Let $A\sim A'$ and $B\sim B'$ then $A\times B\sim A'\times B'$

Question

Let $A\sim A'$ and $B\sim B'$ then $A\times B\sim A'\times B'.$

Well I know that $A\sim A'$ means that exists a bijective function $f$ between the set $A$ and $A'$. Also $B\sim B'$ means that exists a bijective function $g$ between $B$ and $B'$.

I want a bijective function $h:A\times B \rightarrow A'\times B'$. I'm kind of stuck here.

Answer 1

Let $h:A\times B\to A'\times B'$ be defined by $h(a,b)=(f(a),g(b)),$ where $a\in A,b\in B,$ and $f$ and $g$ are the given bijections. Then, it is easy to check $h$ is a bijection (check injectivity and surjectivity.)

Answer 2

Let $f\colon A\to A', g\colon B\to B'$ be bijections. Then $h\colon A\times B\to A'\times B'$ defined by $h(a,b):=(f(a),g(b))$ is also bijective.