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Question
The following relation is defined on the set of real numbers. aRb if |a| ≤ b
Find whether relation is reflexive, symmetric or transitive.
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Solution
Reflexivity :
Let a be an arbitrary element of R. Then,
a ∈ R [Since, |a|=a]
⇒ |a|≮ a
So, R is not reflexive.
Symmetry :
Let (a, b) ∈ R
⇒ |a| ≤ b
⇒ |b| ≰ a for all a, b ∈ R
⇒ (b, a) ∉ R
So, R is not symmetric.
Transitivity :
Let (a, b) ∈ R and (b, c) ∈ R
⇒ |a| ≤ b and |b| ≤ c
Multiplying the corresponding sides, we get
|a| |b| ≤ bc
⇒ |a| ≤ c
⇒ (a, c) ∈ R
Thus, R is transitive.
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