Sets, Relations and FunctionsMCQPYQ Dec. 21Question 1944 of 217
All Questions

If AisrelatedtoBifandonlyifthedifferencein\displaystyle A is related to B if and only if the difference inAand\displaystyle andB$ is an even integer. This relation is

Options

ASymmetric, reflexive but not transitive
BSymmetric, transitive but not reflexive
CTransitive, reflexive but not symmetric
DEquivalence relation
For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option d — Equivalence relation

All Options:

  • ASymmetric, reflexive but not transitive
  • BSymmetric, transitive but not reflexive
  • CTransitive, reflexive but not symmetric
  • DEquivalence relation

Ad

Detailed Solution & Explanation

Let the relation R\displaystyle R be defined on the set of integers Z\displaystyle \mathbb{Z} by:
ARB  ⟺  A−B is an even integerA R B \iff A - B \text{ is an even integer}
Let's check the three properties of an equivalence relation:
1. **Reflexive**: For any integer A\displaystyle A, we have:
A−A=0A - A = 0
Since 0\displaystyle 0 is an even integer (0=2×0\displaystyle 0 = 2 \times 0), ARA\displaystyle A R A holds for all A\displaystyle A. The relation is reflexive.
2. **Symmetric**: If ARB\displaystyle A R B, then A−B=2k\displaystyle A - B = 2k for some integer k\displaystyle k.
B−A=−(A−B)=−2k=2(−k)B - A = -(A - B) = -2k = 2(-k)
Since −k\displaystyle -k is also an integer, B−A\displaystyle B - A is an even integer, which means BRA\displaystyle B R A. The relation is symmetric.
3. **Transitive**: If ARB\displaystyle A R B and BRC\displaystyle B R C, then A−B=2k1\displaystyle A - B = 2k_1 and B−C=2k2\displaystyle B - C = 2k_2 for some integers k1\displaystyle k_1 and k2\displaystyle k_2. Adding these two equations:
(A−B)+(B−C)=2k1+2k2(A - B) + (B - C) = 2k_1 + 2k_2
A−C=2(k1+k2)A - C = 2(k_1 + k_2)
Since k1+k2\displaystyle k_1 + k_2 is an integer, A−C\displaystyle A - C is an even integer, which means ARC\displaystyle A R C. The relation is transitive.
Since R\displaystyle R is reflexive, symmetric, and transitive, it is an **equivalence relation**.
Hence, **Option D** 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.

Related Comparison Tables

Consignment vs Sale: Key Accounting Differences

More Questions from Sets, Relations and Functions

If A={1,2,3,4,5}\displaystyle A = \{1, 2, 3, 4, 5\}, B={2,4}\displaystyle B = \{2, 4\} and C={1,3,5}\displaystyle C = \{1, 3, 5\} then (A−C)×B\displaystyle (A - C) \times B is:

If f(x)=x2−5\displaystyle f(x)=x^2-5, evaluate f(3)\displaystyle f(3), f(−4)\displaystyle f(-4), f(5)\displaystyle f(5), and f(1)\displaystyle f(1).

Out of a group of 20\displaystyle 20 teachers in a school, 10\displaystyle 10 teach Mathematics, 9\displaystyle 9 teach Physics and 7\displaystyle 7 teach Chemistry. 4\displaystyle 4 teach Mathematics and Physics but none teach both Mathematics and Chemistry. How many teach Chemistry and Physics; how many teach only Physics?

If A={1,2,3,4,5,7,8,9}\displaystyle A = \{1, 2, 3, 4, 5, 7, 8, 9\} and B={2,4,6,7,9}\displaystyle B = \{2, 4, 6, 7, 9\} then how many proper subset of A∩B\displaystyle A \cap B can be created?

The number of subsets of the set {0,1,2,3}\displaystyle \{0, 1, 2, 3\} is:

If f(x)=x2+x−1\displaystyle f(x) = x^2+x-1 and 4f(x)=f(2x)\displaystyle 4f(x) = f(2x), then find the value of 'x'.

Ready to Master Sets, Relations and Functions?

Practice all 217 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free