--- title: "Equivalence Relations and Partitions" description: "A reflexive, symmetric and transitive relation divides a set into classes with no overlap. Modular arithmetic, quotient constructions and much of later algebra use this same 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Equivalence Relations and Partitions A reflexive, symmetric and transitive relation divides a set into classes with no overlap. Modular arithmetic, quotient constructions and much of later algebra use this same 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 S specifies ordered pairs of elements of S. Write a ~ b when the pair (a, b) belongs to it. Reflexive means a ~ a for every a. Symmetric means a ~ b implies b ~ a. Transitive means a ~ b and b ~ c imply a ~ c. An equivalence relation has all three properties. Equality satisfies them. The relation a < b on integers fails reflexivity because a < a never holds. **Example.** For a positive integer n, congruence modulo n means n divides a - b. It is reflexive because a - a = 0, symmetric because reversing a difference changes its sign, and transitive because a - c = (a - b) + (b - c). Integers are grouped by remainders 0 through n - 1. Use a = qn + r with 0 <= r < n, including for negative a. For example, -8 = (-3) times 3 + 1. *(drawing: A positive modulus divides zero, the negative of any multiple, and the sum of two multiples. These identities prove the three properties.)* The class [a] is the set of x with x ~ a. Reflexivity puts a in [a]. If c belongs to both [a] and [b], then a ~ c ~ b, so a ~ b. For any x in [a], x ~ a ~ b puts x in [b]. Reversing this argument gives [b] inside [a]. Thus classes with a shared element are equal. For example, 1 and 4 have the same class modulo 3 because their difference is divisible by 3. *(drawing: These classes continue in both directions by steps of 2. Every integer belongs to exactly one of them.)* A partition is a collection of nonempty, disjoint blocks covering a set. Equivalence classes form a partition. Conversely, define a ~ b when a and b lie in the same block; this is an equivalence relation. To find classes from listed pairs, collect every element related to a representative. On the empty set, the empty relation satisfies all three properties: there is no element or pair that can violate them. **Recap.** Check all three properties for the relation, sort integers into remainder classes, and remember that classes sharing an element are equal. *Lesson and questions written for this dataset.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Sets and Functions in the Language of Proof](https://lightmysky.com/learn/mathematics/sets-and-functions-in-the-language-of-proof-mt_XAcHX_3DVz) ## Optional connections - [Proof by Induction and Strong Induction](https://lightmysky.com/learn/mathematics/proof-by-induction-and-strong-induction-mt_7lvC02JBOC) ## 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)