To me, it seems logical that if I have two finite sets of equal size, and there is an injection between them, then that injection must be a bijection.
However, of course, we cannot just claim these things as various tricky counter examples have demonstrated in the past.
However, I am unable to prove the above claim. What would a proof of such a claim look like? I can see that we now need to prove that the mapping is also surjective, but I am not completely sure how to go about doing this.
Also, the condition above has that the sets are finite. What happens if we drop this condition? Does the claim no longer necessarily hold?
Best Answer
Hint: for two finite sets A and B, what would happen if an injection from A to B were not a surjection?
If $A,B$ are both infinite, let $A=B=\Bbb N$, then $f(x)=x+1$ is an injection from $A$ to $B$, but...