Proving the equivalence of a finite set

Question

Let A be a finite set. Prove that if A≈n and A≈m, then n=m.

The answer in the book uses a max function, so I was just wondering if there was a simpler way. If not, it would be appreciated if someone could explain it a little better. Thanks!

Answer 1

You could prove the contrapositive statement.

Suppose $n \neq m$. Then prove there is no bijection between $\{1, \dots, n\}$ and $\{1, \dots, m \}$. If you know about the pigeonhole principle, then this part is easy, because you just invoke it. By the pigeonhole principle, we can't find such a bijection if $n \neq m$, thus we've proven what we want.

Since the contrapositive is true, it follows that if $\{1, \dots, n \}$ is in bijection with $\{1, \dots, m \}$, then $n = m$.