information:extraction, structural:expansion
Power sets

The power set of a set S is the set of all subsets of S. The empty set and S itself are elements of the power set of S, because these are both subsets of S. For example, the power set of {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. The power set of a set S is commonly written as P (S) or 2.

If S has n elements, then P (S) has 2 elements. For example, {1, 2, 3} has three elements, and its power set has 2 = 8 elements, as shown above.

If S is infinite (whether countable or uncountable), then P (S) is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of S with the elements of P (S) will leave some elements of P (S) unpaired. (There is never a bijection from S onto P (S).)