I am trying to prove the following;
If $X$ and $Y$ are finite, then $|X \times Y| = |X||Y|$.
Now, I'll define a bijection $g:\mathbb{N_{n}} \rightarrow X$ and a bijection $f: \mathbb{N_{m}} \rightarrow Y$, so the respective cardinalities of $X$ and $Y$ are $n$ and $m$. So now I try to prove $|X \times Y| = nm$.
I'm thinking along the lines of defining $|X \times Y|$ as the union of $(x_k \times Y)$, and then stating that there are $m$ ordered pairs for a single $x_k$, and similarly $n$ ordered pairs for a single $y_k$, and so $nm$ ordered pairs in total, but this hardly constitutes as a proof so I'm looking for some help on how to prove it properly, thanks.
It is enough to establish a bijection $\mathbb{N_{n}} \times \mathbb{N_{m}} \to \mathbb{N_{nm}}$.
The map $(i,j) \mapsto (i-1)m+j$ is such a bijection.