( {f(x) x A} ) A f (the image of A under f) X f f(x) = {y ( x X)y = f(x)} = {f(x) x X} f (the image of f) f (the range of f) 3 f = f = f X = f(x) = {f(x) x X} = Image(f). 2.2 (Y ),,,, [4], [5] ( ) target 1 x f(x) Y the codomain of f, the target (set) of f, the range of f f ( the range of f ) 4 4 {y x(x X, y = f(x)} ( f(x) ) f Y f 3 4 range f(x) Y 4

5 f(x) Y [1] Y f ( ) range range Y f(x) ( ) f(x) f, f X f X f f, f X 1: f(x) = x, f(x) x 1 f(x) 2. f(x) = x X, Y (a) X X f(x) x (b) Y Y f Y f f(x) R Y = R 2.3 Y = R f(x) = x 2 +2x+3, x R f(x) R, X = R OK ( f : X Y ). f(x) = 1, x R \ {0} f(x) R, X = R \ {0} OK. x f(x) = x, x [0, ) f(x) R, X = [0, ) OK. f(x) = log x, x (0, ) f(x) R, X = (0, ) OK. 1 (x > 0) f(x) = 0 (x = 0). X = R 1 (x < 0) (?) 5

6 2.4 Y 1 ( 1 ) X Y R N ( R N ) X Y ( ) ( ) X id X : X X id X (x) = x (x X) X (the identity mapping of X) 2.6 (Dirichlet ) f : R R, f(x) = { 1 (x Q) 0 (x Q). f(1) = 1, f(1/2) = 1, f( 2) = 0, f(π) = 0. f Dirichlet 2.7 ( ) X, Y X Y pr 1 : X Y X, pr 1 (x, y) = x. pr 2 : X Y Y, pr 2 (x, y) = y. pr j j X Y X = {,, }, Y = {, } (R 2 ) a, b, c, d R f : R 2 f(x, y) = (x, y ) ( ) ( ) ( ) x a b x =. y c d y R 2 f R 2 1 ad bc 0 ( ) 2.9 X :=, Y := R, f(a) := A, f : X Y 2.10 ( ) C (R; R) R R C ( ) X := C (R; R), Y := C (R; R), D : X Y f f D(f) = f (f X) 2.11 ( ) X, Y c Y f : X Y f(x) = c (x X) f (constant map) 6

7 2.12 ( ) X A X χ A : X R χ A (x) = { 1 (x A) 0 (x X \ A) χ A A (the characteristic function of A) 2.13 ( ) X Y i: X Y i(x) = x (x X) i (the inclusion map) 2.14 ( ) N R a 1, a 2,, a n, f : N R, f(n) = a n (n N) ( ) A n (a 1, a 2,, a n, ) N n=1 A n n N n=1 a n A n ( ) f : X Y, g : Y Z h(x) = g(f(x)) (x X) h: X Z ( ) h f g (the composition of f and g), (the composite mapping) g f : g f : X Z, (g f)(x) = g (f(x)) (x X). n {}}{ g f gf f f f f n 3.2 ( ) f : X Y, g : Y Z, h: Z W h (g f) = (h g) f. g f : X Z (g f)(x) = g(f(x)) (x X) h (g f): X W (h (g f))(x) = h ((g f) (x)) = h (g (f(x))). h g : Y W (h g)(y) = h(g(y)) (y Y ) (h g) f : X W ((h g) f) (x) = (h g) (f(x)) = h(g(f(x))). h (g f) = (h g) f. 7

8 1. f : X Y f id X = f, id Y f = f 4,,, ( ) f : X Y (an injection, injective) 1 1 (one to one) ( ) ( x X)( x X) (x x f(x) f(x )) ( ) ( x X)( x X) (f(x) = f(x ) x = x ) ( ) 4.1 ( ) x x prime (dash) - (en-dash), (em-dash) ( ) 4.2 X =, f : X R, f(x) = x, f ( ) 4.3 ( ) (( x 1 X) ( x 2 X) (x 1 < x 2 f(x 1 ) < f(x 2 )) f ) (( x 1 X) ( x 2 X) (x 1 < x 2 f(x 1 ) > f(x 2 )) f ) f : R R, f(x) = x f 1 : R R, f 1 (x) = x 2 1 1, f( 1) = f(1) f 2 : [0, ) R, f 2 (x) = x 2 [0, ) (x < x f 2 (x) < f 2 (x )) f 1 (x), f 2 (x) x 2 (1), (2) 8

