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Question
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 (c) R = {(x, y): x is exactly 7 cm taller than y}
Solution
(i) Reflexive:
R = {(x, y): x is exactly 7 cm taller than y}
Now,
(x, x) ∉ R
Since 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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