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Question
Given an example of a relation. Which is symmetric but neither reflexive nor transitive.
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Solution
Let A = {5, 6, 7}.
Define a relation R on A as R = {(5, 6), (6, 5)}.
Relation R is not reflexive as (5, 5), (6, 6), (7, 7) ∉ R.
Now, as (5, 6) ∈ R and also (6, 5) ∈ R, R is symmetric.
⇒ (5, 6), (6, 5) ∈ R, but (5, 5) ∉ R
∴ R is not transitive.
Hence, relation R is symmetric but not reflexive or transitive.
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