Reflexive Relation
2026-02-28 01:08 Diff
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Last updated on October 28, 2025

A relation is reflexive if each element of a set appears as a pair with itself. This means that for every element a in a set A, the relation R must include the pair (a, a) for it to be reflexive.

What is Reflexive Relation?

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A reflexive relation has each element paired with itself. For instance, if A = {x, y}, then the relation must include (x, x) and (y, y). If any of the self-pairs is missing, then the relation cannot be reflexive. Reflexive relations are important in set theory, mathematics, and logic in concepts such as equality, similarity, and more.

Difference between Reflexive Relation and the Identity Relation

Reflexive relations and identity relations both have a similar condition, that is the presence of self-pairs in the elements of their sets; however, there are a few key differences between them.

Reflexive Relation Identity Relation Every element is related to itself and may be related to other elements as well. Each element is related only to itself. Additional pairs like (a, b), etc., can be included. Only self-pairs and no other pairs are included. Reflexive relations are not a subset of identity relations. Identity relation is always a reflexive relation. For example, R = {(1, 1), (2, 2), (1, 2), (2, 1)} is reflexive on A = {1, 2}. For example, I = {(1, 1), (2, 2)} is the identity relation on A = {1, 2}

How to Prove a Relation Is Reflexive

A relation can be proved as reflexive by following these steps:

  1. List the elements of the base set A that form the basis for relation R.
  2. Make sure that the relation R has a pair (a, a) for all a in the set A. 
  3. The relation is reflexive only if every element has its own pair; if even one of the elements doesn't appear in a self-pair, the relation is not considered reflexive.

For example, for A = {4, 2, 5}, if R = {(4, 4), (2, 2), (5, 4), (4, 2), (5, 5)}, the relation is reflexive. 

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What are the Properties of Reflexive Relations?

Key properties of reflexive relations include:

  • Every element relates to itself.
  • Just because (a, b)(a, b)(a, b) is in R, it doesn't mean (b, a)(b, a)(b, a) has to be. Reflexivity does not guarantee symmetry. 
  • A relation that is reflexive, symmetry, and transitive is called an equivalence relation.
  • If a non-empty set has an empty relation, it is not reflexive.
  • A relation defined on an empty set is always reflexive.
  • A universal relation defined on any set is always reflexive.

Number of Reflexive Relations

To find how many reflexive relations can be formed on a set with n elements, follow these steps:

  • A set A with n elements will have n2 ordered pairs possible in a relation because each element can pair with all elements in the set.
  • To be reflexive, a relation requires all self-pairs (a, a) for every a in set A. 
  • Out of n2, we know that n pairs are self-pairs; this leaves us with n2 - n. The remaining pairs are non-diagonal, like (a, b), where a ≠ b. These remaining pairs are optional, giving us the option to include or not include them. So, the total number of reflexive relations on a set with n elements is 2n2-n.

Let’s take an example for the same: A = {1, 2, 3}, so n = 3

Total possible pairs = 32 = 9

Required self-pairs are (1, 1), (2, 2), (3, 3)

Remaining pairs = 9 - 3 = 6

These remaining 6 pairs can either be included or not.

So, the number of reflexive relations = 26 = 64

Tips and Tricks to Master Reflexive Relation

Reflexive relation can seem like a tricky topic at first, especially when mixed with other types of relations.
However, with the right tips and tricks, you can easily understand and master it with confidence.

  • Always check if all diagonal pairs (a ,a) exist in the relation. Missing even one means it’s not reflexive.
     

  • In a relation matrix, all diagonal elements should be 1. This ensures each element is related to itself.
     

  • In a digraph, reflexive means every vertex has a self-loop. No self-loop = not reflexive.
     

  • Reflexive focuses on “self,” symmetric on “swap,” and transitive on “chain.” Remember this three-word trick to avoid confusion.
     

  • For a set with n elements, a reflexive relation must include at least n pairs. These are the (a,a) pairs for each element.

Common Mistakes and How to Avoid Them in Reflexive Relations

Sometimes students overlook important conditions when identifying reflexive relations. Listed below are some easily avoidable mistakes for reference.

Real-Life Applications of Reflexive Relations

The scope of reflexive relations extends to various fields like computer science, law, and social systems. Some practical applications of reflexive relations are listed below:

  • Identity verification in databases: Reflexive relations ensure that each record in a database can be recognized as linked to itself. For example, a user ID is linked to the same user for authentication.
  • Equality checks in software: The equality relation is reflexive, as a = a. This property helps in object-oriented programming, where any object is equal to itself.
  • Communication on social networks: Social media apps allow users to send messages to their accounts, which shows a reflexive relation as the user is related to themselves.
  • Legal records and citizenship status: Citizen records in legal databases are linked to their identity, making it easy to organize and access their data. This process works on the principle of reflexivity.
  • File ownership in computer operating systems: Every user has ownership of their files in a computer system that they can access. Reflexive relation allows the owners access rights and holds accountability for data security.

Problem 1

Let A = {1, 2, 3}. Define a relation R on A as R = {(1, 1), (2, 2), (3, 3), (1, 2)}. Is R reflexive?

Okay, lets begin

Yes, R is reflexive.

Explanation

A relation is reflexive when all elements a of set A contain self-pairs (a, a), and any extra pairs outside the set do not affect the reflexivity. Since R contains (1, 1), (2, 2), (3, 3), it is a reflexive relation.

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Problem 2

Let the set B = {a, b, c}. Is the relation R = {(a, a), (b, b)} reflexive on the set B?

Okay, lets begin

No, R is not reflexive.

Explanation

We know that all elements must be self-pairs for reflectivity to occur, but here, (c, c) ∉R, so the relation is not reflexive.

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Problem 3

Define a relation R on Z as R = {(a, b)∣a − b is divisible by 5}. Is R reflexive?

Okay, lets begin

Yes.

Explanation

Here, for any integer a, a - a = 0, and zero is divisible by 5. So, (a, a)∈R for all a∈Z, meaning the relation is reflexive.

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Problem 4

Let C = {x∈R ∣ x > 0} and define the relation R = {(a, b) ∣ a < b}. Is R reflexive?

Okay, lets begin

No

Explanation

The conditions for reflexivity state that a relation must include self-pairs for every a∈C. The condition a < a cannot be true for real numbers, and since no element can be related to itself under this condition, the relation is not reflexive.

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Problem 5

Consider the set D = {p, q}. Is the universal relation R = D × D reflexive?

Okay, lets begin

Yes.

Explanation

A universal relation includes all possible ordered pairs of D, so it contains (p,p) and (q,q). Since all self-pairs are present, the relation is reflexive.

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FAQs on Reflexive Relation

1.Can a relation be both symmetric and reflexive?

Yes, some relations can be both symmetric and reflexive, and in some cases even transitive.

2.Is the empty relation reflexive?

No, an empty relation is not reflexive because it does not have any self-pairs (a, a).

3.Can a relation be reflexive if its set has only one element?

Yes, for a set A = {x}, if the relation contains (x, x), it is reflexive.

4.What is a reflexive relation on a set?

A reflexive relation on a set refers to each element in the set having a self-pair (a, a) for every a in the set.

5.How to check for reflexivity using a matrix?

When a relation is represented as a matrix, if all diagonal elements are 1, then the relation is reflexive.