Power set
The power set of a given set is the set
of all subsets of that set.
Contents
Examples
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.
Size comparison
Note that for any nonnegative integer ,
and so for any finite set
,
(where absolute value signs here denote the cardinality of a set). The analogous result is also true for infinite sets (and thus for all sets): for any set
, the cardinality
of the power set is strictly larger than the cardinality
of the set itself.
Proof
There is a natural injection taking
, so
.
Suppose for the sake of contradiction that
. Then there is a bijection
. Let
be defined by
. Then
and since
is a bijection,
.
Now, note that by definition if and only if
, so
if and only if
. This is a clear contradiction. Thus the bijection
cannot really exist and
so
, as desired.
Note that this proof does not rely upon either the Continuum Hypothesis or the Axiom of Choice. It is a good example of a diagonal argument, a method pioneered by the mathematician Georg Cantor.
See Also
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