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Question
Test whether the following relation is (i) reflexive (ii) symmetric and (iii) transitive R on Z defined by (a,b)∈R⇔|a−b|≤5.
R on Z defined by (a,b)∈R⇔|a−b|≤5.
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Solution
Given the relation is R on Z defined by (a,b)∈R⇔|a−b|≤5.
This relation is reflexive as ∀a∈Z,(a,a)∈R since |a−a|=0≤5.Also the relation is symmetric as |b−a|=|a−b|≤5 so (a,b)∈R⇒(b,a)∈R,∀a,b∈Z. But the relation is not transitive as (1,2)∈R,(2,7)∈R but (1,7)∉R
Given the relation is R on Z defined by (a,b)∈R⇔|a−b|≤5.
This relation is reflexive as ∀a∈Z,(a,a)∈R since |a−a|=0≤5.
Also the relation is symmetric as |b−a|=|a−b|≤5 so (a,b)∈R⇒(b,a)∈R,∀a,b∈Z.
But the relation is not transitive as (1,2)∈R,(2,7)∈R but (1,7)∉R
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