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Questions
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 is father of and y}
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 is father of y}
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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