Reflexive Relation

Reflexive Relation

It is a binary element in which every element is related to itself.

Let A be a set and R be the relation defined on it.

R is set to be reflexive, if (p, p) ∈ R for all p ∈ A that is, every element of A is R-related to itself, in other words pRp for every p ∈ A.

A relation R on a set A is not reflexive if there is at least one element p ∈ A such that (p, p) ∉ R.

Consider, for example, a set A = {l,m,n,o}.

The relation R1

But the relation R2

Solved example of reflexive relation on set:

1. A relation R is defined on the set Z (set of all integers) by “aRb if and only if 2a + 3b is divisible by 5”, for all a, b ∈ Z. Examine if R is a reflexive relation on Z.

Solution:

Let b ∈ Z. Now 2b + 3b = 5b, which is divisible by 5. Therefore bRb holds for all b in Z i.e. R is reflexive.

2. A relation R is defined on the set Z by “aRb if a – b is divisible by 5” for a, b ∈ Z. Examine if R is a reflexive relation on Z.

Solution:

Let p ∈ Z. Then p – p is divisible by 5. Therefore pRp holds for all p in Z i.e. R is reflexive.

3. Consider the set Z in which a relation R is defined by ‘aRb if and only if a + 3b is divisible by 4, for a, b ∈ Z. Show that R is a reflexive relation on on setZ.

Solution:

Let a ∈ Z. Now a + 3a = 4a, which is divisible by 4. Therefore aRa holds for all a in Z i.e. R is reflexive.

4. A relation ρ is defined on the set of all real numbers R by ‘xρy’ if and only if |x – y| ≤ y, for x, y ∈ R. Show that the ρ is not reflexive relation.

Solution:

The relation ρ is not reflexive as x = -2 ∈ R but |x – x| = 0 which is not less than -2(= x).