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Define relation R on Real number (R)-0 where aRb if and only if a/b is in Q. Prove R is an equivalence relation.

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R = {(a, b) | a/b ∈ Q, a,b ∈ R - {0}}

Reflexivity :→ Let a ∈ R - {0}

Then a/a = 1 ∈ Q.

∴ (a, a) ∈ R \(\forall\) a ∈ R - {0}.

Which implies that relation R is a reflexive relation.

Symmetricity :→ Let a, b ∈ R - {0} such that

(a, b) ∈ R.

(i.e.) a/b ∈ Q

Now since a ≠ 0 and a ∈ R and b ∈ R - {0}.

Then b/a ∈ Q

⇒ (b, a) ∈ R.

This implies if (a,b) ∈ R then (b, a) ∈ R

\(\forall\) a, b ∈ R - {0}.

i.e., relation R is a symmetric relation.

Transitivity :→ Let a, b, c ∈ R - {0} such that (a, b) ∈ R and (b, c) ∈ R.

i.e., a/b ∈ Q and b/c ∈ Q

Then a/b x b/c ∈ Q (∵ product of two rational number is rational)

⇒ a/c ∈ Q

⇒ (a/c) ∈ R \(\forall\) a, b, c ∈ R - {0}.

Which implies that relation R is transitive relation. Since, relation R is reflexive, symmetric and transitive.

Therefore, relation R is an equivalence relation.

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