Reflexive Relation: Definition, Formula, Examples, FAQs

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 = 2^{n(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 = 2^n(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 $\times$ 2^n(n-1) = 2^n(n-1)

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

Facts on Reflexive Relation

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 = 2^{n(n – 1)}

N = 2^{4(4 – 1)}

N = 2^{4 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

Frequently Asked Questions about Reflexive Relations