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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 wife of y}
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Solution
(i) Reflexivity:
Let x be an element of R.
Then, x is wife of x cannot be true.
⇒ ( x, x) ∉ R
So, R is not a reflexive relation.
(ii) Symmetric:
Let (x, y) ∈ R
⇒ x is wife of y.
⇒ x is female and y is male.
⇒ y cannot be wife of x as y is husband of x.
⇒ (y, x) ∉ R
So, R is not a symmetric relation.
(iii) Transitivity:
⇒ If x is the wife of y, then y is not the wife of z.
⇒ If (x, y) ∈ R and (y, z) ∉ R, then (x, z) ∉ R.
⇒ So R is transitive.
Hence, R is neither reflexive, nor symmetric, but transitive.
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