Sets, Relations and FunctionsPYQ May 25Question 4332 of 145
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Which of the following relations is transitive but not reflexive for the set S={3,4,6}\displaystyle S = \{3, 4, 6\}?

Options

AR={(3,4),(4,6),(3,6)}\displaystyle R = \{(3, 4), (4, 6), (3, 6)\}
BR={(1,2),(1,3),(1,4)}\displaystyle R = \{(1, 2), (1, 3), (1, 4)\}
CR={(3,3),(4,4),(6,6)}\displaystyle R = \{(3, 3), (4, 4), (6, 6)\}
DR={(3,4),(4,3)}\displaystyle R = \{(3, 4), (4, 3)\}
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Correct Answer

✅ Option a — R={(3,4),(4,6),(3,6)}\displaystyle R = \{(3, 4), (4, 6), (3, 6)\}

All Options:

  • AR={(3,4),(4,6),(3,6)}\displaystyle R = \{(3, 4), (4, 6), (3, 6)\}
  • BR={(1,2),(1,3),(1,4)}\displaystyle R = \{(1, 2), (1, 3), (1, 4)\}
  • CR={(3,3),(4,4),(6,6)}\displaystyle R = \{(3, 3), (4, 4), (6, 6)\}
  • DR={(3,4),(4,3)}\displaystyle R = \{(3, 4), (4, 3)\}

Detailed Solution & Explanation

Let the set be S={3,4,6}\displaystyle S = \{3, 4, 6\}.
Recall the definitions of reflexive and transitive relations:
1. **Reflexive**: A relation R\displaystyle R on S\displaystyle S is reflexive if for every x∈S\displaystyle x \in S, (x,x)∈R\displaystyle (x, x) \in R. For S\displaystyle S, this means R\displaystyle R must contain (3,3)\displaystyle (3, 3), (4,4)\displaystyle (4, 4), and (6,6)\displaystyle (6, 6).
2. **Transitive**: A relation R\displaystyle R on S\displaystyle S is transitive if (x,y)∈R\displaystyle (x, y) \in R and (y,z)∈R\displaystyle (y, z) \in R implies (x,z)∈R\displaystyle (x, z) \in R.

Let us analyze the given options:
- **Option a**: R={(3,4),(4,6),(3,6)}\displaystyle R = \{(3, 4), (4, 6), (3, 6)\}
- Reflexive? No, because it does not contain (3,3)\displaystyle (3, 3), (4,4)\displaystyle (4, 4), or (6,6)\displaystyle (6, 6).
- Transitve? Yes, because (3,4)∈R\displaystyle (3, 4) \in R and (4,6)∈R\displaystyle (4, 6) \in R implies (3,6)∈R\displaystyle (3, 6) \in R, which is in R\displaystyle R. No other pairs violate transitivity.
Therefore, Option a is transitive but not reflexive.

- **Option b**: R={(1,2),(1,3),(1,4)}\displaystyle R = \{(1, 2), (1, 3), (1, 4)\}
This is not a relation on S\displaystyle S as it contains elements 1\displaystyle 1 and 2\displaystyle 2 which are not in S\displaystyle S.

- **Option c**: R={(3,3),(4,4),(6,6)}\displaystyle R = \{(3, 3), (4, 4), (6, 6)\}
This is the identity relation. It is both reflexive and transitive.

- **Option d**: R={(3,4),(4,3)}\displaystyle R = \{(3, 4), (4, 3)\}
This relation is not transitive because (3,4)∈R\displaystyle (3, 4) \in R and (4,3)∈R\displaystyle (4, 3) \in R but (3,3)∉R\displaystyle (3, 3) \notin R.

Hence, Option a is transitive but not reflexive.
Hence, **Option A** 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.

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