Indexed Families of Sets — Mathematics Knowledge System

Overview

An indexed family of sets is a collection of sets labeled by an index set I, written {A_i}_{i∈I}, which is formally just a function A that assigns a set A_i to each index i ∈ I. This is a small but powerful generalization of an ordinary list or sequence of sets: when I = {1,2,3} you recover an ordinary finite list A_1, A_2, A_3, and when I = ℕ you recover an infinite sequence of sets — but I can just as well be uncountable, letting you talk about 'a set for every real number' or 'a set for every point in the plane.' Indexed families let set operations like union and intersection generalize cleanly beyond two sets, or even beyond countably many sets, to ANY collection, however large, via the indexed union ⋃_{i∈I} A_i and indexed intersection ⋂_{i∈I} A_i.

Intuition

Ordinary set union and intersection only ever combine two sets at a time, or (with repeated application) finitely many. Indexing lifts a whole collection of sets — possibly infinite, possibly uncountable — under a single label i drawn from an index set I, the way a spreadsheet column header 'i' can range over row numbers. The indexed union ⋃A_i then just asks 'is x in AT LEAST ONE of the sets in the family?' and the indexed intersection ⋂A_i asks 'is x in EVERY set in the family?' — exactly the two-set definitions, just replacing 'this set or that set' with 'some/every set in the family.' A useful mental picture: if A_r is the open disk of radius r centered at the origin for every real r > 0, then ⋃_{r>0} A_r is the whole plane (every point lies in SOME sufficiently large disk), while ⋂_{r>0} A_r is just the single point at the origin (only the origin lies in EVERY disk, however small).

Formal Definition

Definition

Given an index set I, an indexed family of sets is a function A: I → 𝒰 (into some universe of sets), written i ↦ A_i. The indexed union and intersection over the family are defined pointwise via the existential/universal quantifier over the index set.

\{A_i\}_{i \in I}, \quad A: I \to \mathcal{U},\; i \mapsto A_i

An indexed family of sets

\bigcup_{i \in I} A_i = \{x \mid \exists i \in I,\; x \in A_i\}

Indexed union

\bigcap_{i \in I} A_i = \{x \mid \forall i \in I,\; x \in A_i\}

Indexed intersection

\left(\bigcup_{i\in I} A_i\right)^{c} = \bigcap_{i \in I} A_i^{c}, \qquad \left(\bigcap_{i\in I} A_i\right)^{c} = \bigcup_{i \in I} A_i^{c}

De Morgan's laws for indexed families

Notation

Notation	Meaning
$\{A_i\}_{i \in I}$	An indexed family of sets, one set A_i for each index i in the index set I
$\bigcup_{i \in I} A_i$	The indexed union: elements belonging to at least one A_i
$\bigcap_{i \in I} A_i$	The indexed intersection: elements belonging to every A_i
$\prod_{i \in I} A_i$	The indexed (Cartesian) product: the set of all functions choosing one element from each A_i
$\{A_n\}_{n \in \mathbb{N}}$	The special case of a countably infinite sequence of sets

Properties

Indexed De Morgan (union)

\left(\bigcup_{i\in I} A_i\right)^{c} = \bigcap_{i \in I} A_i^{c}

Indexed De Morgan (intersection)

\left(\bigcap_{i\in I} A_i\right)^{c} = \bigcup_{i \in I} A_i^{c}

Distributivity over indexed union

B \cap \bigcup_{i \in I} A_i = \bigcup_{i \in I} (B \cap A_i)

Nested family monotonicity

I' \subseteq I \implies \bigcap_{i \in I} A_i \subseteq \bigcap_{i \in I'} A_i

Example: Intersecting over fewer sets can only make the intersection bigger (or equal)

Empty index set convention

\bigcup_{i \in \emptyset} A_i = \emptyset, \qquad \bigcap_{i \in \emptyset} A_i = \mathcal{U} \text{ (the universe)}

Applications

A partition of a database or distributed system into shards is an indexed family {A_i}_{i∈I} of disjoint subsets whose indexed union recovers the full dataset.

Worked Examples

The indexed union collects everything that appears in at least one A_i: {1,2}∪{2,3}∪{3,4} = {1,2,3,4}.
$\bigcup_{i=1}^{3} A_i = \{1,2,3,4\}$
The indexed intersection keeps only elements in ALL three sets. 1 is missing from A_2 and A_3; 2 is missing from A_3; 3 is missing from A_1; 4 is missing from A_1 and A_2 — no element survives all three.
$\bigcap_{i=1}^{3} A_i = \emptyset$

Answer: ⋃A_i = {1,2,3,4}; ⋂A_i = ∅.

Practice Problems

Difficulty 3/10

Let I = {1,2} with A_1 = {a,b,c} and A_2 = {b,c,d}. Compute ⋃_{i∈I}A_i and ⋂_{i∈I}A_i.

Difficulty 4/10

By De Morgan's law for indexed families, the complement of ⋂_{i∈I} A_i equals:

⋂_{i∈I} A_i^c⋃_{i∈I} A_i^c⋃_{i∈I} A_i∅

Difficulty 6/10

A distributed database shards its 1,000,000 user records into an indexed family {A_i}_{i∈I} of 10 disjoint shards (I={1,...,10}), each shard holding exactly 100,000 records, and ⋃_{i∈I}A_i equal to the full user set. What must ⋂_{i∈I}A_i equal, and why does disjointness force this?

Common Mistakes

Common Mistake

Assuming indexed union/intersection only make sense for finite or countable index sets.

The index set I can be any set at all, including uncountable ones (e.g. I = ℝ) — the quantifier-based definitions (∃i, ∀i) work identically regardless of the size of I.

Common Mistake

Confusing an indexed family {A_i}_{i∈I} with the set {A_i : i∈I} of DISTINCT sets appearing in it.

A family allows repeats — different indices i≠j can point to the SAME set A_i = A_j — so the family retains more information (which index gave which set) than the plain set of its values.

Quiz

Summary

An indexed family {A_i}_{i∈I} is a function from an index set I to a universe of sets, generalizing finite lists and infinite sequences of sets.
Indexed union ⋃A_i = {x : ∃i, x∈A_i} and indexed intersection ⋂A_i = {x : ∀i, x∈A_i} extend binary union/intersection to any collection.
De Morgan's laws extend cleanly: the complement of an indexed union is the indexed intersection of complements, and vice versa.
Partitions are the special case of pairwise-disjoint indexed families whose indexed union is the whole space.

References

WebsiteWikipedia — Indexed family
WebsiteWikipedia — Union (set theory)