Prove by induction that for any two non-empty finite sets M, N, the following holds |M x N| = |M||N|?

Question

non-empty finite sets

Answer 1

If u=|M| and v=|N|, that is, u and v are the number of elements in their respective sets, then we can represent M={ m₁, m₂, m₃, ..., mᵤ } and N={ n₁, n₂, n₃, ..., nᵥ }.

M×N={ (m₁,n₁), (m₁,n₂), (m₁,n₃), ..., (m₂,n₁), (m₂,n₂), (m₂,n₃), ..., (mᵤ,nᵥ) }.

So for element m₁ we have v products, therefore for all u elements of set M, we have a total of uv products in M×N, and so |M×N|=uv=|M||N|.