Editing Set (mathematics) (section)

===Power set===
{{main|Power set}}

The ''power set'' of a set {{tmath|E}} is the set that has all subsets of {{tmath|E}} as elements, including the [[empty set]] and {{tmath|E}} itself.<ref name="Lucas1990" /> It is often denoted {{tmath|\mathcal P(E)}}. For example, 
<math display=block> \mathcal P(\{1,2,3\})=\{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}.</math>

There is a natural one-to-one correspondence ([[bijection]]) between the subsets of {{tmath|E}} and the functions from {{tmath|E}} to {{tmath|\{0,1\} }}; this correspondence associates to each subset the function that takes the value {{tmath|1}} on the subset and {{tmath|0}} elsewhere. Because of this correspondence, the power set of {{tmath|E}} is commonly identified with a set exponentiation:
<math display=block> \mathcal P(E)=\{0,1\}^E.</math>
In this notation, {{tmath|\{0,1\} }} is often abbreviated as {{tmath|2}}, which gives<ref name="Lucas1990" />{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/18/mode/2up 19]}}
<math display=block> \mathcal P(E)=2^E.</math>
In particular, if {{tmath|E}} has {{tmath|n}} elements, then {{tmath|2^E}} has {{tmath|2^n}} elements.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/20/mode/2up 20]}}
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If {{math|''S''}} is infinite (whether [[countable]] or [[uncountable]]), then {{math|''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 {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a [[bijection]] from {{math|''S''}} onto {{math|''P''(''S'')}}.)<ref name="BurgerStarbird2004">{{cite book|author1=Edward B. Burger|author2=Michael Starbird|title=The Heart of Mathematics: An invitation to effective thinking|url=https://books.google.com/books?id=M-qK8anbZmwC&pg=PA183|date =18 August 2004|publisher=Springer Science & Business Media|isbn=978-1-931914-41-3|page=183}}</ref>-->