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Question
Given an example of a relation. Which is reflexive and transitive but not symmetric.
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Solution
Define a relation R in R as:
R = {a, b): a3 ≥ b3}
Clearly (a, a) ∈ R as a3 = a3.
∴ R is reflexive.
Now,
(2, 1) ∈ R (as 23 ≥ 13)
But,
(1, 2) ∉ R (as 13 < 23)
∴ R is not symmetric.
Now,
Let (a, b), (b, c) ∈ R.
⇒ a3 ≥ b3 and b3 ≥ c3
⇒ a3 ≥ c3
⇒ (a, c) ∈ R
∴ R is transitive.
Hence, relation R is reflexive and transitive but not symmetric.
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