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, 3), (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 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

Reflexive Relation: Definition, Formula, Examples, FAQs

Attend this quiz & Test your knowledge.

1

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

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

Frequently Asked Questions about Reflexive Relations

The smallest reflexive relation formed of X = {a, b, c, d} will be {(a, a), (b, b), (c, c), (d, d)}.

Yes, every identity relation is reflexive. An identity relation on a set ‘A’ is defined by the set of ordered pairs where each element ‘a’ is related to itself, i.e., (a, a) for all ‘a’ in ‘A’. This property of identity relations makes them reflexive.

The relation of “LESS THAN OR EQUAL TO” denoted by “≤” is an example of a reflexive relation which is not a symmetric relation. For all a, a ≤ a. But, a ≤ b does not mean that b ≤ a.