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प्रश्न
Let A be the set of all human beings in a town at a particular time. Determine whether of the following relation is reflexive, symmetric and transitive :
R = {(x, y) : x and y live in the same locality}
determination of whether the following relations are reflexive, symmetric, and transitive:
Relation R in the set A of human beings in a town at a particular time given by R = {(x, y) : x and y live in the same locality}
उत्तर
(i) Reflexivity:
Let x be an arbitrary element of R. Then,
x ∈ R
⇒ x and x live in the same locality is true since they are the same.
So, R is a reflexive relation.
(ii) Symmetry:
Let (x, y) ∈ R
⇒ x and y live in the same locality
⇒ y and x live in the same locality
⇒ (y, x) ∈ R
So, R is a symmetric relation.
(iii) Transitivity:
Let (x, y)∈R and (y, z)∈R. Then,
x and y live in the same locality and y and z live in the same locality
⇒ x, y and z all live in the same locality
⇒ x and z live in the same locality
⇒ (x, z) ∈ R
So, R is a transitive relation.
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