Question about power of sets

Question

If two sets are finite and they have the same power, can we say that the two sets are equivalent? Is every finite set countable?

Answer 1

A set is countable if it is void or if it is the image of some function whose domain is the natural numbers. This includes any finite set.

Answer 2

The term "equivalent" requires context. It does not specify the relation. Clearly not all finite sets are the same. We do say that they are equinumerous.

For the second question, the definition of countable slightly vary from place to place. Sometimes it is simpler to exclude finite sets, in which case a countable set would refer to an infinite set of the same cardinality as $\Bbb N$; but in other places we include finite sets in the definition of countable.

This is largely a matter of what your focus is going to be and how inconvenient to say "Finite or countable", or "At most countable" vs. "Countably infinite" when you need to differ one from the other. For introductory purposes it is my experience that including finite sets in the definition might work better. But you should consult whoever is supplying you with the definition in the first place (books, notes, teachers, etc.) and they should give you the correct answer for that context.