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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