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In terms of size: the null set, a finite set, a countably infinite set and an uncountably infinite set.
A countably infinite set is one where each element of the set can be put into a 1-to-1 correspondence with the set of natural numbers. For example, the set of positive even numbers. It is infinite, but each positive even number can me mapped onto one and only one counting number. The set of Real numbers cannot be mapped in such a way (as was proven by Cantor).
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