Class CBSE Class 12 Mathematics Relations and Functions Q #1283
KNOWLEDGE BASED
UNDERSTAND
5 Marks 2024 AISSCE(Board Exam) LA
A relation R is defined on $N\times N$ (where N is the set of natural numbers) as: $(a, b)~R~(c,d)\Leftrightarrow a-c=b-d$ Show that R is an equivalence relation.
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Detailed Solution

Step 1: Reflexivity

To prove R is reflexive, we need to show that $(a, b) ~R~ (a, b)$ for all $(a, b) \in N \times N$. Since $a - a = 0$ and $b - b = 0$, we have $a - a = b - b$, which means $(a, b) ~R~ (a, b)$. Therefore, R is reflexive.

Step 2: Symmetry

To prove R is symmetric, we need to show that if $(a, b) ~R~ (c, d)$, then $(c, d) ~R~ (a, b)$. Given $(a, b) ~R~ (c, d)$, we have $a - c = b - d$. Multiplying both sides by -1, we get $c - a = d - b$. This implies $(c, d) ~R~ (a, b)$. Therefore, R is symmetric.

Step 3: Transitivity

To prove R is transitive, we need to show that if $(a, b) ~R~ (c, d)$ and $(c, d) ~R~ (e, f)$, then $(a, b) ~R~ (e, f)$. Given $(a, b) ~R~ (c, d)$, we have $a - c = b - d$. Given $(c, d) ~R~ (e, f)$, we have $c - e = d - f$. Adding these two equations, we get $(a - c) + (c - e) = (b - d) + (d - f)$, which simplifies to $a - e = b - f$. This implies $(a, b) ~R~ (e, f)$. Therefore, R is transitive.

Step 4: Conclusion

Since R is reflexive, symmetric, and transitive, R is an equivalence relation.

Final Answer: R is an equivalence relation.

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to demonstrate understanding of the properties of equivalence relations (reflexivity, symmetry, and transitivity) and apply these concepts to the given relation.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of relations, Cartesian products, and equivalence relations, including their properties (reflexivity, symmetry, and transitivity). The student must apply these concepts to prove that the given relation is an equivalence relation.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of relations and functions, specifically equivalence relations, which is a core topic in the syllabus. The question is based on textbook knowledge and requires the student to apply definitions and theorems learned in class.

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