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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 is father of y}
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Solution
(i) Reflexivity:
Let x be an arbitrary element of R.
Then, x is father of x cannot be true since no one can be father of himself.
So, R is not a reflexive relation.
(ii) Symmetric:
Let (x, y) ∈ R
⇒ x is father of y.
⇒ y is son/daughter of x.
⇒ (y, x) ∉ R
So, R is not a symmetric relation.
(iii) Transitivity:
Let (x, y) ∈ R and (y, z) ∈ R.
Then, x is father of y and y is father of z.
⇒ x is grandfather of z.
⇒ (x, z) ∉ R
So, R is not a transitive relation.
Hence, R is not reflexive, not symmetric and not transitive.
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