9 4.5 ( ) f : X Y, g : Y Z (1) f g g f (2) g f f (3) g f g (4) g f f g (1) x, x X, x x f f(x) f(x ). g g(f(x)) g(f(x )). (g f)(x) (g f)(x ). g f. ( ) x, x X, g f(x) = g f(x ) f(x), f(x ) Y, g(f(x)) = g(f(x )) g f(x) = f(x ). f x = x. g f (2) x, x X, f(x) = f(x ) (g f)(x) = g(f(x)) = g(f(x )) = g f(x ). g f x = x. f (3) f : [0, ) R, f(x) = x (x [0, )), g : R R, g(y) = y 2 (y N) g f : [0, ) R, (g f)(x) = ( x ) 2 = x. g ( 1 1 g( 1) = 1 = g(1)) (4) y, y Y, y y f x, x X s.t. f(x) = y, f(x ) = y. g(y) = g(f(x)) = g f(x), g(y ) = g(f(x )) = g f(x ). g f g f(x) g f(x ). g(y) g(y ). g ( ) f : X Y (a surjection, surjective) (an onto mapping, onto) ( ) ( y Y ) ( x X) y = f(x) ( ) Y = f(x) ( Y = Image f) 9

10 ( f(x) Y ) ( y Y )( x X) y = f(x) Y f(x) Y = f(x). 4.7 f 1 : R R, f 1 (x) = x 2 f(r) = [0, ) ( f 1 f 1 (0) = 0 lim f 1 (x) =, ) R x f 1 (R) R f 3 : R [0, ), f 3 (x) = x 2 f 3 (R) = [0, ), [0, ) f 3 (R) = [0, ) (1) (2) 4.8 ( ) f : X Y, g : Y Z (1) f g g f (2) g f g (3) g f f (4) g f g f (1) z Z g ( y Y ) g(y) = z. y f ( x X) f(x) = y. (g f)(x) = g(f(x)) = g(y) = z. g f (2) z Z g f ( x X) (g f)(x) = z. x y := f(x) y Y, g(y) = g(f(x)) = (g f)(x) = z. g (3) f : [0, ) R, f(x) = x, g : R [0, ), g(y) = y 2 g f(x) = x (x [0, ]), g f = id [0, ) g f f (f(x) = 1 x ) (4) z Z z := g(y) g f ( x X) (g f)(x) = z. z = g(y) = g(f(x)) g y = f(x). f ( ) f : X Y f (a bijection, bijective) 1 1 (one-to-one correspondence) 10

11 4.2 (1 1 ) (one to one, ) 5 ( ) ( f : X Y ) X Y 1 (X and Y are in one-to-one correspondence) 1 1 (2 ) ( ) (1) f 1 : R R, f 1 (x) = x 2. ( 1 1 f 1 ( 1) = 1 = f 1 (1)) (f 1 (x) = 1 x R ) (2) f 2 : [0, ) R, f 2 (x) = x 2. ( ) (f 2 (x) = 1 x [0, ) ) (3) f 3 : R [0, ), f 3 (x) = x 2. (f 1 ) (y [0, ) x := y x R, f 3 (x) = y.) (4) f 4 : [0, ) [0, ), f 4 (x) = x 2. (f 2 ) (f 3 ) [0, ) [0, ) (X = {1, 2, 3}, Y = {4, 5}) f j (j = 2, 3, 4, 5, 6, 7) f 1 f 8 5 (X = {1, 2, 3}, Y = {4, 5, 6}) 3. X = {1, 2}, Y = {3, 4, 5} X Y 4.11 ( ) X, Y X, Y (1) X Y X Y. (2) X Y X Y. (3) X = Y f : X Y (i), (ii), (iii) 11

