Bijections?

Question

Prove that if A and B are finite and f : A -> b and g : B -> A are injections, then |A| = |B| and f and g are bijections.

Answer 1

>> |A| = |B|

This is a direct result of the injections f : A -> B and g : B -> A
An injection is a 1-to-1 relationship.

For f, this means more specifically that the number of elements in A is smaller than or equal to the number of elements in B, or |A| <= |B|
Similarly for g, we get |B| <= |A|

Combine these to get |A| = |B|

>> f and g are bijections

This is true because f and g are injections (given) as well as surjections (because we already established that |A| = |B|).