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Question
Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:
R = {(x, y) : x and y live in the same locality}
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Solution
(i) Reflexivity:
(x, x) ∈ R because x and x live in the same locality.
∴ R is reflexive.
(ii) Symmetry:
Let (x, y) ∈ R
⇒ x and y live in the same locality.
⇒ y and x also live in the same locality.
⇒ (y, x) ∈ R
Thus, R is symmetric.
(iii) Transitivity:
Let (x, y) ∈ R and (y, z) ∈ R
⇒ x and y live in the same locality and y and z live in the same locality.
⇒ x and z live in the same locality.
⇒ (x, z) ∈ R
Thus, R is transitive.
Hence, R is reflexive, symmetric and transitive.
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