Editing Exponentiation (section)

===Sets as exponents===
{{See also|Function (mathematics)#Set exponentiation}}
A {{mvar|n}}-tuple <math>(x_1, \ldots, x_n)</math> of elements of {{mvar|S}} can be considered as a [[function (mathematics)|function]] from <math>\{1,\ldots, n\}.</math> This generalizes to the following notation.

Given two sets {{mvar|S}} and {{mvar|T}}, the set of all functions from {{mvar|T}} to {{mvar|S}} is denoted <math>S^T</math>. This exponential notation is justified by the following canonical isomorphisms (for the first one, see [[Currying]]):
: <math>(S^T)^U\cong S^{T\times U},</math>
: <math>S^{T\sqcup U}\cong S^T\times S^U,</math>
where <math>\times</math> denotes the Cartesian product, and <math>\sqcup</math> the [[disjoint union]].

One can use sets as exponents for other operations on sets, typically for [[direct sum]]s of [[abelian group]]s, [[vector space]]s, or [[module (mathematics)|modules]]. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, <math>\R^\N</math> denotes the vector space of the [[infinite sequence]]s of real numbers, and <math>\R^{(\N)}</math> the vector space of those sequences that have a finite number of nonzero elements. The latter has a [[basis (linear algebra)|basis]] consisting of the sequences with exactly one nonzero element that equals {{math|1}}, while the [[Hamel basis|Hamel base]]s of the former cannot be explicitly described (because their existence involves [[Zorn's lemma]]).

In this context, {{math|2}} can represents the set <math>\{0,1\}.</math> So, <math>2^S</math> denotes the [[power set]] of {{mvar|S}}, that is the set of the functions from {{mvar|S}} to <math>\{0,1\},</math> which can be identified with the set of the [[subset]]s of {{mvar|S}}, by mapping each function to the [[inverse image]] of {{math|1}}.

This fits in with the [[Cardinal exponentiation|exponentiation of cardinal numbers]], in the sense that {{math|1={{abs|''S''<sup>''T''</sup>}} = {{abs|''S''}}<sup>{{abs|''T''}}</sup>}}, where {{math|{{abs|''X''}}}} is the cardinality of {{math|''X''}}.