Theorem 5.5: Let A and B be sets. There is a one to one correspondence

if and only if #(A ) = #(B ).

Proof: Assume that there is a one to one correspondence

Let #(A ) = n. Then, by Definition 4.4, there exists a one to one correspondence

Then by Theorem 3.4,

is a one to one correspondence, and by Definition 4.4,

For the converse, assume that #(A ) = #(B ) = n. Then, by Definition 4.4, there exist one to one correspondences

and

Then, by Theorem 3.7, g^-1 is a one to one correspondence, so by Theorem 3.4,

is a one to one correspondence.