Reflexive relation
A reflexive relation on a set is one that relates every element to itself. Covers definition, examples and non-examples, connections to equivalence relations and orders, and basic properties.
A reflexive relation is a basic concept in set theory and discrete mathematics. Informally, a relation is reflexive when every element of the underlying set stands in the relation to itself. Reflexivity is one of several standard properties used to classify relations and to build more structured concepts such as equivalence relations and partial orders.
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Definition
Given a set A and a relation R on A (that is, R is a set of ordered pairs whose members come from A), R is called reflexive if for every element x in A the pair (x,x) belongs to R. In symbolic form: R is reflexive on A if ∀x∈A, (x,x)∈R. The word "relation" here can be explored further: relation is the general concept that includes reflexive and many other kinds of pairings.
Examples and non-examples
- Equality on any set is reflexive because every element equals itself.
- The universal relation (all possible ordered pairs from A×A) is reflexive because it contains every (x,x).
- For the set A={1}, the relation {(1,1)} is reflexive; this simple instance can be linked to a concrete ordered pair example: (1,1).
- Strict less-than (<) on the real numbers is not reflexive because no number is strictly less than itself.
- A relation that misses even one self-pair (x,x) for x in A is not reflexive on A.
Related properties and constructions
Reflexivity often appears together with other properties. An equivalence relation is reflexive, symmetric and transitive; a partial order is reflexive, antisymmetric and transitive. If a relation is missing some self-pairs, its reflexive closure is obtained by adding all missing (x,x) for x in A. Opposite notions include irreflexive (no element is related to itself) and non-reflexive (not reflexive, meaning at least one self-pair is missing).
Importance and representations
Reflexivity plays a role in classification, algebraic structures and computer science. In directed graphs, reflexivity corresponds to loops at every vertex. When relations are represented by a boolean matrix, reflexivity means all diagonal entries are 1. Identifying whether a relation is reflexive is a simple but essential step in recognizing more complex structures and in algorithms that manipulate relations.
Recognizing reflexive relations is straightforward: check the presence of (x,x) for every element of the set. This property, although elementary, underpins many definitions and constructions throughout mathematics and theoretical computer science.
Formal definition
If 
a set and 
a two-digit relation on , then one defines (using infix notation):
is reflexive : 
is irreflexive : 
Examples
Reflexive
- The less than or equal relation ≤
on the real numbers is reflexive, since always 
holds. Moreover, it is a total order. The same holds for the relation ≥ 
.
- The ordinary equality
on the real numbers is reflexive, since always 
holds. Moreover, it is an equivalence relation.
- The subset relation
between sets is reflexive, since always 
holds. Moreover, it is a half-order.
Irreflexive
- The Kleiner relation
on the real numbers is irreflexive, since never 
holds. Moreover, it is a strict total order. The same is true for the relation 
.
- The inequality
on the real numbers is irreflexive, since never 
holds.
- The real subset relation
between sets is irreflexive, since never 
holds. Moreover, it is a strict half-order.
Neither reflexive nor irreflexive
The following relation on the set of real numbers is neither reflexive nor irreflexive:
Reason: For 
holds 
, for 
holds ¬ 
.
Related articles
Author
AlegsaOnline.com Reflexive relation Leandro Alegsa
URL: https://en.alegsaonline.com/art/81804