Editing Set (mathematics) (section)

===Indexed families===

Intuitively, an [[indexed family]] is a set whose elements are labelled with the elements of another set, the index set. These labels allow the same element to occur several times in the family.

Formally, an indexed family is a function that has the index set as its domain. Generally, the usual [[functional notation]] {{tmath|f(x)}} is not used for indexed families. Instead, the element of the index set is written as a subscript of the name of the family, such as in {{tmath|a_i}}.

When the index set is {{tmath|\{1,2\} }}, an indexed family is called an [[ordered pair]]. When the index set is the set of the {{tmath|n}} first natural numbers, an indexed family is called an {{tmath|n}}-[[tuple]]. When the index set is the set of all natural numbers an indexed family is called a [[sequence]].

In all these cases, the natural order of the natural numbers allows omitting indices for explicit indexed families. For example, {{tmath|(b,2,b)}} denotes the 3-tuple {{tmath|A}} such that {{tmath|1=A_1=b, A_2=2, A_3=b}}.

The above notations <math display=inline>\bigcup_{A\in \mathcal S} A</math> and <math display=inline>\bigcap_{A\in \mathcal S} A</math> are commonly replaced with a notation involving indexed families, namely
<math display=block>\bigcup_{i\in \mathcal I} A_i=\{x\mid (\exists i\in \mathcal I)\; x\in A_i\}</math> 
and
<math display=block>\bigcap_{i\in \mathcal I} A_i=\{x\mid (\forall i\in \mathcal I)\; x\in A_i\}.</math>

The formulas of the above sections are special cases of the formulas for indexed families, where {{tmath|1=\mathcal S = \mathcal I}} and {{tmath|1=i = A =A_i}}. The formulas remain correct, even in the case where {{tmath|1=A_i=A_j}} for some {{tmath|i\neq j}}, since {{tmath|1=A=A\cup A= A\cap A.}}