Reflexive Relation: Definition, Formula, Types & Examples

Muskan Shafi Education Content Expert

Education Content Expert

Reflexive Relation is a relation between elements of a Set A in which each element is related to itself. In a reflexive relation, each element of the set has its own reflection.

Reflexive Relation is an integral concept in the Set Theory.
Since every set is a subset of itself, thus, the relation “is a subset of” on a group of sets denotes a Reflexive Relation.
There are different types of relations in discrete mathematics such as Reflexive, Symmetric, Transitive, etc.

A relation is considered to be a Reflexive Relation if

(a, a) ∈ R ∀ a ∈ A

Where a refers to the Element, A denotes the Set and R is the Relation.

Table of Content

What is Reflexive Relation?
Examples of Reflexive Relations
Number of Reflexive Relations
Reflexive Relation Formula
Types of Reflexive Relations
Reflexive Relation Solved Examples
Things to Remember
Previous Year Questions
Sample Questions

Key Terms: Reflexive Relation, Reflexivity, Relations, Set Theory, Sets, Symmetry, Transitivity, Reflexive Relation Formula, Ordered Pair

What is Reflexive Relation?

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Reflexive Relation is defined as a relation in which every element maps to itself.

It is said to have the reflexive property or possess reflexivity.
It is one of the three properties defining Equivalence Relations along with Symmetry and Transitivity.

Example: Consider a set A = {1, 2}. The reflexive relation will be R = {(1, 1), (2, 2), (1, 2), (2, 1)}.

Thus, a relation is a reflexive relation if

(a, a) ∈ R ∀ a ∈ A

Where

a denotes the Element.
A denotes the Set.
R denotes the Relation.

Relations and Functions Detailed Video Explanation

Reflexive Relation Definition

In Set Theory, a binary relation R defined on Set A is said to be a reflexive relation if every element of the set is related to itself.

If there is even a single element of the set that is not related to itself, then R is not a reflexive relation.
For instance, if for b ∈ A and b are not related to itself (Denoted by (b, b) ∉ R or 'not bRb'), then R is not a reflexive relation.
Reflexive Relation on a Set A is also expressed as I_A = {(a, a): a ∈ A}, where I_A ⊆ R and R is a relation defined on Set A.

Example: Consider Set A = {a, b, c, d, e} and R is a relation defined on A. Thus,

R = {(a, a), (a, b), (b, b), (c, c), (d, d), (e, e), (c, e)}.

As (a, a), (b, b), (c, c), (d, d), (e, e) ∈ R, thus, R is a reflexive relation as every element of A is related to itself in R.

Read More:

Relevant Concepts
Relations and Functions	Real Valued Functions	What is a Function?
Types of Functions	Bijective Function	Difference between Relation and Function

Examples of Reflexive Relations

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Given below are the examples of Reflexive Relations:

Statement	Symbol	Explanation
“is equal to” (Equality)	=	Every element of a set is equal to itself.
“is a subset of” (Set Inclusion)	⊆	Every set is a subset of itself.
“divides” (Divisibility)	÷ or /	Every number divides itself.
“is greater than or equal to”	≥	Every element of a set is greater than or equal to itself.
“is less than or equal to”	≤	Every element of a set is less than or equal to itself.

Number of Reflexive Relations

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The number of Reflexive Relations on a Set A can also be determined.

Relation R defined on a Set A with n elements has ordered pairs of the form of (a, b).
It is known that element 'a' and element ‘b’ can be chosen in n ways.
It means that there are n² ordered pairs (a, b) in R.
In a Reflexive Relation, ordered pairs of the form (a, a) are required.
There are n ordered pairs of the form (a, a), thus, it can be said that there are n² - n ordered pairs for a reflexive relation.
Thus, the total number of reflexive relations will be 2^n(n-1).

Reflexive Relations Formula

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Reflexive Relations Formula is used to find the Number of Reflexive Relations on a Set A with ‘n’ number of elements. The theory behind the formula has already been described in detail above.

Reflexive Relations Formula is given as:

N = 2^n(n-1)

Where

N: Number of Reflexive Relations
n: Number of Elements in Set

Solved Example

Example: What will be the number of Reflexive Relations from Set A to A, defined as A = a, b, c?

Solution: Given that, A = a, b, c

Thus, the number of elements in Set A is 3.

If Set A has n elements, then, the total number of reflexive relations is given by the Reflexive Relations Formula

N = 2^n(n-1)

Here, n = 3. Substituting the value, we get

N = 2^3(3-1) = 2^9-3 = 2⁶ = 64

Therefore, the total number of reflexive relations from set A to A is 64.

