section_path: Powers of sets
section_name: Powers of sets
paragraph_text: Powers of sets
  
  The Cartesian product of two sets S and T is the set of the ordered pairs (x, y) {\displaystyle (x,y)} such that x ∈ S {\displaystyle x\in S} and y ∈ T. {\displaystyle y\in T.} This operation is not properly commutative nor associative, but has these properties up to canonical isomorphisms, that allow identifying, for example, (x, (y, z)), {\displaystyle (x,(y,z)),} ((x, y), z), {\displaystyle ((x,y),z),} and (x, y, z). {\displaystyle (x,y,z).}
  
  This allows defining the n th power S n {\displaystyle S^{n}} of a set S as the set of all n - tuples (x 1, …, x n) {\displaystyle (x_{1},\ldots,x_{n})} of elements of S.
  
  When S is endowed with some structure, it is frequent that S n {\displaystyle S^{n}} is naturally endowed with a similar structure. In this case, the term " direct product " is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example R n {\displaystyle \mathbb {R} ^{n}} (where R {\displaystyle \mathbb {R} } denotes the real numbers) denotes the Cartesian product of n copies of R, {\displaystyle \mathbb {R},} as well as their direct product as vector space, topological spaces, rings, etc.