12 (i) f (ii) f (iii) f (4) X Y X = Y. n := X, m := Y, X = {x 1,, x n }, Y = {y 1,, y m } ( ) (1) f : X Y f(x i ) (1 i n) {f(x i ) 1 i n} Y X = n Y. n m f(x i ) = y i (1 i n) f : X Y f (2) f : X Y {f(x i ) 1 i n} = Y X = n Y. n m f(x i ) = y i (1 i m), f(x i ) = y 1 (m < j n) f : X Y f (3) (i) (ii) f : X Y f(x i ) (1 i n) {f(x i ) 1 i n} = n, Y {f(x i ) 1 i n} Y {f(x i ) 1 i n} = Y. f f : X Y {f(x i ) 1 i n} = Y. n = Y f(x i ) (1 i n) f (4) X Y (1) X Y, (2) X Y X = Y. X = Y (1) X Y f (3) f 4.12 ( ) X, Y K dim X, dim Y (1) X Y dim X dim Y. (2) X Y dim X dim Y. (3) dim X = dim Y f : X Y (i) f (ii) f (iii) f (4) X Y dim X = dim Y. 12

13 4.4 ( id X ) f : X Y f y Y f(x) = y x X x f (1 y ) 1 y = f(x) = f(x ) f x = x. y Y f(x) = y x X! (! 1 )! P (x) x x(p (x) y(p (y) y = x)). (!x) P (x) 1 1 f : X Y ( ) ( y Y )(!x X) f(x) = y. ( ) f 4. ( ) f 4.13 f : X Y (i) f (ii) ( y Y ) (!x X) f(x) = y. y Y f(x) = y x X y Y (1) g(y) := f(x) = y x (f y x) g : Y X g f (the inverse mapping of f) f 1 f 1 f inverse f : X Y (2) f 1 (y) := f(x) = y x X (y Y ) f 1 : Y X f ( ) ( : f : X Y Y f(x) ) ( ) : 13

14 f 1 = f, f 1 = f. f (3) ( x X)( y Y ) (y = f(x) x = f 1 (y)) f 1 (f y y = f(x) x f 1 (y) ) f x X, y Y y = f(x) x = f 1 (y) f : R R f(x) = x 2 (x R) f(2) = 4 f 1 (4) = 2 ( f 1 ) (3) f 1 (f(x)) = x (x X), f(f 1 (y)) = y (y Y ) 5 (f 1 f)(x) = x (x X), (f f 1 )(y) = y (y Y ). ( f 1 f : X X, f f 1 : Y Y ) (4) f 1 f = id X, f f 1 = id Y f : X Y (5) f 1 f = id X, f f 1 = id Y. g f 4.15 f : X Y, g : Y X ( ) g f = id X, f g = id Y ( : f ) f g = f 1 ( f g g f = g 1 ) 4.16 ( ( )) f : X Y ( ) g f ( ) 5 : x X y := f(x) x = f 1 (y) f 1 (f(x)) = f 1 (y) = x. y Y x := f 1 (y) y = f(x) f(f 1 (y)) = f(x) = y. 14

15 g f = id X f ( 4.5) f g = id Y f ( 4.8) f f 1 y x = f 1 (y) y = f(x) g(y) = g(f(x)) = id X (x) = x. g(y) = f 1 (y). g = f 1 g f = id X (g f) f 1 = id X f 1. g = f ( ) n X := {1, 2,, n} n 1 f : X X f g : X X g f = id X, f g = id X 4.15 f g = f 1. ( ) f g g g 1 = f 4.18 f f f 1 f 1 f ( f 1 ) 1 = f f 4 : [0, ) [0, ), f 4 (x) = x 2 f4 1 : [0, ) [0, ) f4 1 (y) = y 5. f(x) = e x f : X Y (X, Y (X f ) f ) f 1 f 1 (y) sin f : X Y, g : Y Z g f : X Z (g f) 1 = f 1 g 1. g f : X Z, f 1 g 1 : Z X (g f) (f 1 g 1 ) = ((g f) f 1 ) g 1 = (g (f f 1 )) g 1 = (g id Y ) g 1 = g g 1 = id Z, (f 1 g 1 ) (g f) = ( (f 1 g 1 ) g ) f = (f 1 (g 1 g)) f = (f 1 id Y ) f = f 1 f = id X. f 1 g 1 g f : (g f) 1 = f 1 g ( ) A, B n (BA) 1 = A 1 B 1. 5 f(x) f A f(a) 15

