Question

Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.

Answer 1

$$\begin{gathered} A\mathrm{and}B\mathrm{are}\mathrm{two}\mathrm{non}\mathrm{empty}\mathrm{sets}. \\ \mathrm{Let}f\mathrm{be}\mathrm{a}\mathrm{function}\mathrm{from}A\mathrm{to}B. \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{there}\mathrm{is}\mathrm{injective}\mathrm{map}\mathrm{from}A\mathrm{to}B. \\ \mathrm{That}\mathrm{means}f\mathrm{is}\mathrm{one}-\mathrm{one}\mathrm{function}. \\ \mathrm{It}\mathrm{is}\mathrm{also}\mathrm{given}\mathrm{that}\mathrm{there}\mathrm{is}\mathrm{injective}\mathrm{map}\mathrm{from}B\mathrm{to}A. \\ \mathrm{That}\mathrm{means}\mathrm{every}\mathrm{element}\mathrm{of}\mathrm{set}B\mathrm{has}\mathrm{its}\mathrm{image}\mathrm{in}\mathrm{set}A. \\ \Rightarrow f\mathrm{is}\mathrm{onto}\mathrm{function}\mathrm{or}\mathrm{surjective}. \\ \therefore f\mathrm{is}\mathrm{bijective}. \\ \left(\mathrm{If}\mathrm{a}\mathrm{function}\mathrm{is}\mathrm{both}\mathrm{injective}\mathrm{and}\mathrm{surjective},\mathrm{then}\mathrm{the}\mathrm{function}\mathrm{is}\mathrm{bijective}.\right) \end{gathered}$$