(1)Functions and are defined as follows:

Find functions and . Is the inverse of ? Is injective or surjective? How about ?

So (i.e. the identity element ).

I said is the inverse function of . I said is surjective because (hence not injective). I am not sure whether is injective or surjective.

(2)Suppose that and are surjections. Prove that the composite is also a surjection. So I started out by saying that . Then how would I go from this to showing that ?

(3)Let be a function. Prove that there exists a function such that if and only if is a surjection. So we want to prove the following: is a surjection. This is equivalent to saying, from my earlier post, that a function is surjective if and only if it has a right inverse. So given that , points back to the domain of . Then . Is this correct?

Thanks