# Reflexive relation
A reflexive relation is a binary relation R on a set A in which every element is related to itself: (a, a) ∈ R for every a ∈ A, written a R a.[1](https://mathworld.wolfram.com/Reflexive.html) • [2](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) A binary relation on A is a set of [ordered pairs](https://www.edgepedia.org/ordered-pair) of elements of A, that is, a subset of A × A, so reflexivity is the requirement that the relation contain the whole diagonal of the [Cartesian product](https://www.edgepedia.org/cartesian-product).[2](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html)
| Key fact | Detail |
|---|---|
| Definition | R on A is reflexive if (a, a) ∈ R for every a ∈ A[1](https://mathworld.wolfram.com/Reflexive.html) |
| Companion properties | Symmetric: (b, a) ∈ R whenever (a, b) ∈ R; transitive: (a, c) ∈ R whenever (a, b) ∈ R and (b, c) ∈ R[2](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) |
| Equivalence relation | Reflexive, symmetric, and transitive; the identity relation on any set is the paradigmatic example[2](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) |
| Partial order | Reflexive, antisymmetric, and transitive; ⊆ on any set of sets is an example[2](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) |
| Reflexive closure | R ∪ {(a, a) : a ∈ A}, the smallest reflexive relation containing R[3](https://cse.sc.edu/~pfu/document/teaching/cs2112/lecture-note.pdf) |
| Matrix test | A reflexive relation has an adjacency matrix with every diagonal entry equal to 1; an irreflexive relation has diagonal entries all 0[4](https://www.cse.psu.edu/~gik2/teach/Discrete_Math-4.pdf) |
| Empty set | Every relation on the empty set is vacuously reflexive, symmetric, and transitive[5](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_%28Ernst%29/07%3A_New_Page/7.01%3A_New_Page) |
## Definition and examples
To prove that a relation on a finite set is reflexive, one checks each element a of A and confirms that (a, a) ∈ R; to disprove it, a single missing diagonal pair suffices.[6](https://louis.pressbooks.pub/discretemathematics/chapter/properties-of-relations/) On A = {1, 2, 3}, the relation R = {(1, 1), (2, 2), (3, 3)} is reflexive, and it is also symmetric and transitive.[7](https://runestone.academy/ns/books/published/DiscreteMathText/rstrelations8-2.html) The identity relation on any set is the paradigmatic example of a reflexive relation, and another example is the relation on the set of all finite sets of natural numbers consisting of pairs (a, b) such that a and b have the same number of elements.[2](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html)
**Checking the quantifier matters.** The definition quantifies over all elements of A, so the empty relation on a non-empty set is not reflexive, since diagonal pairs are missing; on the empty set itself, however, every relation is vacuously reflexive, symmetric, and transitive because there is no element a for which (a, a) could fail to be in R.[5](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_%28Ernst%29/07%3A_New_Page/7.01%3A_New_Page)
## Role among relation properties
Reflexivity is one of the three properties most often used to classify relations, alongside symmetry and transitivity.[7](https://runestone.academy/ns/books/published/DiscreteMathText/rstrelations8-2.html) A relation that is reflexive, symmetric, and transitive is an equivalence relation; if R is such a relation and (a, b) ∈ R, then a and b are called R-equivalent, and the equivalence class of a, denoted [a]_R, is the set of all elements of A that are R-equivalent to a.[2](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) A relation that is reflexive, antisymmetric, and transitive is called a (reflexive) partial order; removing all pairs (a, a) from a partial order yields a strict partial order, and the subset relation ⊆ on any set of sets is an example of a partial order.[2](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) Relations that are reflexive and transitive but not necessarily antisymmetric are called quasiorders or preorders.[8](https://www.cs.yale.edu/homes/aspnes/pinewiki/Relations.html)
## Representations and testing
Reflexivity has a direct visual and matrix form. In the directed graph of a relation, reflexivity means every vertex carries a loop back to itself, and directed graphs are a standard tool for identifying the reflexive, symmetric, and transitive properties.[7](https://runestone.academy/ns/books/published/DiscreteMathText/rstrelations8-2.html) In the adjacency matrix of a reflexive relation, every cell on the diagonal from upper left to lower right contains the value True, equivalently all diagonal entries are 1, while an irreflexive relation has a diagonal of all 0s.[4](https://www.cse.psu.edu/~gik2/teach/Discrete_Math-4.pdf)
## Reflexive closure
A relation need not be reflexive, and closure operations repair this. The reflexive closure of R is defined as {(a, a) | a ∈ A} ∪ R, the smallest reflexive relation on A that contains R.[3](https://cse.sc.edu/~pfu/document/teaching/cs2112/lecture-note.pdf) The diagonal relation Δ = {(a, a) : a ∈ A} is itself the smallest reflexive relation on a set, so the closure simply adds any missing self-pairs.[9](https://ww2.coastal.edu/dweathers/edm/basic-closures.html) In matrix terms, computing the reflexive closure amounts to setting every diagonal entry to 1.[3](https://cse.sc.edu/~pfu/document/teaching/cs2112/lecture-note.pdf)
Closures combine to build structured relations from arbitrary ones. Taking the reflexive, symmetric, and transitive closure of any relation R produces an equivalence relation containing R, sometimes called the equivalence closure.[8](https://www.cs.yale.edu/homes/aspnes/pinewiki/Relations.html) • [10](https://datafield.dev/discrete-mathematics/part-02/chapter-12/) Antisymmetry, by contrast, is not a closure property: adding pairs to a non-antisymmetric relation cannot make it antisymmetric, which is why turning an arbitrary relation into a partial order is trickier than turning it into an equivalence relation.[8](https://www.cs.yale.edu/homes/aspnes/pinewiki/Relations.html)
## References
1. [Reflexive, Wolfram MathWorld](https://mathworld.wolfram.com/Reflexive.html)
2. [Set Theory > Basic Set Theory, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html)
3. [Discrete Mathematics lecture notes, CS 2112, University of South Carolina](https://cse.sc.edu/~pfu/document/teaching/cs2112/lecture-note.pdf)
4. [Discrete Math slides, Penn State CSE](https://www.cse.psu.edu/~gik2/teach/Discrete_Math-4.pdf)
5. [7.1: Relations, Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_%28Ernst%29/07%3A_New_Page/7.01%3A_New_Page)
6. [4.2 Properties of Relations, Logical Thinking through Discrete Mathematics](https://louis.pressbooks.pub/discretemathematics/chapter/properties-of-relations/)
7. [Section 8.2 Reflexive, Symmetric, Transitive Properties, Runestone Discrete Mathematics](https://runestone.academy/ns/books/published/DiscreteMathText/rstrelations8-2.html)
8. [Relations, Yale CS notes (James Aspnes)](https://www.cs.yale.edu/homes/aspnes/pinewiki/Relations.html)
9. [Basic closures, Coastal Carolina University](https://ww2.coastal.edu/dweathers/edm/basic-closures.html)
10. [Chapter 12: Relations, Discrete Mathematics, DataField.Dev](https://datafield.dev/discrete-mathematics/part-02/chapter-12/)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Set theory › Elementary set theory*
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