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Question
Determine whether the following relation is 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 exactly 7 cm taller than y}.
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Solution
(i) Reflexive:
R = {(x, y) : x is exactly 7 cm taller than y}
Now, (x, x) ∉ R
Since a human being (x) cannot be taller than himself.
∴ R is not reflexive.
(ii) Symmetric:
Now, let (x, y) ∈ R
⇒ x is exactly 7 cm taller than y.
Then, y is not taller than x.
∴ (y, x) ∉ R
Indeed, if x is exactly 7 cm taller than y, then y is exactly 7 cm shorter than x.
∴ R is not symmetric.
(iii) Transitive:
Now, let (x, y), (y, z) ∈ R
⇒ x is exactly 7 cm taller than y, and y is exactly 7 cm taller than z.
⇒ x is exactly 14 cm taller than z.
∴ (x, z) ∉ R
∴ R is not transitive.
Hence, R is not reflexive, not symmetric and not transitive.
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