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Home / Discrete mathematics / Let R be a symmetric and transitive relation on a set. Show that for every a in A there exist b in A such that (a, b) ∊ R then R is an equivalence relation.

Let R be a symmetric and transitive relation on a set. Show that for every a in A there exist b in A such that (a, b) ∊ R then R is an equivalence relation.

Proof: For all a ∊ A, we have (a, b) ∊ R.
Since R is symmetric, so (b, a) ∊ R
Now, (a, b) ∊ R and (b, a) ∊ R imply that (a, a) ∊ R.
Since R is transitive hence R is reflexive.
Therefore R is an equivalence relation. (Proved)

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