%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 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