Statement : If  , where  is odd  then  exists.

The statement is true.

Because, if  is odd  then only the fuction has an inverse function since if  is even then the function is not an one to one.

Verify by the followin description.

Consider .

.

Therefore, the function  is not an one to one.

If  is odd then every element is mapped with different element and therefore the function is an one to one.

And it is strictly monotonic.

From the theorem 5.7 :

If the function  is strictly monotonic and it is one to one therefore the function has an inverse function.

Therefore, if  , where  is odd  then  exists.

The statement is true.

True.