This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 f1 ( 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 (onetoone) ƒ :A → B is onetoone 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(onetoone 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, onetoone ƒ : R → Z, ƒ (x)= x , onto ƒ : R → R , ƒ (x)= 2x1 ， bijection ƒ : R × R → R × R , ƒ (<x,y>) = <x+y, xy>, 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 2y 2  ƒ (N × {0}) ={ n 2 n ∈ N}, ƒ1 ({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...
View
Full
Document
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.
 Spring '08
 Tao

Click to edit the document details