Reflexive Relation on Set

Reflexive relation on set is a binary element in which every
element is related to itself.

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

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

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

Consider, for example, a set A = {p, q, r, s}.

The relation R1

1 = {(p, p), (p, r), (q, q), (r, r), (r, s), (s, s)} in A is reflexive, since every element in A is R1

1-related to itself.

But the relation R2

2 = {(p, p), (p, r), (q, r), (q, s), (r, s)} is not reflexive in A since q, r, s ∈ A but (q, q) ∉ R2

2, (r, r) ∉ R2

2 and (s, s) ∉ 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 a ∈ Z. Now 2a + 3a = 5a, which is divisible by 5. Therefore
aRa holds for all a 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 a ∈ Z. Then a – a is divisible by 5. Therefore aRa holds
for all a 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).