1. The problem statement, all variables and given/known data Let f: A --> B be an injection and suppose that the set A is countably infinite; how can I prove that there is an injection from B to A if and only if B is countably infinite? Also, if we would suppose that A is uncountable, can B be countable? 2. Relevant equations 3. The attempt at a solution Here is what I have thus far, First direction: Suppose B is countably infinite. Then, by definition, there is a bijection from B to the naturals. Since A is also countably infinite, there is a bijection from A to the naturals, and hence a bijection between B and A (and hence injection from B to A). Next direction: First show that since f is an injection of a countably infinite set to B, then B must be infinite. Now, if there is an injection from B to A, then there is a bijection from B to a subset of A, call this subset S. So B and S have the same cardinality. But any infinite subset S of a countably infinite set A is countably infinite, so B has the same cardinality as a countably infinite set S.