---
title: "Equivalence Relations and Partitions"
description: "A relation that is reflexive, symmetric and transitive cuts a set into classes with no overlap. Modular arithmetic, quotient constructions and much of later algebra are this one idea."
canonical: https://lightmysky.com/learn/mathematics/equivalence-relations-and-partitions-mt_wgEupYwEUF
source: https://lightmysky.com/learn/mathematics/equivalence-relations-and-partitions-mt_wgEupYwEUF.md
retrieved: 2026-09-12
---

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# Equivalence Relations and Partitions

A relation that is reflexive, symmetric and transitive cuts a set into classes with no overlap. Modular arithmetic, quotient constructions and much of later algebra are this one idea.

Subject: Mathematics · Area: Mathematical Thinking · Ages 18 to 19
Page: https://lightmysky.com/learn/mathematics/equivalence-relations-and-partitions-mt_wgEupYwEUF

## Ready when they can

- Verify the three properties for a proposed relation
- Describe the classes of congruence modulo n
- Show that equivalence classes either coincide or are disjoint

## Lesson: Sorting sets with equivalence relations

A relation on a set is a collection of ordered pairs saying which elements stand related. It is reflexive when every element relates to itself, symmetric when each pair appears both ways, and transitive when chains link through. An equivalence relation meets all three at once. Equality passes: a equals a, a equals b gives b equals a, and chains link up. But a less than b fails at once, since a is never less than itself.

**Example.** Congruence modulo n groups integers by remainder. Two numbers share a class exactly when n divides their difference. Modulo 3 there are three classes: multiples of 3, numbers leaving remainder 1, and numbers leaving remainder 2. Modulo 5 the possible remainders are 0, 1, 2, 3 and 4, so there are five classes. Every integer lands in exactly one class.

**Tip.** Two equivalence classes either coincide or are disjoint, with no partial overlap. If the classes of a and b share an element c, symmetry and transitivity link every member of one class to the other, so each class sits inside the other. Lee is right about 2 and 5 modulo 3: 5 minus 2 equals 3, so 5 lies in the class of 2 and the classes coincide. A relation can be reflexive and symmetric yet fail transitivity, and that near miss is not an equivalence relation.

**Recap.** Check all three properties for the relation, sort integers into remainder classes, and remember that overlapping classes collapse into one.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Proof by Induction and Strong Induction](https://lightmysky.com/learn/mathematics/proof-by-induction-and-strong-induction-mt_7lvC02JBOC)
- [Sets and Functions in the Language of Proof](https://lightmysky.com/learn/mathematics/sets-and-functions-in-the-language-of-proof-mt_XAcHX_3DVz)

## Opens up

- [Groups, Subgroups and Symmetry](https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L)
- [Modular Arithmetic and Congruence Classes](https://lightmysky.com/learn/mathematics/modular-arithmetic-and-congruence-classes-mt_Lt4_Aazx2j)
- [Normal Subgroups and Quotient Groups](https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5)
- [Countable and Uncountable Sets](https://lightmysky.com/learn/mathematics/countable-and-uncountable-sets-mt_VTHPloNBbJ)
- [Homomorphisms, Cosets and Lagrange's Theorem](https://lightmysky.com/learn/mathematics/homomorphisms-cosets-and-lagranges-theorem-mt_yKiZ8sg7yN)