Well, there is a clear definition, and at least in theory you can always determine whether a number is a primer number or not, so I would say, yes.
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2 is the only member of both the set of all prime numbers and the set of all even numbers.
Finite, no.
Any set of prime numbers has a GCF of 1, which is the technical definition of being relatively prime but it's a bit redundant. The members of that set are already prime.
The set of counting numbers greater than one.
Prime numbers have only 2 factors and their set is not well defined because they do not follow an orderly mathematical pattern.