CUET · MATHS · PYQ PAPER 2023

The relation \(R=\{(a, b):\) both \(a\) and \(b\) are either odd or even \(\}\) on the set \(\{1,2,4,5,7,8\}\) is

A reflexive and symmetric but not transitive
B reflexive and transitive but not symmetric
C symmetric and transitive but not reflexive
D an equivalence relation

Verified Solution

Answer & Solution

Correct Answer

(D) an equivalence relation

Step-by-step Solution

Detailed explanation

\((a,a)\) are both odd or both even for any \(a\). Reflexive. If \((a,b) \in R\), then \(a,b\) are both odd or both even. Then \(b,a\) are both odd or both even. Symmetric. If \((a,b) \in R\) and \((b,c) \in R\), then \(a,b\) have same parity and \(b,c\) have same parity. Thus…

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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