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 R\(_{1}\) = {(p, p), (p, r), (q, q), (r, r), (r, s), (s, s)} in A is reflexive, since every element in A is R\(_{1}\)-related to itself.

But the relation R\(_{2}\) = {(p, p), (p, r), (q, r), (q, s), (r, s)} is not reflexive in A since q, r, s ∈ A but (q, q) ∉ R\(_{2}\), (r, r) ∉ R\(_{2}\) and (s, s) ∉ R\(_{2}\)

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).

Set Theory

Sets

Representation of a Set

Types of Sets

Pairs of Sets

Subset

Practice Test on Sets and Subsets

Complement of a Set

Problems on Operation on Sets

Operations on Sets

Practice Test on Operations on Sets

Word Problems on Sets

Venn Diagrams

Venn Diagrams in Different Situations

Relationship in Sets using Venn Diagram

Examples on Venn Diagram

Practice Test on Venn Diagrams

Cardinal Properties of Sets






7th Grade Math Problems

8th Grade Math Practice

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