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
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