Editing Set (mathematics) (section)

==External operations==
In {{alink|Basic operations}}, all elements of sets produced by set operations belong to previously defined sets. In this section, other set operations are considered, which produce sets whose elements can be outside all previously considered sets. These operations are [[Cartesian product]], [[disjoint union]], [[set exponentiation]] and [[power set]].

===Cartesian product===
{{main|Cartesian product|Direct product}}
The Cartesian product of two sets has already be used for defining functions.

Given two sets {{tmath|A_1}} and {{tmath|A_2}}, their ''Cartesian product'', denoted {{tmath|A_1\times A_2}} is the set formed by all ordered pairs {{tmath|(a_1, a_2)}} such that {{tmath|a_1\in A_1}} and {{tmath|a_i\in A_1}}; that is,
<math display=block>A_1\times A_2 = \{(a_1, a_2) \mid a_1\in A_1 \land a_2\in A_2\}.</math>

This definition does not supposes that the two sets are different. In particular,
<math display=block>A\times A = \{(a_1, a_2) \mid a_1\in A \land a_2\in A\}.</math>  

Since this definition involves a pair of indices (1,2), it generalizes straightforwardly to the Cartesian product or [[direct product]] of any indexed family of sets:
<math display=block>\prod_{i\in \mathcal I} A_i= \{(a_i)_{i\in \mathcal I}\mid (\forall i\in \mathcal I) \;a_i\in A_i\}.</math>
That is, the elements of the Cartesian product of a family of sets are all families of elements such that each one belongs to the set of the same index. The fact that, for every indexed family of nonempty sets, the Cartesian product is a nonempty set is insured by the [[axiom of choice]].

===Set exponentiation===
{{main|Set exponentiation}}

Given two sets {{tmath|E}} and {{tmath|F}}, the ''set exponentiation'', denoted {{tmath|F^E}}, is the set that has as elements all functions from {{tmath|E}} to {{tmath|F}}.

Equivalently, {{tmath|F^E}} can be viewed as the Cartesian product of a family, indexed by {{tmath|E}}, of sets that are all equal to {{tmath|F}}. This explains the terminology and the notation, since [[exponentiation]] with integer exponents is a product where all factors are equal to the base.

===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>-->

===Disjoint union===
{{main|Disjoint union}}
The ''disjoint union'' of two or more sets is similar to the union, but, if two sets have elements in common, these elements are considered as distinct in the disjoint union. This is obtained by labelling the elements by the indexes of the set they are coming from.

The disjoint union of two sets {{tmath|A}} and {{tmath|B}} is commonly denoted {{tmath|A\sqcup B}} and is thus defined as
<math display=block>A\sqcup B=\{(a,i)\mid (i=1 \land a\in A)\lor (i=2 \land a\in B\}.</math>

If {{tmath|1=A=B}} is a set with {{tmath|n}} elements, then {{tmath|1=A\cup A =A}} has {{tmath|n}} elements, while {{tmath|1=A\sqcup A}} has {{tmath|2n}} elements.

The disjoint union of two sets is a particular case of the disjoint union of an indexed family of sets, which is defined as 
<math display=block>\bigsqcup_{i \in \mathcal I}=\{(a,i)\mid i\in \mathcal I \land a\in A_i\}.</math>

The disjoint union is the [[coproduct]] in the [[category (mathematics)|category]] of sets. Therefore  the notation 
<math display=block>\coprod_{i \in \mathcal I}=\{(a,i)\mid i\in \mathcal I \land a\in A_i\}</math>
is commonly used.

==== Internal disjoint union ====

Given an indexed family of sets {{tmath|(A_i)_{i\in \mathcal I} }}, there is a [[canonical map|natural map]]
<math display=block>\begin{align}
\bigsqcup_{i\in \mathcal I} A_i&\to \bigcup_{i\in \mathcal I} A_i\\
(a,i)&\mapsto a ,
\end{align}</math>
which consists in "forgetting" the indices.

This maps is always surjective; it is bijective if and only if the {{tmath|A_i}} are [[pairwise disjoint]], that is, all intersections of two sets of the family are empty. In this case, <math display=inline>\bigcup_{i\in \mathcal I} A_i</math> and <math display=inline>\bigsqcup_{i\in \mathcal I} A_i</math> are commonly identified, and one says that their union is the ''disjoint union'' of the members of the family.

If a set is the disjoint union of a family of subsets, one says also that the family is a [[partition of a set|partition]] of the set.