Reflexive relation

A binary relation on a set is reflexive when every element of the set is related to itself. Formally, a relation R on A is reflexive if (a, a) ∈ R for every a ∈ A.

Binary relation: A relation between elements of two sets, represented by ordered pairs. Reflexivity is a property of binary relations whose two arguments come from the same set.
Relation matrix: A matrix encoding a relation on a finite set, with entries indicating which ordered pairs belong to it. Reflexivity appears as 1s in every diagonal entry.
Equivalence relation: A relation that is reflexive, symmetric, and transitive. Reflexivity ensures every object belongs to its own equivalence class.
Antisymmetric relation: A relation R is antisymmetric if aRb and bRa together imply a = b. Antisymmetry restricts pairs of distinct elements; it neither requires nor forbids self-pairs.
Equality relation: The relation that holds between two objects exactly when they are equal. Equality relates every element to itself and is a standard reflexive example.
Set: A collection of distinct objects treated as a single mathematical object. The elements that must relate to themselves are exactly the members of the underlying set.
Reflexive closure: The smallest reflexive relation containing a given relation, formed by adding missing self-pairs. It repairs a relation's failures of reflexivity without adding other pairs.
Partial order: A relation that is reflexive, antisymmetric, and transitive. Reflexivity makes every object comparable to itself in the order.
Asymmetric relation: A relation R is asymmetric if aRb implies that bRa does not hold. Asymmetry rules out self-pairs, so a nonempty set cannot have an asymmetric reflexive relation.
Empty relation: A relation containing no ordered pairs. On a nonempty set it lacks every required self-pair, so it is not reflexive.

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