---
tag: 05Q6G4EH2G
kind: definition
name: Family of sets
msc: ["03E20"]
defines: 0C21Q694AG
uses: [0SJ3RX9TXA]
sources:
  - url: https://en.wikipedia.org/wiki/Family_of_sets
    locator: lead section
---

## Definition

Let $I$ be a set.
A family of sets indexed by $I$ assigns to each $i \in I$ a [[set|0SJ3RX9TXA]] $A_i$. It is written $(A_i)_{i \in I}$.
The set $I$ is the index set.

## Remarks

Formally, a family indexed by $I$ is a [[function|0XHDTE5YMZ]] with [[domain|093CM69AGS]] $I$.
The same set can appear for two different indices: $A_i = A_j$ with $i \neq j$ is allowed.
Example: $A_n = \{0, 1, \dots, n\}$ for $n \in \mathbb{N}$ is a family indexed by $\mathbb{N}$.
Some authors use "family of sets" for any set of sets, without an index.

## Explanations

### L1 human

A family is a labelled list of sets, where the labels can be any set.
With labels $1, 2, 3$, you get three sets $A_1, A_2, A_3$.
With labels in $\mathbb{N}$, you get an infinite sequence of sets.