# Reflexive Relation: Definition, Formula, Examples, FAQs

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## What Is a Reflexive Relation?

A Relation R defined on Set A is said to be reflexive if each element of the set is mapped to itself. In other words, aRa for all a R.

Reflexive Relation Example: The relation R = {(0, 0), (1, 2), (1, 1), (2, 2)} defined on A = {0, 1, 2} is a reflexive relation.

## Definition of Reflexive Relation

A binary Relation R defined on a Set A is said to be reflexive if for each element a ∈ A, we have aRa.

R is reflexive if for each a ∈ A, (a, a) ∈ R.

This proves that when each element of a set is related to itself, only then can a relation be said to be reflexive. R will not be a reflexive relation if even one element of the set is not related to itself.

## Reflexive Relation Formula

The formula for reflexive relation states the number of reflexive relations in a given set.

The formula is:

N = 2n(n – 1)

where

N = Number of reflexive relations

n = the number of elements in the set

## How to Prove a Relation Is Reflexive

To prove that a Relation R defined on Set A is reflexive, first identify the elements of Set A.

For each element x of Set A, we must have an ordered pair (x, x) in R.

If for a single element in A, R does not meet this condition, R is not reflexive.

Example: A = {1, 2, 3}

R1 = {(1,1), (1, 2), (2, 2), (2, 3)} is not reflexive but R2 = {(1,1), (1,2), (2, 2), (3, 3), (1, 3)} is reflexive.

## Properties of a Reflexive Relation

1. Co-reflexive: 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. A combination of a co-reflexive and a transitive relation will always result in a transitive.
2. Anti-reflexive: In set A, a relation R is anti-reflexive (or irreflexive) if no element of the set is related to itself. Thus, R is anti-reflexive if (a, a) $\notin$ R for all a A.
3. Quasi-reflexive: For a given set A, the relation R will be quasi-reflexive if (a, b) ∈ R implies that (a, a)∈ R and (b, b)∈ R for all the elements a and b of A.
4. Left Quasi-reflexive: For a given set A, the relation R will be left quasi-reflexive if (a, b)∈ R implies that (a, a) ∈ R for all the elements a, b ∈ A.
5. Right Quasi-reflexive: For a given set A, the relation R will be right quasi-reflexive if (a, b) ∈ R implies that (b, b) ∈ R for all the elements a, b ∈ A.
6. If there is a non-empty set A, any reflexive relation R cannot be anti-reflexive, anti-transitive, or asymmetric.

## Number of Reflexive Relations

Consider a relation R defined on set A, where the set A has ‘n’ number of elements. The elements of the relation R are ordered pairs of the form of (a, b), where a and b are elements of A.

• Relation R is a subset of A $\times$ A.
If A has n elements, then A $\times$ A has n2 elements.
Here, in (a, b), the element “a” can be chosen in n ways. The element “b” can be chosen in n ways. So, there are n2 ordered pairs possible for R.
• For the Relation R to be a reflexive relation, R must have ordered pairs of the form (a, a) for each a∈ A. Since there are n elements in A, there are n such ordered pairs possible.
Thus, out of n2 ordered pairs, n pairs must be present for a reflexive relation. We can do this straightforward selection in only 1 way. So, we select n ordered pairs in 1 way.
• The remaining ordered pairs are n2 – n = n (n-1) ordered pairs, which may or may not be present. So, each such ordered pair has 2 choices (present or not present). There are 2 ways to select each element out of (n2 – n) elements.
(n2 – n) elements can be selected in 2 $\times$ 2 $\times$ 2$\times$ …. 2 = 2n(n-1), since 2 is multiplied (n2 – n) times.
• Number of reflexive relations = Number of ways to select n elements $\times$ Number of ways to select remaining (n2 – n) elements
Number of reflexive relations = 1 $\times$ 2n(n-1) = 2n(n-1)

Therefore, the total number of reflexive relations on set A with n elements is given by N = 2n(n-1)).

## Facts on Reflexive Relation

• If any relation is symmetric, transitive and reflexive, it is known as an equivalence relation.
• Number of relations on set A having n elements = 2Number of elements in A x A = $2^{n^{2}}$

## Conclusion

In this article, we learned about reflexive relations, how to identify reflexive relations, and the formula for the number of reflexive relations on a set. Let’s solve a few examples and practice problems based on these concepts.

## Solved Examples on Reflexive Relation

1. Is the relation R = {(0,0), (0,1)} defined on A = {0, 1} reflexive?

Solution:

For R to be reflexive, we must have (m, m) R for all m A.

0 ∈ A and (0, 0) ∈ R

1 ∈ A but (1, 1) ∉ R

Hence, R is not a reflexive relation.

2. If A = {w, x, y, z}, then find the number of reflexive relations on set A.

Solution:

A = {w, x, y, z}

Number of elements in set A = n = 4

Number of reflexive relations is given by the formula:

N = 2n(n – 1)

N = 24(4 – 1)

N = 24 x 3

N = 212

N = 4096

Therefore, the number of reflexive relations in Set A = 4096.

3. A relation R is defined on the set N of natural numbers as iRj if i ≥ j. Find out if R is a reflexive relation or not.

Solution:

A relation R is defined on the set N as iRj if i ≥ j.

i = i, which satisfies i ≥ i for every i ∈ N.

This implies that iRi.

If i is any random element of N, we have (i, i) ∈ R as all i ∈ N.

Hence, the relation R defined on the set N is reflexive.

4. In a set of natural numbers, N, a relation R is defined as mRn only when 7m + 9n is divisible by 8. Find whether R is a reflexive relation or not.

Solution:

For the relation R to be reflexive, we need (a, a) ∈ R for all m ∈ N

R is defined as mRn only when 7m + 9n is divisible by 8.

For m ∈ N, we have

7m + 9m = 16, which is divisible by 8.

⇒ mRm.

We know that m is a random element of set N, thus (m, m) ∈ R for all m ∈ N.

Hence, R is a reflexive relation.

## Practice Problems on Reflexive Relation

1

### What will be the total number of reflexive relations on a set B having 3 elements?

32
64
128
8
CorrectIncorrect
$N = 2^{n(n - 1)} = 2^{3(3 - 1)} = 2^{3\times 2} = 2^{6} = 64$.
2

### What is the formula for finding the number of reflexive relations in a set?

$N = 2^{n(n + 1)}$
$N = 2^{n}$
$N = 2^{n(n \;-\; 1)}$
$N = 2^{n(n \;-\; 2)}$
CorrectIncorrect
Correct answer is: $N = 2^{n(n \;-\; 1)}$
The formula for the number of reflexive relations is $N = 2^{n(n\;-\;1)}$, where n is the number of elements in the set.
3

### A relation R is defined for a set of integers (Z) such that aRb if and only if 2a + 4b is divisible by 3, where a, b $\in$ Z. Is R a reflexive relation?

R is not a reflexive relation.
It depends on the value of a.
R is a reflexive relation.
It depends on the value of both a and b.
CorrectIncorrect
Correct answer is: R is a reflexive relation.
For all a ∈ Z, we have, 2a + 4a = 6a, which is divisible by 3.
Here, ‘a’ will take an integer value.
Thus, regardless of what value ‘a’ takes, R will be a reflexive relation.