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Question
Give an example of a relation which is reflexive and symmetric but not transitive?
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Solution
Let A = {4, 6, 8}
Let the relation R defined on a set A be as follows:
R = {(4, 4), (6, 6), (8, 8), (4, 6), (6, 4), (6, 8), (8, 6)}
⇒ The relation R is reflexive because for every element a ∈ A, (a, a) ∈ R, i.e., (4, 4), (6, 6), (8, 8) ∈ R.
∴ R is Reflexive.
⇒ The relation R is symmetric, because (a, b) ∈ R and (b, a) ∈ R, for all a, b ∈ R.
∴ R is symmetric.
⇒ The relation R is not transitive, because (4, 6), (6, 8) ∈ R, but (4, 8) ∉ R.
∴ R is not transitive.
Thus, R is reflexive and symmetric but not transitive.
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