Q2131: Let $A = \text{set of workers in a publication house}$. Let ...

CBSE Class 12 Mathematics Relations and Functions Q #2131

COMPETENCY BASED

APPLY

3 Marks Generic-Curriculum SA

Let $A = \text{set of workers in a publication house}$. Let $a R b$, if and only if $a$ and $b$ tell lie ; $a, b \in A$. (i) Is $R$ symmetric relation? (ii) Is this relation an equivalence relation?

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AI Tutor Explanation

Detailed Solution

Step 1: Define the Relation

The relation $R$ is defined on the set $A$ such that $aRb$ if and only if both $a$ and $b$ tell lies. Let $L$ be the subset of $A$ consisting of all workers who tell lies. Then $aRb$ holds if and only if $a \in L$ and $b \in L$.

Step 2: Check for Symmetry

A relation $R$ is symmetric if $aRb \implies bRa$. If $aRb$ is true, then both $a$ and $b$ tell lies. Consequently, $b$ and $a$ also tell lies, which means $bRa$ is true. Thus, the relation is symmetric.

Step 3: Check for Equivalence

For $R$ to be an equivalence relation, it must be reflexive, symmetric, and transitive.
Check Reflexivity: $aRa$ requires $a$ to tell a lie. If there exists a worker $c \in A$ who does not tell lies (i.e., $c \notin L$), then $cRc$ is false. Since the relation is not reflexive for all $a \in A$, it is not an equivalence relation.

Final Answer: (i) Yes, it is symmetric. (ii) No, it 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 formal definitions of symmetry and equivalence relations to a non-mathematical set scenario.

Knowledge Dimension: CONCEPTUAL

Justification: The student must understand the underlying logic of set theory and relation properties rather than just performing rote calculations.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the student's ability to interpret logical conditions within the framework of Relations and Functions (Chapter 1).

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