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Home / Discrete mathematics / Let R be a transitive reflexive relation on A. Let T be a relation on A such that (a, b) ∊T if and only if both (a, b) ∊ T and (b, a) ∊ R. Show that T is equivalence relation.

Let R be a transitive reflexive relation on A. Let T be a relation on A such that (a, b) ∊T if and only if both (a, b) ∊ T and (b, a) ∊ R. Show that T is equivalence relation.

Proof: For all a ∊ R, (a, a) ∊ R. Since R is reflexive. So according to problem
(a, a) ∊ T.
Suppose (a, b) ∊ T.
Then by problem (a, b) ∊ R and (b, a) ∊ R
⇒ (b, a) ∊R and (a, b) ∊R
⇒ (b, a)∊ T.
Let (a, b) ∊ T and (b, c) ∊ T, then by problem (a, b) ∊ R and (b, c) ∊ R
And (b, c) ∊ R and (c, a) ∊ R.
⇒ (a, b) ∊ R and (b, c) ∊ R and (c, b) ∊ R and (b, a) ∊ R
⇒ (a, c) ∊ R and (b, a) ∊ R
Then T is transitive.
Therefore T is equivalence relation.(proved)

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