# Functions whose Inverses are Functions

Recall that we defined inverses for any relation:

Definition. If is a relation from to , then is a relation from to , defined by iff .

Thus (in our formalism) every function has an inverse. The inverse of a function so defined, however, may not be a function itself! (Remember that most operations we do to sets or to relations will not preserve functionhood.) So let's discover when taking the inverse (swapping and ") actually does result in a function.

Recall that means that two conditions hold:

1. .

Exercise 1. Suppose . By substituting for , for , and for , write what it means for .

Exercise 2. Use the definition of to rewrite your conditions solely in terms of (with no anywhere).

Definition.  Let . We call onto or surjective if . We call one-to-one or injective if for any , we have that guarantees .

Definition. A function which is both one-to-one and onto is called a one-to-one correspondence or a bijection.

The following theorem is important, but its proof is just the combination of Exercises 1 and 2.

Theorem. Let . Then iff is both one-to-one and onto.

Theorem.  Suppose is one-to-one and is one-to-one. Then is one-to-one.

Proof. Note that the condition that be defined on the range of is necessary in order to compose the two functions as functions.

Let . Suppose . Applying the fact that the composition of functions is a function, we have

Since is one-to-one, we have . Since is one-to-one, we have . Since were arbitrary, we've shown is one-to-one. .

Theorem. Suppose is onto and is onto. Then is onto.

Proof. This is an exercise for you.