Question

Let A and B be two sets with a finite number of elements. Assume that there is injective mapping from A to B and that there is an injective mapping from B to A. Prove that there is a bijective mapping from A to B.

Answer 1

A function is said to be injective if $f\left(a\right)=f\left(a^{\prime }\right)$ implies that $a=a^{\prime }$.

A function is said to be surjective if for all $b\in B$, there exists $a\in A$ such

that $f\left(a\right)=b$.

A function is said to be bijective if it is injective and surjective.

We say that two sets A and B are equivalent,

written $A\sim B$ if and only if there exists a function $f:A\to B$ which is a

bijection.