%o Proof:(b)(1)Iff- is an everywhere function, then f Is one-to-one correspondence. ☆( if is onto For any b∈B, there exists a∈ A such that f(a)=2b ☆(i) f is one to one ☆ If there exist a1;a2∈ A such that f(a1)f(a2)=b∈B, then a,?=a 2 4(2)ffis one-to-one correspondence, then f-lis a everywhere function %- is an everywhere function, for any beB, there exists one and only a∈ A so that(b,a)∈f1. For any b∈B, there exists a∈ A such that(ba)∈?f1 Forb∈B, If there exist a1,a2∈ A such that(b,a1)∈f1 and(b, a2)ef-, then a1?=a2
❖ Proof: (b)(1)If f –1 is an everywhere function, then f is one-to-one correspondence. ❖ (i)f is onto. ❖ For any bB，there exists aA such that f (a)=?b ❖ (ii)f is one to one. ❖ If there exist a1 ,a2A such that f (a1 )=f (a2 )=bB, 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 bB，there exists one and only aA so that (b,a) f -1 . ❖ For any bB, there exists aA such that (b,a)?f -1 . ❖ For bB，If there exist a1 ,a2A such that (b,a1 )f -1 and (b,a2 ) f -1 ,then a1?=a2
Definition 3.5: Let f be one-to-one correspondence between A and B We say that inverse relation f-l is the everywhere inverse function of f. We denoted f B-A And iff(a)=b then f-(b)=a &o Theorem 3.8: Letf be one-to-one correspondence between A and B. Then the inverse function f-lis also one-to-one correspondence. 8 Proof: (1)f-is onto(f- is a function from B to A For any a∈A, there exists b∈ B such that f(b=a) 冷(2f- is one to one For any b1,b2∈B,ifb1≠b2 then f1(b1)≠f1(b2) 冷IffA→ B is one-to-0 ne correspondence, then f1 B-A is also one-to-one correspondence. The function f is called invertible
❖ 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 –1 is onto (f –1 is a function from B to A ❖ For any aA,there exists bB such that f -1 (b)=a) ❖ (2)f –1 is one to one ❖ For any b1 ,b2B, if b1b2 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