16 5.1 (, ( )) f : X Y (1) A X A f f(a) A f (the image of A under f) (the direct image of A under f) f(a) := {y ( x A) y = f(x)} = {f(x) x A}. (2) f X f f(x) f (the image of f) f (the range of f) Image(f) Image(f) := f(x) = {y ( x X) y = f(x)} = {f(x) x X}. f : X Y, a X, A X f(a) f(a) f f(a) Y f(a) Y ( x A) y = f(x) y {y P (y)} (P (y) y ) {f(x) x A} {y ( x A) y = f(x)} f f(x) = Image(f) = f = f = {f(x) x X}. f f 5.2 ( ) f : X Y B Y X f B f 1 (B) B f (the inverse image of B, pull-back) f 1 (B) := {x X f(x) B}. f f f 1 f 1 (B) [1] 5.3 f : R R, f(x) = sin x ({ f ({0}) = {0}, f 0, π }) ([ = {0, 1}, f ({0, π}) = {0}, f 0, π ]) = [0, 1], f ([0, π]) = [0, 1]. 2 2 ({ }) 1 { π } ( f 1 ({0}) = {x R ( n Z)z = nπ}, f 1 ({2}) = ϕ, f 1 =, f 1 [ 1 ) 2 6 2, 6] =.. f 1 f 1 (2), f 1 (0) 5.4 ( [1] ) [1] f(a) f 1 (B) f(a) f (A), f 1 (B) f (B) f(a) = f (A) = A f = {f(a) a A} = {y a(a A y = f(a))}. f 1 (B) = f (B) = B f = {x X f(x) B} = {x x X f(x) B}. f(a) f 1 (B) 16

17 X x f f(x) X A f f(a) ( ) f (B) f (A) f 1 f 1 (B) f 1 B f B f 1 f 1 B = { x X ( y) ( y B x = f 1 (y) )} = {x X ( y) (y B y = f(x))} = {x X f(x) B} = f B. (f 1 ) (B) = f (B) f 1 (B) f f ( ) Image(f), f (X), f(x) 3 (Image(f) ) f : X Y f(x) = Y f : X Y B Y f 1 : Y X B f {f 1 (b) b B} = { y b(b B y = f 1 (b)) } = {y b(b B b = f(y))} = {y f(y) B }. 5.5 f : X Y (1) f( ) =, f 1 ( ) =. (2) f 1 (Y ) = X. (3) ( x X) f({x}) = {f(x)}. (4) ( y Y ) f 1 ({y}) = {x X f(x) = y}

18 5.6 f : X Y, A 1, A 2 X (1) A 1 A 2 f(a 1 ) f(a 2 ). (2) f(a 1 A 2 ) f(a 1 ) f(a 2 ). (3) f(a 1 A 2 ) = f(a 1 ) f(a 2 ). (4) f(a 1 \ A 2 ) f(a 1 ) \ f(a 2 ). (5) f (1) (2) (4) ( ) f(a 1 ) f(a 2 ) A 1 A 2. ( ) f(a 1 A 2 ) = f(a 1 ) f(a 2 ). ( ) f(a 1 \ A 2 ) = f(a 1 ) \ f(a 2 ). (6) f ( ), ( ), ( ) A 1, A 2 (3) (2), (4) (5) (6) (1) A 1 A 2 y f(a 1 ) ( x A 1 ) y = f(x). A 1 A 2 x A 2. y f(a 2 ). f(a 1 ) f(a 2 ). (2) ( y f(a 1 A 2 ) (1) ) A 1 A 2 A 1 A 1 A 2 A 2 (1) f(a 1 A 2 ) f(a 1 ) f(a 1 A 2 ) f(a 2 ). f(a 1 A 2 ) f(a 1 ) f(a 2 ). (3) ( (1) ) (a) A 1 A 1 A 2 A 2 A 1 A 2 (1) f(a 1 ) f(a 1 A 2 ) f(a 2 ) f(a 1 A 2 ). f(a 1 ) f(a 2 ) f(a 1 A 2 ). 18

