The product of two bijections is a bijection

Statement

Let $f : A \to B$ and $g : C \to D$ be two bijections. Prove that the function $h : A \times C \to B \times D$ defined by $h(a, c) = \big(f(a),\, g(c)\big)$ is a bijection.

Solution

Since $f$ and $g$ are bijections, they have inverses $f^{-1} : B \to A$ and $g^{-1} : D \to C$. Define $k : B \times D \to A \times C, \qquad k(b, d) = \big(f^{-1}(b),\, g^{-1}(d)\big).$ We check that $k$ is the inverse of $h$: $k\big(h(a,c)\big) = k\big(f(a), g(c)\big) = \big(f^{-1}(f(a)),\, g^{-1}(g(c))\big) = (a, c),$ $h\big(k(b,d)\big) = h\big(f^{-1}(b), g^{-1}(d)\big) = \big(f(f^{-1}(b)),\, g(g^{-1}(d))\big) = (b, d).$ Having a (two-sided) inverse, $h$ is a bijection. $\blacksquare$