JEE Mains · Maths · STD 12 - 1. relation and function

Let \(R _{1}=\{( a , b ) \in N \times N :| a - b | \leq 13\}\) and \(R _{2}=\{( a , b ) \in N \times N :| a - b | \neq 13\} .\) Thenon \(N\)

A Both \(R_{1}\) and \(R_{2}\) are equivalence relations
B Neither \(R_{1}\) nor \(R_{2}\) is an equivalence relation.
C \(R_{1}\) is an equivalence relation but \(R_{2}\) is not
D \(R_{2}\) is an equivalence relation but \(R_{1}\) is not

Verified Solution

Answer & Solution

Correct Answer

(B) Neither \(R_{1}\) nor \(R_{2}\) is an equivalence relation.

Step-by-step Solution

Detailed explanation

\(R_{1}=\{(a, b) \in N \times N:|a-b| \leq 13\}\) \(R_{2}=\{(a, b) \in N \times N:|a-b| \neq 13\}\). For \(R_{1}\) : \((i)\,Reflexive \,\,relation\) \((a, a) \in N \times N:|a-a| \leq 13\) \((ii)\, Symmetric\,\, relation\)…

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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