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Home / Discrete mathematics / Let R be a reflexive relation on a set A. Show that R is an equivalence relation if and only if (a,b) and (a,c) in R implies that (b,c) ∊R.

Let R be a reflexive relation on a set A. Show that R is an equivalence relation if and only if (a,b) and (a,c) in R implies that (b,c) ∊R.

Proof: First we say that R is an equivalence relation. We have (a,b)∊R and (a,c) ∊R

⟹(b,a) ∊R and (a,c) ∊R [∵ R is symmetric]

⟹(b,c) ∊R

Conversely, we suppose that

(b,c) ∊R

Given that R is reflexive. Let (a,b) ∊R, then there exist x∊A such that (x,a) ∊R and (x,b) ∊R

⟹(x,b) ∊R and (x,a) ∊R

⟹(b,a) ∊R

∴ R is symmetric.

Since (a,b) ∊R and (a,c) ∊R

⟹ (b.c) ∊R

∴ R is an equivalence relation.(proved)

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