Sets, Relations and FunctionsMCQPYQ Jan. 21Question 1943 of 217
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In the set of all straight lines on a plane which of the following is not "TRUE"?

Options

AParallel to is an equivalence relation
BPerpendicular to is a symmetric relation
CPerpendicular to is an equivalence relation
DParallel to a reflexive relation
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Correct Answer

Option cPerpendicular to is an equivalence relation

All Options:

  • AParallel to is an equivalence relation
  • BPerpendicular to is a symmetric relation
  • CPerpendicular to is an equivalence relation
  • DParallel to a reflexive relation

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Detailed Solution & Explanation

Let us analyze each statement regarding straight lines in a plane:
1. **"Parallel to" relation**:
- **Reflexive**: Every line L\displaystyle L is parallel to itself (LL\displaystyle L \parallel L).
- **Symmetric**: If L1L2\displaystyle L_1 \parallel L_2, then L2L1\displaystyle L_2 \parallel L_1.
- **Transitive**: If L1L2\displaystyle L_1 \parallel L_2 and L2L3\displaystyle L_2 \parallel L_3, then L1L3\displaystyle L_1 \parallel L_3.
Since it is reflexive, symmetric, and transitive, "Parallel to" is an equivalence relation. Hence, **Option A and Option D are TRUE**.
2. **"Perpendicular to" relation**:
- **Reflexive**: A line L\displaystyle L cannot be perpendicular to itself (L⊥̸L\displaystyle L \not\perp L). Thus, it is not reflexive.
- **Symmetric**: If L1L2\displaystyle L_1 \perp L_2, then L2L1\displaystyle L_2 \perp L_1. This is TRUE. Hence, **Option B is TRUE**.
- **Transitive**: If L1L2\displaystyle L_1 \perp L_2 and L2L3\displaystyle L_2 \perp L_3, then L1\displaystyle L_1 and L3\displaystyle L_3 are parallel to each other (L1L3\displaystyle L_1 \parallel L_3), not perpendicular. Thus, it is not transitive.
Since it is not reflexive or transitive, "Perpendicular to" is **NOT** an equivalence relation. Therefore, the statement "Perpendicular to is an equivalence relation" is **FALSE** (not TRUE).
*Note: The question asks which statement is NOT TRUE. Thus, Option C is the correct answer. The textbook answer key incorrectly lists Option B as correct.*
Hence, **Option C** is the correct answer.

About This Chapter: Sets, Relations and Functions

Paper

Paper 3: Quantitative Aptitude

Weightage

3-5 Marks

Key Topics

Sets, Relations, Functions

This chapter covers Sets, Relations, Functions and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 3-5 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

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