section_path: Powers of sets/Sets as exponents
section_name: Sets as exponents
paragraph_text: Sets as exponents
  
  A n -tuple (x 1, …, x n) {\displaystyle (x_{1},\ldots,x_{n})} of elements of S can be considered as a function from { 1, …, n }. {\displaystyle \{1,\ldots,n\}.} This generalizes to the following notation.
  
  Given two sets S and T, the set of all functions from T to S is denoted S T {\displaystyle S^{T}}. This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying):
  
  where × {\displaystyle \times } denotes the Cartesian product, and ⊔ {\displaystyle \sqcup } the disjoint union.
  
  One can use sets as exponents for other operations on sets, typically for direct sums of abelian groups, vector spaces, or modules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} denotes the vector space of the infinite sequences of real numbers, and R (N) {\displaystyle \mathbb {R} ^{(\mathbb {N})}} the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis consisting of the sequences with exactly one nonzero element that equals 1, while the Hamel bases of the former cannot be explicitly described (because their existence involves Zorn's lemma).
  
  In this context, 2 can represents the set { 0, 1 }. {\displaystyle \{0,1\}.} So, 2 S {\displaystyle 2^{S}} denotes the power set of S, that is the set of the functions from S to { 0, 1 }, {\displaystyle \{0,1\},} which can be identified with the set of the subsets of S, by mapping each function to the inverse image of 1.
  
  This fits in with the exponentiation of cardinal numbers, in the sense that | S | = | S |, where | X | is the cardinality of X.