Injective function $h:A\times C→B\times D$

Question

Let $f:A→B,\;g:C→D$ be functions. Prove that if $f, g$ are injective, there's a function $h:A\times C→B\times D$, that is also injective.

How do I prove it?
I started by saying that $f(x)=x$, and $g(x)=x+1$ (so they are injective) but how do I continue?

Answer 1

You can easily check the function $h$, also denoted $f\times g$ \begin{align} h:A\times C&\longrightarrow B\times D, \\ (x,y)&\longmapsto \bigl(f(x),g(y)\bigr), \end{align} is injective.