19 (b) y f(a 1 A 2 ) ( x A 1 A 2 ) y = f(x). x A 1 y f(a 1 ) y f(a 1 ) f(a 2 ). x A 2 y f(a 2 ) y f(a 1 ) f(a 2 ). y f(a 1 ) f(a 2 ). f(a 1 A 2 ) f(a 1 ) f(a 2 ). (a), (b) f(a 1 A 2 ) = f(a 1 ) f(a 2 ). ( ) ( ) y Y y f(a 1 A 2 ) ( x) (x A 1 A 2 y = f(x)) ( x) ((x A 1 x A 2 ) y = f(x)) ( x) ((x A 1 y = f(x)) (x A 2 y = f(x))) (( x) (x A 1 y = f(x)) ( x) (x A 2 y = f(x))) y f(a 1 ) y f(a 2 ) y f(a 1 ) f(a 2 ). f(a 1 A 2 ) f(a 1 ) f(a 2 ). ( ) ( (( x)p (x) Q(x)) (( x)p (x)) (( x)q(x)) 6 ) (4) y f(a 1 ) \ f(a 2 ) y f(a 1 ) y f(a 2 ). y f(a 1 ) ( x A 1 ) y = f(x). x A 2 x A 2 y f(a 2 ) x A 1 \ A 2 y f(a 1 \ A 2 ). (5) [1] pp (6) f ( x 1 X) ( x 2 X) x 1 x 2 f(x 1 ) = f(x 2 ). A 1 := {x 1 }, A 2 := {x 2 } f(a 1 ) = {f(x 1 )}, f(a 2 ) = {f(x 2 )} = {f(x 1 )} = f(a 1 ), f(a 1 A 2 ) = f( ) =, f(a 1 ) f(a 2 ) = {f(x 1 )} = f(a 1 ) f(a 2 ) A 1 A 2, f(a 1 A 2 ) f(a 1 ) f(a 2 ), A 1 := {x 1, x 2 }, A 2 := {x 2 } f(a 1 \ A 2 ) = f ({x 1 }) = {f(x 1 )}, f(a 1 ) \ f(a 2 ) = {f(x 1 ), f(x 2 )} \ {f(x 1 )} = {f(x 1 )} \ {f(x 1 )} = f(a 1 \ A 2 ) f(a 1 ) \ f(a 2 ). f f 1 ( ) 6 (( x)p (x) Q(x) (( x)p (x)) (( x)q(x))) 19

20 5.7 f : X Y, B 1, B 2 Y (1) B 1 B 2 f 1 (B 1 ) f 1 (B 2 ). (2) f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ). (3) f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ). (4) f 1 (B 1 \ B 2 ) = f 1 (B 1 ) \ f 1 (B 2 ). B Y f 1 (B c ) = (f 1 (B)) c. (5) f (1) (1) B 1 B 2 B 1 B 2 f 1 (B 1 ) f 1 (B 2 ). f 1 (B 1 ) = {x X f(x) B 1 } {x X f(x) B 2 } = f 1 (B 2 ). (2) x X x f 1 (B 1 F 2 ) f(x) B 1 B 2 f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ). (3) (2) (4) (2) f(x) B 1 f(x) B 2 x f 1 (B 1 ) x f 1 (B 2 ) x f 1 (B 1 ) f 1 (B 2 ). (5) (1) f f 1 (B 1 ) f 1 (B 2 ) B 1 B 2 y B 1 f ( x X) y = f(x). f(x) B 1 x f 1 (B 1 ). f 1 (B 1 ) f 1 (B 2 ) x f 1 (B 2 ). y = f(x) B 2. B 1 B f : X Y, A X, B Y (1) A X f 1 (f (A)) A. f f 1 (f (A)) = A. (2) B Y f (f 1 (B)) B. f f (f 1 (B)) = B. (3) f (A f 1 (B)) = f(a) B. 6 f : X Y graph f := {(x, y) x X y = f(x)} = {(x, f(x)) x X} 20

21 f graph f X Y G X Y ( ) x X!z G pr 1 (z) = x X Y X Y G ( ) [1], (2012). [2], II, (1965). [3], (1968),. [4], (1979). [5], (1972). 21

22 bijection, 10 bijective, 10 composite mapping, 7 composition (of mapping), 7 Dirichlet, 6, 7, 7 identity mapping, 6 inclusion map, 7 injection, 8 injective, 8 one to one, 8 one-to-one correspondence, 10 onto, 9 onto mapping, 9 surjection, 9 surjective, 9, 4 1, ( ), 8 1 1, 10, 9, 8, 8 ( ), 7, 7, 6, 6, 2, 9, 10 ( ), 4, 10, 8, 4, 7, 6, 6 22

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