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Question
The following relation is defined on the set of real numbers.
aRb if a – b > 0
Find whether relation is reflexive, symmetric or transitive.
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Solution
(i) Reflexivity:
Let a be an arbitrary element of R. Then,
a ∈ R
But a−a = 0 ≯ 0
So, this relation is not reflexive.
Symmetry:
Let (a, b) ∈ R
⇒ a−b > 0
⇒ −(b−a) >0
⇒ b−a < 0
So, the given relation is not symmetric.
Transitivity:
Let (a, b)∈R and (b, c)∈R. Then,
a−b > 0 and b−c >0
Adding the two, we get
a − b+b − c > 0
⇒ a − c> 0
⇒ (a, c) ∈ R.
So, the given relation is transitive.
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