ધારોકે \(R\) પરના બે સંબંધો \(R_{1}\) અને \(R_{2}\) નીયે મુજબ વ્યાખ્યાયિત છે: \(a R_{1} b \Leftrightarrow a b \geq 0\) અને \(a R_{2} b \Leftrightarrow a \geq b\), તો

A \(R_{1}\) એ સામ્ય સંબંધ છે, પરંતુ \(R_{2}\) નથી
B \(R_{2}\) એ સામ્ય સંબંધ છે, પરંતુ \(R_{1}\) નથી
C \(R_{1}\) અને \(R_{2}\) બંને સામ્ય સંબંધો છે
D \(R_{1}\) પણ સામ્ય સંબંધ નથી અને \(R_{2}\) પણ સામ્ય સંબંધ નથી

Verified Solution

Answer & Solution

Correct Answer

(D) \(R_{1}\) પણ સામ્ય સંબંધ નથી અને \(R_{2}\) પણ સામ્ય સંબંધ નથી

Step-by-step Solution

Detailed explanation

\(R_{1}=\{x y \geq 0, x, y \in R\}\) For reflexive \(x \times x \geq 0\) which is true. For symmetric If \(x y \geq 0 \Rightarrow y x \geq 0\) If \(x =2, y =0\) and \(z =-2\) Then \(x . y \geq 0 \& y . z \geq 0\) but \(x . z \geq 0\) is not true \(\Rightarrow\) 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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