Class CBSE Class 12 Mathematics Relations and Functions Q #1771
COMPETENCY BASED
APPLY
3 Marks 2026 AISSCE(Board Exam) SA
Let n be a fixed positive integer. A relation R is defined in set Z such that $R=\{(x,y):(x-y) \text{ is divisible by } n, x, y\in Z\}$. Determine if R is an equivalence relation.
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Detailed Solution

Step 1: Reflexivity

For any $x \in Z$, we observe that $(x - x) = 0$. Since $0$ is divisible by any positive integer $n$ (as $0 = n \times 0$), the pair $(x, x) \in R$. Thus, $R$ is reflexive.

Step 2: Symmetry

Let $(x, y) \in R$. This implies $(x - y) = kn$ for some integer $k$. Then, $(y - x) = -(x - y) = -(kn) = (-k)n$. Since $-k$ is an integer, $(y - x)$ is also divisible by $n$. Thus, $(y, x) \in R$, and $R$ is symmetric.

Step 3: Transitivity

Let $(x, y) \in R$ and $(y, z) \in R$. Then $(x - y) = kn$ and $(y - z) = mn$ for some integers $k, m$. Adding these equations: $$(x - y) + (y - z) = kn + mn$$ $$(x - z) = (k + m)n$$ Since $(k + m)$ is an integer, $(x - z)$ is divisible by $n$. Thus, $(x, z) \in R$, and $R$ is transitive.

Final Answer: Yes, R is an equivalence relation as it satisfies reflexivity, symmetry, and transitivity.

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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 algebraic relation.
Knowledge Dimension: CONCEPTUAL
Justification: The student must understand the underlying concept of equivalence relations and modular arithmetic properties to construct a formal proof.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the student's ability to perform a formal mathematical proof, which is a core competency requirement for the Relations and Functions chapter.

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