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.