Question

Let T= {ti: i exists in N} and S be sets that T is denumerable and there exists a surjection f: T->S. Show that there exists an injection of S->N.

I started off the problems applying the definitions. I must have made a big mistake somewhere because my teacher told me it was nonsense... Also I wrongfully assumed that S was denumerable. It might be possible to prove that it is, but I had just assumed and that was incorrect. Thank you!

Answer 1

f: T->S is a surjection, so a function g: S->T exists. g takes elements of S to t_i, i in N injectively since f exists. Intuitively g can be rewritten to take elements of S to N. Formally, let h:T->N, with h(t_i) = i. Then h(g): S->N.