Can I show that two sets are equinumerous by showing a bijection between them?

For a homework assignment, I am asked to show that two sets A and B are equinumerous. I am wondering, if showing that the cardinality of A is equal to the cardinality of B, is enough to say a bijection exists which implies they are equinumerous? It is important to note that in the problems, the sets A and B are finite and countable. One of the problems is shown to give you a feel:

Prove that the following couples of sets are equinumerous by finding a one-to-one relation between them.

A is the fingers of one hand, B is the set of vowels

P.S. I don't want a solution, this is homework, I just want to know if I am on the right track on tackling the problem, thank you.

• You mean you only finitely-many fingers? Weird :). – Passing By Dec 7 '14 at 1:44

which says that if, given two sets $A,B$ , if you can find an injection $f: A \rightarrow B$ and an injection into $g: B \rightarrow A$, then you can conclude the two sets have the same cardinality. Of course, if $f,g$ as in the theorem exists, then you can construct an actual bijection, but this is not necessary if you can come up with $f,g$ as in the theorem.