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