Let A be a finite set Suppose that f A rightarrow A is an i

Let A be a finite set. Suppose that f : A rightarrow A is an injective function. Show that f is also surjective.

Solution

A is a finite set.

f : A ----> A is an injective function.

Injective function is also called one one function.

Injective function definition :  if every element of the codomain is mapped to by at most one element of the domain.

Now every element of domain of f is mapped to atmost one element of domain.

Domain of f is set A. Range of f is also the same set A.

Since number of elements in domain of f is same as number of elements of codomain and as it is one one function every element of domain of f is mapped to one element of domain.

So, every element of codomain of f is mapped to exactly one element of domain. Thus every element of codomain has a preimage. It is a surjective function.

Thus f is both one one and onto i.e. bijective.