What is Reflexive Relation: Definitions, Number of Reflexive Relations and Examples

Any relation can be termed as reflexive relation on a particular set when every element of that set is related to itself. 

According to the set theory, binary relation on A will be a reflexive relation if each element of the set is related to itself. In mathematical terms,  for every element a ∈ A, we have aRa, that is, (a, a) ∈ R. Even if there is a single element of set that is not related to itself, R will not be reflexive relation. Mathematically,  for c ∈ A and c is not related to itself (denoted as (c, c) ∉ R or 'not cRc'). In such a case, R will NOT be reflexive.

In mathematics, R is the binary relation across the set. X can be termed as reflexive when every set X element is linked or related to itself. Reflexivity, transitivity, and symmetry are three distinct properties that represent equivalent relations 

A reflexive relation in relation and function is where each element maps with itself. For instance, if set A = {1,2} thus, the reflexive relation R = {(1,1), (2,2) , (1,2) , (2,1)}. Therefore, the relation is reflexive when :

(a, a) ∈ R ∀ a ∈ A

Here a is an element, R is the relation, and A is the set. 

The reflexive relations are mentioned below, where the statements depict reflexivity.

Characteristics of Reflexive Relation 

• Anti-reflexive: A relation is irreflexive or anti-reflexive if and only if the set's elements do not relate to itself.
• Quasi-reflexive: If each element that is related to some element is also related to itself, such that relation ~ on a set R is stated formally: ∀ a, b ∈R: a ~ b ⇒ (a ~ a ∧ b ~ b).
• Co-reflexive: A relation ~  is co-reflexive for ∀a and y in set A holds that if a ~ b, then a = b. 
• A reflexive relation on the non-empty set B can neither be irreflexive, nor asymmetric, nor anti-transitive.

Reflexive relation is an important topic in relation and functions; students must study the topic thoroughly as it will help higher education. This chapter is covered in class 12 and those who want to practice questions related to this topic can refer to Maths Relations and Functions Class 12 Solutions.

Let us take a look at some reflexive relations examples that are important for JEE Main and IIT JAM exams: 

1. Let A = {0, 1, 2, 3} and Let a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Show whether R is reflexive, Symmetric, or Transitive?

Solution: R is reflexive and symmetric relation but not the transitive relation since for (1, 0) ∈ R and (0,3)∈R, whereas the pair (1,3) ∉R.

2. Let set B in a relation P be defined by ‘xPy iff x + 3y, which is divisible by 4, for x, y ∈ B. Prove that P is a reflexive relation on the set B.

Solution: Let x ∈ B. Now x + 3x = 4x, which is divisible by 4. Therefore xRx holds for all x in B, i.e., P is reflexive.

3. A relation P is defined on all real numbers R by ‘aPc’ iff |a – b| ≤ b, for a, b ∈ R. Prove that the P is not a reflexive relation.
Solution: The P is not reflexive as a = -2 ∈ R but |a – a| = 0 which is not less than -2(= a).

4. What is the formula of reflexive relations?

Solution: N = $2^{n(n-1)}$ , where n is the total number of elements in a set.

Let us now take a look at number of reflexive relations. The no of reflexive relations on set A. Relation R defined on set A with n elements has ordered pairs of form of (a,b). Element 'a' can be chose in 'n' number of ways and element 'b' can be chosen in 'n' number of ways. This implies that there are $n^2$ ordered pairs (a,b) in R. We require ordered pairs of form (a,a) for reflexive relation. There are 'n' ordered paird of (a,a) form. So, there are $n^2$ -  n ordered pairs for reflexive relation. Therefore, number of reflexive relations formula will be $2^{n(n-1)}$ .

Let us take a look at some of the related reflexive relation definition that students of CBSE board must remember:

• Co-reflexive relation: As per this type of reflexive relation definition, R relation defined on set A will be a co-reflexive relation when if (a, b) ∈ R ⇒ a = b for all a, b ∈ A.
• Quasi-reflexive Relation: If (a, b) ∈ R ⇒ (a, a) ∈ R and (b, b) ∈ R for all a, b ∈ A for a relation R defined on set A
• Left Quasi-reflexive Relation:  If (a, b) ∈ R ⇒ (a, a) ∈ R for all a, b ∈ A for a relation R defined on set A.
• Right Quasi-reflexive Relation: If (a, b) ∈ R ⇒ (b, b) ∈ R for all a,b ∈ A for a relation R defined on set A.
• Anti-reflexive Relation: If no element of A is related to itself in a relation R defined on set A.