Function - Functions Definition of Function Definition Let...

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Unformatted text preview: Functions Definition of Function Definition: Let A and B be nonempty sets. A function f from A to B , which is denoted f : A → B, is a relation from A to B such that for all a ∈ A , f ( a ) contains just one element of B . A special kind of binary relation Under f , each element in the domain of f has a unique image. Note: the domain of ƒ :A → B is A , but the range is not necessarily equal to B. Image and counterimage Let f : A → B , A ’ ⊆ A , then ƒ ( A ’ )={ y | y = f ( x ) , x ∈ A ’ } is called the image of A ’ under f . An element in Dom( f ) corresponds a value A subset of Dom( f ) corresponds an image Let B ’ ⊆ B , then f-1 ( B ’)={ x | x ∈ A , f ( x ) ∈ B ’} is called the counterimage of B ’ under f . What is f -1 ( f ( A ’ )) ? A ’ ⊆ f -1 ( f ( A ’ )) ? f -1 ( f ( A ’ )) ⊆ A ’ ? B A B’ A’ f Image and Counterimage Special Types of Functions Surjection ƒ :A → B is a surjection or “ onto ” iff. Ran( ƒ ) =B, iff. 2200 y ∈ B, 5 x ∈ A, such that f (x)=y Injection (one-to-one) ƒ :A → B is one-to-one iff. 2200 y ∈ Ran( f ) , there is at most one x ∈ A, such that f (x)=y iff. 2200 x 1 ,x 2 ∈ A, if x 1 ≠ x 2 , then ƒ (x 1 ) ≠ ƒ (x 2 ) iff. 2200 x 1 ,x 2 ∈ A, if ƒ (x 1 ) = ƒ (x 2 ) , then x 1 =x 2 Bijection(one-to-one correspondence) surjection plus injection If A , B are nonempty sets how many different functions from A to B are there? |B| |A| how many Injection from A to B are there? If |A|>|B| then 0 else |A|!* |A| C |B| how many Bijection from A to B are there? If |A|= |B| = m then M! else 0 Special Types of Functions: Examples ƒ : Z + → R , ƒ (x)= ln x, one-to-one ƒ : R → Z, ƒ (x)= x , onto ƒ : R → R , ƒ (x)= 2x-1 , bijection ƒ : R × R → R × R , ƒ (<x,y>) = <x+y, x-y>, bijection Try to prove it What is ƒ ({<x,y>|x,y ∈ R, y=x+1})? R × {-1} ƒ :N × N → N, ƒ (<x,y>) = | x 2-y 2 | ƒ (N × {0}) ={ n 2 |n ∈ N}, ƒ-1 ({0}) ={<n,n>|n ∈ N} Characteristic Function of Set Let U be the universal set, for any A ⊆ U , the characteristic function of A , f A : U → {0,1} is defined as f A ( x )=1 iff. x ∈ A Natural Function R is an equivalence relation on set A, g : A → A/ R , for all a ∈ A, g ( a )= R (a), then G is called a natural function on A Natural function is surjection For any R ( a ) ∈ A/ R, there exists some x ∈ A , such that g ( x )= R (x) Images of Union and Intersection...
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This note was uploaded on 03/31/2010 for the course SE C0229 taught by Professor Tao during the Spring '08 term at Nanjing University.

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Function - Functions Definition of Function Definition Let...

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