CBSE Class 12 Mathematics Relations and Functions Q #2137
COMPETENCY BASED
APPLY
2 Marks Generic-Curriculum VSA
For the set $A = \{1, 2, 3\}$, define a relation $R$ in the set $A$ as follows: $R = \{(1, 1), (2, 2), (3, 3), (1, 3)\}$. Write the ordered pairs to be added to $R$ to make it the smallest equivalence relation.
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Detailed Solution

Step 1: Understand Equivalence Relation Requirements

For a relation $R$ on a set $A$ to be an equivalence relation, it must satisfy three properties: Reflexivity, Symmetry, and Transitivity.

Step 2: Check Current Properties

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

  • Reflexivity: Since $(1, 1), (2, 2), (3, 3) \in R$, it is reflexive.
  • Symmetry: For symmetry, if $(a, b) \in R$, then $(b, a)$ must be in $R$. Here, $(1, 3) \in R$, but $(3, 1) \notin R$. Thus, it is not symmetric.
  • Transitivity: Since $(1, 3) \in R$ and there are no other pairs starting with 3, it currently satisfies transitivity.

Step 3: Determine Necessary Additions

To make $R$ symmetric, we must add the ordered pair $(3, 1)$. After adding $(3, 1)$, the relation becomes $R' = \{(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)\}$. Checking transitivity again: $(1, 3) \in R'$ and $(3, 1) \in R'$ implies $(1, 1) \in R'$, which is already present. No further pairs are required.

Final Answer: (3, 1)

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the definitions of equivalence relations to identify missing elements in a set.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a step-by-step logical procedure to verify the three conditions of an equivalence relation.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question tests the conceptual understanding of relations rather than rote memorization of textbook examples.

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