Reflexive Relation

Reflexive relation is a relation of elements of a set A such that each element of the set is related to itself. As it suggests, the image of every element of the set is its own reflection. Reflexive relation is an important concept in set theory. For example, the relation "is a subset of" on a group of sets is a reflexive relation as every set is a subset of itself.f

There are different types of relations that we study in discrete mathematics such as reflexive, transitive, symmetric, etc. In this lesson, we will understand the concept of reflexive relations and the formula to determine the number of such relations along with some solved examples for a better understanding.

In set theory, a binary relation on A is said to be a reflexive relation if every element of the set is related to itself. Let us consider a mathematical example to understand the meaning this concept. Define a relation on the set of integers Z as ' is equal to'. Now, we know that each integer is equal to itself such as 0 = 0, -1 = -1, 2 = 2, and so on. This implies every integer is related to itself. Hence, the relation 'is equal to' on the set of integers is a reflexive relation.

Reflexive Relation Definition

A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R. This implies that a relation defined on a set is a reflexive relation if and only if every element of the set is related to itself. If there is a single element of the set that is not related to itself, then R is not a reflexive relation. For example, if for b ∈ A and b is not related to itself (it is denoted by (b, b) ∉ R or 'not bRb') then R is NOT reflexive.

A reflexive relation on a set A is also represented as I_A = {(a, a): a ∈ A}, where I_A ⊆ R and R is a relation defined on the set A. Let us consider an example. Let A = {a, b, c, d, e} and R is a relation defined on A as R = {(a, a), (a, b), (b, b), (c, c), (d, d), (e, e), (c, e)}. Since, (a, a), (b, b), (c, c), (d, d), (e, e) ∈ R, therefore R is a reflexive relation as every element of A is related to itself in R.

Examples of Reflexive Relations

• The relation 'Is equal to' is a reflexive defined on a set A as every element of a set is equal to itself. aRa as a = a for all a ∈ A
• The relation 'greater than or equal to' is reflexive defined on a set A of numbers as every element of a set is greater than or equal to itself. aRa as a ≥ a for all a ∈ A
• The relation 'less than or equal to' is reflexive defined on a set A of numbers as every element of a set is less than or equal to itself. aRa as a ≤ a for all a ∈ A
• The relation 'divides' is reflexive defined on a set A of numbers as every number divides itself. aRa as a / a for all a ∈ A

We can determine the number of reflexive relations on a set A. A relation R defined on a set A with n elements has ordered pairs of the form of (a, b). Now, we know that element 'a' can be chosen in n ways and similarly, element 'b' can be chosen in n ways. This implies we have n² ordered pairs (a, b) in R. For a reflexive relation, we need ordered pairs of the form (a, a). There are n ordered pairs of the form (a, a), so there are n² - n ordered pairs for a reflexive relation. Hence, the total number of reflexive relations is 2^n(n-1).

Reflexive Relation Formula

The number of reflexive relations on a set with the ‘n’ number of elements is given by N = 2^n(n-1), where N is the number of reflexive relations and n is the number of elements in the  set.

• Anti-Reflexive Relation - A relation R defined on a set A is said to be an anti-reflexive relation if no element of A is related to itself. Mathematically, it is represented as (a, a) ∉ R for every a ∈ A. It is also known as irreflexive relation.
• Co-reflexive Relation - A relation R defined on a set A is said to be a co-reflexive relation if (a, b) ∈ R ⇒ a = b for all a, b ∈ A.
• Quasi-reflexive Relation - A relation R defined on a set A is said to be a quasi-reflexive relation if (a, b) ∈ R ⇒ (a, a) ∈ R and (b, b) ∈ R for all a, b ∈ A.
• Left Quasi-reflexive Relation - A relation R defined on a set A is said to be a left quasi-reflexive relation if (a, b) ∈ R ⇒ (a, a) ∈ R for all a, b ∈ A.
• Right Quasi-reflexive Relation - A relation R defined on a set A is said to be a right quasi-reflexive relation if (a, b) ∈ R ⇒ (b, b) ∈ R for all a, b ∈ A.

Important Notes on Reflexive Relation

• A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R.
• The number of reflexive relations on a set with the ‘n’ number of elements is given by 2^n(n-1)
• A relation R defined on a set A is said to be an anti-reflexive relation if no element of A is related to itself.

☛ Related Articles:

• Relations and Function Worksheets
• Inverse Relation
• Transitive Relations

What is Reflexive Relation in Discrete Mathematics?

A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R. This implies that a relation defined on a set is said to be a reflexive relation if and only if every element of the set is related to itself.

What is the Difference Between a Reflexive Relation and an Identity Relation?

A relation R is an identity relation if R relates every element of a set to itself only. In other words, an identity relation cannot relate an element to any element other than itself. A relation R is a reflexive if R relates every element of a set to itself. In other words, a reflexive relation can relate an element to other elements along with relating the element with itself.

How to Find the Number of Reflexive Relations?

The number of reflexive relations on a set with the ‘n’ number of elements is given by N = 2^n(n-1), where

• N is the number of reflexive relations and
• n is the number of elements in the set.

What is the Difference Between an Antisymmetric and Reflexive relation?

A relation R defined on a set A is said to be antisymmetric if (a, b) ∈ R ⇒ (b, a) ∉ R for every pair of distinct elements a, b ∈ A. A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R.

Is a Reflexive Relation also Symmetric?

A reflexive relation may or may not be symmetric. For example a relation R = {(a, a), (b, b), (c, c), (a, b), (a, c), (c, a)} defined on set A = {a, b, c} is reflexive but not a symmetric relation as (a, b) ∈ R but (b, a) ∉ R.

What is the Difference Between Irreflexive and Reflexive Relations?

A binary relation on A is said to be reflexive if every element of the set is related to itself. A relation R defined on a set A is said to be an irreflexive relation if for at least one element of A is NOT related to itself.

What is Reflexive Relations Example in Real Life?

Consider a group of boys of different heights. Define a relation R on the group as the height of a boy is greater than or equal to the height of another boy. Since the height of every boy is equal to his height. Therefore, it satisfies the relation that the height of a boy is greater than or equal to his own height. Hence, the relation is reflexive.