Let f A Right arrow B be some function where A and B are fi

Let f : A Right arrow B be some function, where A and B are finite sets and |A| = |B|. Prove that f is one-to-one if and only if it is onto.

Solution

Let n = |A| =|B| i. e. A and B have same cardinality.f is 1-1 .As f is one-to-one, so every element of A gets a unique element of B.

Lets say, in addition to the range of A there were another element in B, then |B| would be at least one greater than |A|. In particular, there is an element of A for each element of B.

If two or more elements of A were mapped to the same element of B, then |A| would be at least one greater than the |B|.

This is not possible, which proves that f is one-to-one.