Power set
The power set of a given set is the set of subsets of that set.
The empty set has only one subset, itself. Thus .
A set with a single element has two subsets, the empty set and the entire set. Thus .
A set with two elements has four subsets, and .
Similarly, for any finite set with elements, the power set has elements.
Note that for any set such that , , so the power set of any set has a cardinality at least as large as that of itself. Specifically, sets of cardinality 1 or 0 are the only sets that have power sets of the same cardinality, since if is a finite set with cardinality at least 2, then clearly has cardinality greater than 2. A similar result holds for infinite sets: for no infinite set is there a bijection between and .
Proof
See Also
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