CBSE Class 12 Mathematics Relations and Functions Q #2144
KNOWLEDGE BASED
APPLY
3 Marks Generic-Curriculum SA
Determine whether $R$ in the set $A = \{1, 2, 3, 4\}$ defined by $R = \{(a, b) : |a - b| \text{ is odd}\}$ is an equivalence relation. Justify your answer. [All India 2014]
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Detailed Solution

Step 1: Check for Reflexivity

A relation $R$ is reflexive if $(a, a) \in R$ for all $a \in A$. Here, $|a - a| = 0$, which is an even number. Since $0$ is not odd, $(a, a) \notin R$. Thus, $R$ is not reflexive.

Step 2: Check for Symmetry

A relation $R$ is symmetric if $(a, b) \in R \implies (b, a) \in R$. If $|a - b|$ is odd, then $|b - a| = |-(a - b)| = |a - b|$ is also odd. Thus, $R$ is symmetric.

Step 3: Check for Transitivity

A relation $R$ is transitive if $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$. Let $a=1, b=2, c=3$. Here, $|1-2|=1$ (odd) and $|2-3|=1$ (odd), so $(1, 2) \in R$ and $(2, 3) \in R$. However, $|1-3|=2$ (even), so $(1, 3) \notin R$. Thus, $R$ is not transitive.

Step 4: Conclusion

Since the relation is not reflexive and not transitive, it is not an equivalence relation.

Final Answer: The relation R is not an equivalence relation.

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the definitions of reflexivity, symmetry, and transitivity to a specific set and rule.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a step-by-step algorithmic process to verify the three conditions required for an equivalence relation.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE (Textbook). This is a standard problem type found in Chapter 1 (Relations and Functions) of the NCERT textbook, testing foundational understanding of set theory and relations.

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