Definition:Equivalence Relation

Definition

A relation on a set $S$ which is:

• reflexive,
• symmetric and
• transitive

is called an equivalence relation, or an equivalence, on $S$.

When discussing equivalence relations, various notations are used for $\left({x, y}\right) \in \mathcal R$. Examples are:

• $x \equiv y \left({\mathcal R}\right)$
• $x \sim y$

and so on.

Specialised equivalence relations generally have their own symbols, which can be defined as they are needed.

Also see

• Equivalence Class
• Quotient Set
• Quotient Mapping, also known as the Canonical Surjection

• Relation Partitions a Set iff Equivalence

• Results about equivalence relations can be found here.

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 3$
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 7$: Relations
• W.E. Deskins: Abstract Algebra (1964): $\S 1.2$: Definition $1.5$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 2.2$: Definition $4$
• Seth Warner: Modern Algebra (1965): $\S 10$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.3$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.4$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.6$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 17$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 6$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.3$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 16$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$
• John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Definition $2.24$