Prove that if A is a denumerable set and there exists a surj

Prove that if A is a denumerable set, and there exists a surjective function from A to D (and B is infinite), then B is denumerable.

Solution

The case A is empty is trivial, so we may dispose of it. Based on the formulation of what you are trying to prove, it seems the meaning of denumerable for you is that of a set that is either finite or has the same cardinality as N. So, prove first that a set S denumerable if, and only if, there exists a surjection f:NA.

Now, if g:AB is surjective and A is denumerable, then there exists a surjection f:NA. The composition of surjective functions is surjective, thus gf:NB is a surjection, so B is denumerable.