माना \(N\) प्राकतिक संख्याओं का समुच्चय है और \(N\) पर एक संबंध \(R\) निम्न द्वारा परिभाषित है : \(R=\left\{(x, y) \in N \times N: x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0\right\} \mid\) तो संबंध \(R\)

A सममित है, परन्तु न तो स्वतुल्य है और न ही संक्रामक है
B स्वतुल्य है, परन्तु न तो सममित है और न ही संक्रामक है
C स्वतुल्य और सममित है, परन्तु संक्रामक नहीं है
D एक तुल्यता संबंध है

Verified Solution

Answer & Solution

Correct Answer

(B) स्वतुल्य है, परन्तु न तो सममित है और न ही संक्रामक है

Step-by-step Solution

Detailed explanation

\(x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0\) \(\Rightarrow x(x-y)(x+y)-3 y(x-y)(x+y)=0\) \(\Rightarrow(x-3 y)(x-y)(x+y)=0\) Now, \(x=y \forall(x, y) \in N \times N\) so reflexive But not symmetric \& transitive See, \((3,1)\) satisfies but \((1,3)\) does not. Also…

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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