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 Rrelated 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
● Subset
● Practice Test on Sets and Subsets
● Problems on Operation on Sets
● Practice Test on Operations on Sets
● Venn Diagrams in Different Situations
● Relationship in Sets using Venn Diagram
● Practice Test on Venn Diagrams
8th Grade Math Practice
From Reflexive Relation on Set to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
