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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 work at the same place}
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 work at the same place}
उत्तर
(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
So, 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
So, 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
So, R is a transitive relation.
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