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Question
Test whether the following relation R2 is (i) reflexive (ii) symmetric and (iii) transitive:
R2 on Z defined by (a, b) ∈ R2 ⇔ |a – b| ≤ 5
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Solution
Reflexivity:-
Let a be an arbitrary element of R2. Then,
a ∈ R2
⇒ | a−a | = 0 ≤ 5
So, R1 is reflexive.
Symmetry:-
Let (a, b) ∈ R2
⇒ |a−b| ≤ 5 [ Since, |a−b| = |b−a| ]
⇒ |b−a| ≤ 5
⇒ (b, a) ∈ R2
So, R2 is symmetric.
Transitivity:-
Let (1, 3) ∈ R2 and (3, 7) ∈R2
⇒|1−3|≤5 and |3−7|≤5
But |1−7| ≰5
⇒ (1,7) ∉ R2
So, R2 is not transitive.
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