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An indexed family is a collection of objects that are indexed by some indices. They are denoted as $\{A_i\}_{i\in I}$ .

## Definition

Let $S$ be a set. Let $I$ be a set.

Then, a family of elements of $S$ indexed by elements of $I$ is a mapping $a:I\to S$ .

$I$ is called the indexing set.

### Notation

The image of an element of $I$ is denoted as $S_i$ .

That is, $a(i)=S_i$ .

A a family of elements of $S$ indexed by $I$ is denoted as $\{S_i\}_{i\in I}$ . If the indexing set is known, they may be denoted as $\{S_i\}$ .