RA×B,R is a relation from A to B ， DomRA。

(a,b)R (a, c)R (a,b)R (a, c)R unless b=c function DomR=A， (everywhere)function。

Chapter 3 Functions

3.1 Introduction Definition3.1: Let A and B be nonempty sets. A

relation is a (everywhere)function from A to B, denoted by f : AB, if for every aA, there is one and only b B so that (a,b) f, we say that b=f (a). The set A is called the domain of the function f. If XA, then f(X)={f(a)|aX} is called the image of X. The image of A itself is called the range of f, we write Rf. If YB, then f -1(Y)={a|f(a)Y} is called the preimage of Y. A function f : AB is called a mapping. If (a,b) f so that b= f (a), then we say that the element a is mapped to the element b.

(everywhere)function: (1)Domf=A； (2)if (a,b) and (a,b')f, then b=b‘ Relation: (a,b),(a,b')R, function : if (a,b) and (a,b')f, then b=b‘ Relation: DomRA (everywhere)function: DomR=A

Example ： Let A={1,2,3,4},B={a,b,c},R1={(1,a),(2,b),(3,c)},R2={(1,a),(1,b),(2,b),(3,c),(4,c)},R3={(1,a),(2,b),(3,b),(4,a)}Example: Let A ={-2,-1, 0,1,2} and

B={0,1,2,3,4,5}. Let f={(-2,0),(-1,1), (0,0),(1,3),(2,5)}. f is a

(everywhere)function. X={-2,0,1}, f(X)=? Y={0,5}, f -1(Y)=?

Theorem 3.1: Let f be a (everywhere) function from A to B, and A1 and A2 be subsets of A. Then

(1)If A1A2, then f(A1) f(A2)

(2) f(A1∩A2) f(A1)∩f(A2)

(3) f(A1 A∪ 2)= f(A1)∪f(A2)

(4) f(A1)- f(A2) f(A1-A2) Proof: (3)(a) f(A1)∪f (A2) f(A1 A∪ 2)

(b) f(A1 A∪ 2) f(A1)∪f (A2)

(4) f (A1)- f (A2) f (A1-A2) for any y f (A1)-f (A2)

Theorem 3.2 ： Let f be a (everywhere) function from A to B, and AiA(i=1,2,…n). Then

n

ii

n

ii AfAf

11

)()()1(

n

ii

n

ii AfAf

11

)()()2(

2. Special Types of functions Definition 3.2 ： Let A be an arbitrary nonempty

set. The identity function on A, denoted by IA, is defined by IA(a)=a.

Definition 3.3.: Let f be an everywhere function from A to B. Then we say that f is onto(surjective) if Rf=B. We say that f is one to one(injective) if we cannot have f(a1)=f(a2) for two distinct elements a1 and a2 of A. Finally, we say that f is one-to-one correspondence(bijection), if f is onto and one-to-one.

The definition of one to one may be restated in the following equivalent form:

If f(a1)=f(a2) then a1=a2 for all a1, a2A Or If a1a2 then f(a1)f(a2) for all a1, a2A

Example:1) Let f: R(the set of real numbers)→C(the set of complex number), f(a)=i|a|;

2)Let g: R(the set of real numbers)→C(the set of complex number), g(a)=ia;

3)Let h:Z→Zm={0,1,…m-1}, h(a)=a mod m

onto ,one to one?

3.2 Composite functions and Inverse functions 1.Composite functions Relation ,Composition, Theorem3.3: Let g be a (everywhere)function

from A to B, and f be a (everywhere)function from B to C. Then composite relation f g is a (everywhere)function from A to C.

Proof: (1)For any aA, there exists cC such that (a,c) f g?

(2)For every aA, If there exist x,yC such that (a,x) f gand (a,y) f g ， then x=y?

Definition 3.4: Let g be a (everywhere) function from A to B, and f be a (everywhere) function from B to C. Then composite relation f g is called a (everywhere) function from A to C, we write f g:A→C. If aA, then(f g)(a)=f(g(a)).

Since composition of relations has been shown to be associative (Theorem 2.), we have as a special case the following theorem.

Theorem 3.4: Let f be a (everywhere) function from A to B, and g be a (everywhere) function from B to C, and h be a (everywhere) function from C to D. Then h(gf )=(hg)f

Exercise: P176 2,9,10,13,14, 28,37,38

Next: Inverse functionsThe Characteristic function of the set

P178 5.2CardinalityParadox