Indexed family

In mathematics, an indexed family is a collection of values associated with indices. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers.

Formally, an indexed family is the same thing as a mathematical function; a function with domain J and codomain X is equivalent to a family of elements of X indexed by elements of J. The difference is conceptual; indexed families are interpreted as collections instead of as functions. Every element of the image of the family's underlying function is an element of the family.

When a function f : J → X is treated as a family, J is called the index set of the family, the function image f(j) for j ∈ J is denoted x_j, and the mapping f is denoted {x_j}_j∈J or simply {x_j}.

Next, if the set X is the power set of a set U, then the family {x_j}_j∈J is called a family of sets indexed by J .

Definition. Let I and X be sets. The function

is called a family of elements in X indexed by I .

An indexed family can be turned into a set by considering the set ${\displaystyle {\mathcal {X}}:=\{x_{i}:i\in I\}}$, that is, the range of x. However, the mapping x does not need to be injective, that is, there may exist ${\displaystyle i,j\in I}$ with ${\displaystyle i\neq j}$ but ${\displaystyle x_{i}=x_{j}}$. Thus, ${\displaystyle |{\mathcal {X}}|\leq |\{x_{i}\}_{i\in I}|}$ where |A| denotes the cardinality of the set.

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