Types of Reflexive Relations

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Some important types of Reflexive Relations are listed below:

Anti-Reflexive Relation: Relation R defined on a Set A is considered an anti-reflexive relation if no element of A is related to itself. It is represented as (a, a) ∉ R for every a ∈ A. It is also referred to as Irreflexive Relation.
Co-reflexive Relation: Relation R defined on a Set A is considered a co-reflexive relation if (a, b) ∈ R ⇒ a = b for all a, b ∈ A.
Quasi-reflexive Relation: Relation R defined on a Set A is considered a quasi-reflexive relation if (a, b) ∈ R ⇒ (a, a) ∈ R and (b, b) ∈ R for all a, b ∈ A.
Left Quasi-reflexive Relation: Relation R defined on a Set A is considered a left quasi-reflexive relation if (a, b) ∈ R ⇒ (a, a) ∈ R for all a, b ∈ A.
Right Quasi-reflexive Relation: Relation R defined on a Set A is considered a right quasi-reflexive relation if (a, b) ∈ R ⇒ (b, b) ∈ R for all a, b ∈ A.

Reflexive Relations Solved Examples

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Here are some solved examples on Reflexive Relation for better concept clarity:

Example 1: A relation R is defined on the set of integers Z as aRb if and only if 3a + 4b is divisible by 7. Check if R is a reflexive relation.

Solution: For a ∈ Z, 3a + 4a = 7a which is clearly divisible by 7.

⇒ aRa

Since a is an arbitrary element of Z, therefore (a, a) ∈ R for all a ∈ Z.

Hence, R is a reflexive relation.

Thus, R is defined on Z as aRb if and only if 3a + 4b is divisible by 7 is a reflexive relation.

Example 2: A relation R is defined on the set of natural numbers N as aRb if and only if a ≥ b. Check if R is a reflexive relation.

Solution: For a ∈ N, a = a, which satisfies a ≥ a for every a ∈ N.

⇒ aRa

Since a is an arbitrary element of N, therefore (a, a) ∈ R for all a ∈ N

Hence, R is a reflexive relation.

Thus, R is defined on N as aRb if and only if a ≥ b is a reflexive relation.

Check More:

Related Topics
Function Notation Formula	Domain and Range Relations	Period of a Function
Antisymmetric Relation	Polynomial Functions	Linear Functions

Things to Remember

Reflexive Relation is a relation in which every element of the set is related to itself.
It possesses a reflexive property and is said to hold reflexivity.
It is one of the three relations describing equivalence relations along with Symmetric and Transitive relations.
A relation is said to be a Reflexive Relation if (a, a) ∈ R ∀ a ∈ A.
Number of Reflexive Relations on a set with the 'n' number of elements is calculated by the Reflexive Relation Formula.
Reflexive Relation Formula is N = 2^n(n-1), where N is the number of reflexive relations and n is the number of elements in the set.

Previous Year Questions

Let R be a reflexive relation on a finite set A having n-elements...
(x − 1) (x²− 5x + 7) < (x −,1) then x belongs to… (BITSAT – 2007)
Consider the following lists… (AP EAPCET)
If a + π/2 < 2 tan⁻¹… (KCET – 2019)
The domain and range of relation R = {x,y}/x,y...
If A = {x | x∈N, x ≤ 5}, B = {x | x∈Z, x²− 5x + 6 = 0}… (KCET – 2019)
f : R → R and g : [0, ∞) → R is defined by… (KCET – 2019)
cos[2sin⁻¹3/4 + cos⁻¹3/4]… (KCET – 2019)
On the set of positive rationals, a binary operation… (KCET – 2019)
R is a relation on N given by R = {(x,y) ∣ 4x… (KCET - 2008)

Sample Questions

Ques. A relation R is defined on Set A, a Set of All Integers, by “x R y if and only if 2x + 3y is divisible by 5”, for all x, y ∈ A. Check whether R is a reflexive relation on A or not. (2 Marks)

Ans. Assume that x ∈ A.

Now, 2x + 3x = 5x, which can be divided by 5.

It means xRx holds for all ‘x’ in A

Thus, R is reflexive relation.

Ques. A relation R is defined on Set N, a Set of all Real Numbers by ‘a R b’ if and only if |a-b| ≤ b, for a, b ∈ N. Prove that R is not Reflexive Relation. (2 Marks)

Ans. We know that a relationship is not reflexive if a = -2 ∈ R

However, |a – a| = 0 which is not less than -2(= a).

Thus, the relation R is not reflexive.

Ques. A relation R is defined on the Set A by “x R y if x – y is divisible by 5” for x, y ∈ A. Check whether R is a reflexive relation on Set A or not. (2 Marks)

Ans. Assume that x ∈ A.

Then, x – x is divisible by 5.

It means x R x holds for all x in A

Thus, R is a reflexive relation.

Ques. How to Calculate the Number of Reflexive Relations on a Set? (3 Marks)

Ans. The number of reflexive relations on a set with the ‘n’ number of elements can be calculated using the Reflexive Relation Formula.

Reflexive Relation Formula is given as

N = 2^n(n-1)

Where

N denotes the number of reflexive relations.
n denotes the number of elements in the set.

Ques. Consider Set A in which a relation R is defined by ‘x R y if and only if x + 3y is divisible by 4, for x, y ∈ A. Prove that R is a reflexive relation on set A. (2 Marks)

Ans. Assume that x ∈ A.

