CBSE Class 12 Mathematics Relations and Functions Q #2135
KNOWLEDGE BASED
APPLY
2 Marks Generic-Curriculum VSA
Let $R$ be the equivalence relation in the set $Z$ of integers given by $R = \{(a, b) : 2 \text{ divides } a - b\}$. Write the equivalence class $[0]$. [All India 2018]
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Detailed Solution

Step 1: Understand the definition of the equivalence class

The equivalence class $[0]$ is defined as the set of all elements $x$ in the set of integers $Z$ such that $(x, 0)$ belongs to the relation $R$. Mathematically, $[0] = \{x \in Z : (x, 0) \in R\}$.

Step 2: Apply the condition of the relation

The relation $R$ is defined as $(a, b) \in R$ if $2$ divides $a - b$. Substituting $a = x$ and $b = 0$, we require that $2$ divides $x - 0$. This simplifies to $2$ divides $x$.

Step 3: Identify the set of integers

An integer $x$ is divisible by $2$ if and only if $x$ is an even integer. Therefore, the set of all such integers is the set of all even integers, which can be represented as $\{..., -4, -2, 0, 2, 4, ...\}$.

Final Answer: {x \in Z : x \text{ is an even integer}}

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the formal definition of an equivalence class to a specific relation provided in the problem statement.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a step-by-step logical procedure to translate the set-builder notation into a recognizable set of integers.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. This question tests the fundamental understanding of Relations and Functions, specifically the concept of equivalence classes, which is a core topic in the NCERT curriculum.

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