Q2141: Check whether $f : Z \times Z \to Z \times Z$ (where $Z$ is ...

CBSE Class 12 Mathematics Relations and Functions Q #2141

COMPETENCY BASED

APPLY

2 Marks Generic-Curriculum VSA

Check whether $f : Z \times Z \to Z \times Z$ (where $Z$ is the set of integers) defined as $f(x, y) = (2y, 3x)$ is injective or not. [Delhi 2026]

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AI Tutor Explanation

Detailed Solution

Step 1: Understand the definition of Injective (One-to-One)

A function $f: A \to B$ is injective if $f(x_1, y_1) = f(x_2, y_2)$ implies $(x_1, y_1) = (x_2, y_2)$ for all $(x_1, y_1), (x_2, y_2) \in Z \times Z$.

Step 2: Set up the equality condition

Let $(x_1, y_1)$ and $(x_2, y_2)$ be two elements in $Z \times Z$ such that $f(x_1, y_1) = f(x_2, y_2)$.

Step 3: Apply the function definition

Given $f(x, y) = (2y, 3x)$, the equality becomes: $$(2y_1, 3x_1) = (2y_2, 3x_2)$$

Step 4: Equate the components

By comparing the corresponding components: $$2y_1 = 2y_2 \implies y_1 = y_2$$ $$3x_1 = 3x_2 \implies x_1 = x_2$$

Step 5: Conclusion

Since $f(x_1, y_1) = f(x_2, y_2)$ leads to $(x_1, y_1) = (x_2, y_2)$, the function is injective.

Final Answer: The function is injective.

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student must apply the formal definition of an injective function to a specific algebraic mapping involving Cartesian products of integers.

Knowledge Dimension: PROCEDURAL

Justification: The student follows a standard algorithmic procedure (setting up equality, equating components, and solving) to verify the property of the function.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the foundational understanding of Relations and Functions (Chapter 1), specifically the ability to handle mappings on sets other than simple real numbers.

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