નીચે આપલે પૈકી ક્યો સંબંધ \(\mathrm{R}\) એ વાસ્તવિક સંખ્યા પર સાચો નથી ?

A \((\mathrm{x}, \mathrm{y}) \in \mathrm{R} \Leftrightarrow 0<|\mathrm{x}|-|\mathrm{y}| \leq 1\) એ પરંપરિત અને સંમિત નથી
B \((x, y) \in R \Leftrightarrow 0<|x-y| \leq 1\) એ પરંપરિત અને સંમિત છે
C \((x, y) \in R \Leftrightarrow|x|-|y| \leq 1\) એ સ્વવાચક છે પરંતુ સંમિત નથી
D \((\mathrm{x}, \mathrm{y}) \in \mathrm{R} \Leftrightarrow|\mathrm{x}-\mathrm{y}| \leq 1\) એ સ્વવાચક અને સંમિત છે,

Verified Solution

Answer & Solution

Correct Answer

(B) \((x, y) \in R \Leftrightarrow 0<|x-y| \leq 1\) એ પરંપરિત અને સંમિત છે

Step-by-step Solution

Detailed explanation

Note that \((1,2)\) and \((2,3)\) satisfy \(0<|x-y| \leq 1\) but \((1,3)\) does not satisfy it so \(0 \leq|\mathrm{x}-\mathrm{y}| \leq 1\) is symmetric but not transitive

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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