Difference between revisions of "Power set"
m (→Size comparison) |
IntrepidMath (talk | contribs) m (added all- now states: set of all subsets) |
||
Line 1: | Line 1: | ||
− | The '''power set''' of a given [[set]] <math>S</math> is the set <math>\mathcal{P}(S)</math> of [[subset]]s of that set. | + | The '''power set''' of a given [[set]] <math>S</math> is the set <math>\mathcal{P}(S)</math> of all [[subset]]s of that set. |
==Examples== | ==Examples== |
Revision as of 15:34, 2 September 2006
The power set of a given set is the set of all subsets of that set.
Contents
[hide]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.
See Also
This article is a stub. Help us out by expanding it.