How to Show that cartesian product of finite sets is a finite set

Question

It asks to show it by using definitions and theorem of sets, but not using cardinality.

Suppose $A$, $B$ is finite sets, by definition it means

There is a bijective function $f : \mathbb{N}_n \rightarrow A$

There is a bijective function $g : \mathbb{N}_m \rightarrow B$

If I want to show that $A×B$ is finite sets, then I need to show that

There is a bijective function $h : \mathbb{N}_{mn} \rightarrow A×B $

But I'm stuck after defining $A$ and $B$

Answer 1

Using Euclidean division, any $k \in \mathbb{N}_{nm}$ may be written as $$k = qn + r$$ for unique $q \in \mathbb{N}_m$ and $r \in \mathbb{N}_n$. Then your desired bijection is given by $h(k) = (f(r), g(q))$.