Suppose A and B are both countably infinite sets. Prove there is a 1-1 correspondence between A and B.
The Attempt at a Solution
Since A is countably infinite, there exists a mapping f such that f maps ℕto A that is 1-1 and onto.
Similarly for B, there exists a mapping g such that g maps ℕto B that is 1-1 and onto.
Since g is 1-1 and onto, g^-1 exists and g^-1 maps B into ℕ
Hence g^-1(b)=n for all b in B.
Since f(n)=a for all n in ℕ, then f(g^-1(b))=a for all b in B since g^-1(b) yields its corresponding number from ℕ.
Since f and g are bijective, we can define h as a mapping from B to A by h(b)=a for all b in B by using the ordering created by the mapping g.
Does this make sense?