Question

On \(\mathbb{R}\), a relation \(\rho\) is defined by \(x \rho y\) if and only if \(x-y\) is zero or irrational. Then
(A) \(\rho\) is equivalence relation
(B) \(\rho\) is reflexive but neither symmetric nor transitive
(C) \(\rho\) is reflexive and symmetric but not transitive
(D) \(\rho\) is symmetric and transitive but not reflexive

Answer 1

Ans: (C)
Hint : On the set \(\mathbb{R}\)
\(x \rho y \Leftrightarrow x-y=0\) or \(x-y \in Q^{c} \because x-x=0 \Rightarrow x \rho x\) (Reflexive)
if \(x-y=0 \Rightarrow y-x=0\) or \(x-y \in Q^{c} \Rightarrow y-x \in Q^{c}\) (Symmetric)
Take \(x=1+\sqrt{2} ; y=\sqrt{2}+\sqrt{3} ; z=\sqrt{2}+2\)
\(x-y=1-\sqrt{3} \in Q^{c}\) and \(y-z=\sqrt{3}-2 \in Q^{\circ}\)
Here \(x R y\) and \(y R z\) but \(x\) is not related to \(z \therefore\) Not transitive

Answer 2

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Answer 3

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