1. Chapter 3 Functions
P181(Sixth Edition)
P168(Fifth Edition) 1

2. (1)If A1A2, then f(A1) f(A2) (2) f(A1∩A2) f(A1)∩f(A2)
Theorem 3.1: Let f be an everywhere function from A to B, and A1 and A2 be subsets of A. Then
(1)If A1A2, then f(A1) f(A2)
(2) f(A1∩A2) f(A1)∩f(A2)
(3) f(A1∪A2)= f(A1)∪f(A2)
(4) f(A1)- f(A2) f(A1-A2)
Proof: (3)(a) f(A1)∪f (A2) f(A1∪A2)
(b) f(A1∪A2) f(A1)∪f (A2)

3. (4) f (A1)- f (A2) f (A1-A2)
for any y f (A1)-f (A2)

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

5. 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, a2A Or
If a1a2 then f(a1)f(a2) for all a1, a2A

6. 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?

7. 3.2 Composite functions and Inverse 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.

8. Proof: (1)For every aA, If there exist x,yC such that (a,x) f gand (a,y) f g，then x=y?
(2)For any aA, there exists cC such that (a,c) f g ?
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 aA, then(f g)(a)=f(g(a)).

9. 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(gf )=(hg)f 9

10. Theorem 3.5: Let f be an everywhere function from A to B. Then
(i)f IA=f.
(ii)IB  f = f.
Proof. Concerning(i), let aA, (f  IA)(a) ?=f(a).
Property (ii) is proved similarly to property (i). 10

11. (1)if f and g are onto , then f  g is also onto.
Theorem 3.6: Let g be an everywhere function from A to B, and f be an everywhere function from B to C. Then
(1)if f and g are onto , then f  g is also onto.
(2)If f and g are one to one, then f  g is also one to one.
(3)If f and g are one-to-one correspondence, then f  g is also one-to-one correspondence
Proof: (1) for every cC, there exists aA such that f  g(a)=c 11

12. (2)one to one:if ab，then f  g(a)? f  g(b)
(3)f and g are one-to-one correspondence, f and g are onto. By (1), f  g are onto.
By (2),  f  g are also one to one. Thus f  g is also one-to-one correspondence. 12

13. Is the function’s inverse relation a function? No
2. Inverse functions
Inverse relation,
Function is a relation
Is the function’s inverse relation a function? No
Example: A={1,2,3},B={a,b}, f:A→B, f ={(1,a),(2,b),(3,b)} is a function, but inverse relation f -1={(a,1),(b,2),(b,3)} is not a function. 13

14. Proof: (a)(1)If f –1 is a function, then f is one to one
Theorem 3.7: :(a) Let f be a function from A to B, Inverse relation f -1 is a function from B to A if only if f is one to one
(b) Let f be an everywhere function from A to B, Inverse relation f -1 is an everywhere function from B to A if only if f is one-to-one correspondence.
Proof: (a)(1)If f –1 is a function, then f is one to one
If there exist a1,a2A such that f(a1)=f(a2)=bB, then a1?=a2
(2)If f is one to one，then f –1 is a function
f -1 is a function
For bB，If there exist a1,a2A such that (b,a1)f -1 and (b,a2) f -1,then a1?=a2 14

15. For any bB，there exists aA such that f (a)=?b (ii)f is one to one.
Proof: (b)(1)If f –1 is an everywhere function, then f is one-to-one correspondence.
(i)f is onto.
For any bB，there exists aA such that f (a)=?b
(ii)f is one to one.
If there exist a1,a2A such that f (a1)=f (a2)=bB, then a1?=a2
(2)If f is one-to-one correspondence，then f –1 is an everywhere function
f -1 is an everywhere function, for any bB，there exists one and only aA so that (b,a) f -1.
For any bB, there exists aA such that (b,a)?f -1.
For bB，If there exist a1,a2A such that (b,a1)f -1 and (b,a2) f -1,then a1?=a2 15

16. Proof: (1) f –1is onto (f –1 is an everywhere function from B to A
Definition 3.5: Let f be one-to-one (correspondence) between A and B. We say that inverse relation f -1 is the (everywhere) inverse function of f. We denoted f -1：B→A. And if f (a)=b then f -1(b)=a.
Theorem 3.8: Let f be one-to-one correspondence between A and B. Then the inverse function f -1 is also one-to-one correspondence.
Proof: (1) f –1is onto (f –1 is an everywhere function from B to A
For any aA,there exists bB such that f -1(b)=a)
(2)f –1 is one to one
For any b1,b2B, if b1b2 then f -1(b1) f -1(b2).
If f:A→B is one-to-one correspondence, then f -1：B→A is also one-to-one correspondence. The function f is called invertible. 16

17. Theorem 3.9: Let f be one-to-one correspondence between A and B.
Then
(1)(f -1)-1= f
(2)f -1  f=IA
(3)f  f -1=IB
Proof: (1)(f -1)-1= f
Let f:A→B and g:B→A，
Is g the inverse function of f ?
f g?=IB and g  f ?=IA

18. Theorem 3.10：Let g be one-to-one correspondence between A and B, and f be one-to-one correspondence between B and C. Then (fg)-1= g-1f -1
Proof: By Theorem 3.6, f g is one-to-one correspondence from A to C
Similarly, By theorem 3.7, 3.8, g-1 is a one-to-one correspondence function from B to A, and f –1 is a one-to-one correspondence function from C to B.

19. Theorem 3.11: Let A and B be two finite set with |A|=|B|, and let f be an everywhere function from A to B. Then
(1)If f is one to one, then f is onto.
(2)If f is onto, then f is one to one.
The prove are left your exercises(p189.36)

20. 3.3 The Characteristic function of the set
function from universal set to {0,1}
Definition 3.6: Let U be the universal set, and let AU. The characteristic function of A is defined as a function from U to {0,1} by the following:

21. Theorem 3. 12: Let A and B be subsets of the universal set
Theorem 3.12: Let A and B be subsets of the universal set. Then, for any xU, we have
(1)A(x)0 if only if A=
(2)A(x)  1 if only if A=U;
(3)A(x)≦B(x) if only if AB;
(4)A(x)  B(x) if only if A=B;
(5)A∩B(x)=A(x)B(x);
(6)A∪B(x)=A(x)+B(x)-A∩B(x);

22. Next: Cardinality, Paradox
Exercise:P188 9,10,13,14, 21,22,28, 32,33,36,37,38(Sixth)
OR P176 9,10,13,14, 21,22,28, 32,33,36,37,38(Fifth)
Next: Cardinality, Paradox
Pigeonhole principle P100(Sixth)P (Fifth)