Prove that for all natural numbers k,m, Nk X Nm is finite.
I know the lemma "for every k belong to N, every subset of Nk is finite", could we just say that Nk X Nm is a subset of Nk, so it is finite....?
The hint is "consider the function f: NkXNm--->Nkm given by f(a,b)=(a-1)m+b. Use the Division Algorithm to show f is a one-to-one correspondence. This is a formal version of the product rule that says the number of elements in NkXNm is km
This makes it more complicated than I thought.