CBSE Class 12 Mathematics Relations and Functions Q #2136
COMPETENCY BASED
APPLY
2 Marks Generic-Curriculum VSA
A relation $R$ in set $A = \{1, 2, 3\}$ is defined as $R = \{(1, 1), (1, 2), (2, 2), (3, 3)\}$. Find ordered pair in $R$ shall be removed to make it an equivalence relation in $A$.
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Detailed Solution

Step 1: Understand Equivalence Relation Requirements

A relation R on a set A is an equivalence relation if and only if it is reflexive, symmetric, and transitive.

Step 2: Analyze the given relation

Given A = {1, 2, 3} and R = {(1, 1), (1, 2), (2, 2), (3, 3)}.

Step 3: Check for Reflexivity

For R to be reflexive, (a, a) must be in R for all a in A. Here, (1, 1), (2, 2), (3, 3) are present. Thus, R is reflexive.

Step 4: Check for Symmetry

For R to be symmetric, if (a, b) is in R, then (b, a) must be in R. We observe (1, 2) is in R, but (2, 1) is not in R. Therefore, R is not symmetric.

Step 5: Identify the problematic pair

To make the relation symmetric (and thus an equivalence relation, as removing (1, 2) leaves the identity relation {(1, 1), (2, 2), (3, 3)} which is reflexive, symmetric, and transitive), we must remove the pair that violates symmetry.

Final Answer: (1, 2)

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the definitions of reflexive, symmetric, and transitive relations to modify a given set.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a step-by-step logical procedure to verify properties of relations and identify the element causing the failure of symmetry.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question tests the conceptual understanding of equivalence relations rather than rote memorization of textbook examples.

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