Theorem 3.5: Let f be an everywhere function from A to B. Then (i) f  I A = f .

1. 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).

2. 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

3. (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.

4. 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.

5. 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

6. 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 a 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

7. 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 a 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.

8. 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 • (2)f -1 f=IA • Let f:A→B and g:B→A， • Is g the inverse function of f ? • f g?=IB and g  f ?=IA

9. 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.

10. 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.

11. 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:

12. 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);

13. 3.4 Cardinality • Definition 3.7: The empty set is a finite set of cardinality 0. If there is a one-to-one correspondence between A and the set {0,1,2,3,…, n-1}, then A is a finite set of cardinality n. • Definition 3.8: A set A is countably infinite if there is a one-to-one correspondence between A and the set N of natural number. We write |A|=|N|=0. A set is countable if it is finite or countably infinite.

14. Example: The set Z is countably infinite • Proof: f:N→Z,for any nN,

15. The set Q of rational number is countably infinite, i.e. |Q|=|N|=0. • |[0，1]|0. • Theorem 3.13: The R of real numbers is not countably infinite. And |R|=|[0，1]|. • Theorem 3.14: The power set P(N) of the set N of natural number is not countably infinite. And |P(N)|=|R|=1. • Theorem 3.15(Cantor’s Theorem): For any set, the power set of X is cardinally larger than X. • N, P (N),P (P (N)),…

16. 3.5 Paradox • 1.Russell’s paradox • AA,AA。 • Russell’s paradox: Let S={A|AA}. The question is, does SS? • i.e. SS or SS? • If SS, • If SS, • The statements " SS " and " SS " cannot both be true, thus the contradiction.

17. Exercise: P176 21,22,32,33 • P181 4, • Prove Theorem 3.11 • Prove:(1)|(0,1)|=|R| (2)|[0,1]|=|R| • Next: • Pigeonhole principle P88 3.3 • Permutations of sets P79-81, 3.1