# Functions Problem 5

### Hints

It's not just you: this is a hard problem.
This proof isn't long, but it may take you a while to understand what
you need to do. It may help to work on the problem for a bit, then
put it away for a while (e.g. have dinner, go sleep), and come back to it.

Try drawing a bubble diagram showing the function r going in one
direction and the function s going in the opposite direction.
Add some extra elements to A and follow the arrows around to
see what additional elements you have to add to the diagram.

According to the definition of retraction, you'll have
\(r\circ s = id_A\). Use the definition of function
composition and the definition of \(id_A\) to make
a more concrete equation involving a specific element y.
What set does y need to belong to?

You need to prove that r is onto. Recall the basic outline for an
onto proof: pick an element t from r's co-domain and find an element of
r's domain that maps onto t (i.e. a pre-image of t). Sketch out this
outline, making sure you're clear on which set is r's co-domain and
which set is r's domain.

Another thing to remember about onto proofs: the hard part is
understanding how to take an element of the co-domain and find a
corresponding pre-image. Look at the equations you got from the
definition of retraction. Find the object t that lives in
r's co-domain. Look for an object (e.g. a variable or some
short expression) that lives in r's domain and maps onto t.