Thus, x + 3x = 4x is divisible by 4.

As x R x holds for all x in A.

Hence, R is a reflexive relation.

Ques. State the Difference between a Reflexive Relation and an Identity Relation. (3 Marks)

Ans. Relation R is considered to be an Identity Relation if R relates every element of a set to itself only. In simpler terms, an identity relation cannot relate an element to any element other than itself.

Relation R is reflexive relation if R relates every element of a set to itself. In simpler terms, a reflexive relation can relate an element to other elements along with relating the element with itself.

Ques. A relation R on set A (Set of Integers) is defined by “x R y if 5x + 9x is divisible by 7x” for all x, y ∈ A. Check if R is a reflexive relation on A. (2 Marks)

Ans. Consider x ∈ A.

Now, 5x + 9x = 14x, which is divisible by 7x.

Therefore, x R y holds for all the elements in set A.

Hence, R is a reflexive relation.

Ques. Is Reflexive Relation also Symmetric? (1 Mark)

Ans. Reflexive Relation may or may not be symmetric in nature. For instance, 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 symmetric as (a, b) ∈ R but (b, a) ∉ R.

Ques. A relation R is defined on the set of all real numbers N by ‘a R b’ if |a-a| ≤ b, for a, b ∈ N. Show that the R is not a reflexive relation. (2 Marks)

Ans. N is a set of all real numbers. So, b =-2 ∈ N is possible.

Now |a – a| = 0. Zero is not equal to nor is it less than -2 (=b).

So, |a-a| ≤ b is false.

Therefore, the relation R is not reflexive.

Ques. A relation R on the set S by “x R y if x – y is divisible by 5” for x, y ∈ A. Confirm that R is a reflexive relation on set A. (2 Marks)

Ans. Consider, x ∈ S.

Then x – x= 0. Zero is divisible by 5.

Since x R x holds for all the elements in set S, R is a reflexive relation.

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Reflexive Relation: Definition, Formula, Types & Examples

What is Reflexive Relation?

Reflexive Relation Definition

Examples of Reflexive Relations

Number of Reflexive Relations

Reflexive Relations Formula

Solved Example

Types of Reflexive Relations

Reflexive Relations Solved Examples

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1.
Solve the following linear programming problem graphically: Maximise \( Z = x + 2y \) Subject to the constraints: \[ x - y \geq 0 \] \[ x - 2y \geq -2 \] \[ x \geq 0, \, y \geq 0 \]

2.
The diagonals of a parallelogram are given by \( \mathbf{a} = 2 \hat{i} - \hat{j} + \hat{k} \) and \( \mathbf{b} = \hat{i} + 3 \hat{j} - \hat{k}\) . Find the area of the parallelogram.

3.
The values of $\lambda$ so that $f(x) = \sin x - \cos x - \lambda x + C$ decreases for all real values of $x$ are :

5.
Let \( A \) be a matrix of order \( m \times n \) and \( B \) be a matrix such that \( A^T B \) and \( B A^T \) are defined. Then, the order of \( B \) is:

Similar Mathematics Concepts

Comments

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Reflexive Relation: Definition, Formula, Types & Examples

What is Reflexive Relation?

Reflexive Relation Definition

Examples of Reflexive Relations

Number of Reflexive Relations

Reflexive Relations Formula

Solved Example

Types of Reflexive Relations

Reflexive Relations Solved Examples

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1.Solve the following linear programming problem graphically: Maximise \( Z = x + 2y \) Subject to the constraints: \[ x - y \geq 0 \] \[ x - 2y \geq -2 \] \[ x \geq 0, \, y \geq 0 \]

2.The diagonals of a parallelogram are given by \( \mathbf{a} = 2 \hat{i} - \hat{j} + \hat{k} \) and \( \mathbf{b} = \hat{i} + 3 \hat{j} - \hat{k}\) . Find the area of the parallelogram.

3.The values of $\lambda$ so that $f(x) = \sin x - \cos x - \lambda x + C$ decreases for all real values of $x$ are :

5.Let \( A \) be a matrix of order \( m \times n \) and \( B \) be a matrix such that \( A^T B \) and \( B A^T \) are defined. Then, the order of \( B \) is:

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Solve the following linear programming problem graphically: Maximise \( Z = x + 2y \) Subject to the constraints: \[ x - y \geq 0 \] \[ x - 2y \geq -2 \] \[ x \geq 0, \, y \geq 0 \]

2.
The diagonals of a parallelogram are given by \( \mathbf{a} = 2 \hat{i} - \hat{j} + \hat{k} \) and \( \mathbf{b} = \hat{i} + 3 \hat{j} - \hat{k}\) . Find the area of the parallelogram.

3.
The values of $\lambda$ so that $f(x) = \sin x - \cos x - \lambda x + C$ decreases for all real values of $x$ are :

5.
Let \( A \) be a matrix of order \( m \times n \) and \( B \) be a matrix such that \( A^T B \) and \( B A^T \) are defined. Then, the order of \( B \) is: