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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 work at the same place}
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Solution
(i) Reflexivity:
Let x be an arbitrary element of R.
Then, x ∈ R
⇒ x and x work at the same place,
Which is true since they are the same.
⇒ (x, x) ∈ R
∴ R is a reflexive relation.
(ii) Symmetry:
Let (x, y) ∈ R
⇒ x and y work at the same place.
⇒ y and x work at the same place.
⇒ (y, x) ∈ R
∴ R is a symmetric relation.
(iii) Transitivity:
Let (x, y) ∈ R and (y, z) ∈ R.
Then, x and y work at the same place.
y and z also work at the same place.
⇒ x, y and z all work at the same place.
⇒ x and z work at the same place.
⇒ (x, z) ∈ R
∴ R is a transitive relation.
Hence, R is reflexive, symmetric and transitive.
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