Reflexive relation

In maths, any relation R over a set X is called reflexive if every element of X is related to itself. In formal terms, this may be written as ∀x ∈ X : x R x.

The relation “is equal to” on the set of real numbers is an example of a reflexive, since every real number is equal to itself. A reflexive relation is said to possess reflexivity or said to have the reflexive property.

If (a, a) ∈ R for all a ∈ A then R is set to be reflexive. It means every element of A is R-related to itself. In other words, a relation R in A is said to be reflexive if aRa for all a∈A. Any relation which is reflexive along with being symmetric and transitive is called an equivalence relation. To explain Symmetry and transitivity, R is said to be symmetric if aRb⇒bRa, ∀ a, b ∈ A. Also it is said to be transitive if aRb and bRc⇒aRc∀ a, b, c ∈ 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 R11 = {(p, p), (p, r), (q, q), (r, r), (r, s), (s, s)} in A is reflexive, since every element in A is R11-related to itself.

But the relation R22 = {(p, p), (p, r), (q, r), (q, s), (r, s)} is not reflexive in A since q, r, s ∈ A but (q, q) ∉ R22, (r, r) ∉ R22 and (s, s) ∉ R2

Solved Examples:

1. 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 set Z.

Solution:

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

2. 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).

 

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