Sets, Relations and FunctionsMCQPYQ Dec. 21Question 1944 of 222
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If $A is related to B if and only if the difference in $Aand\displaystyle andB$ is an even integer. This relation is

Select Your Answer (Keyboard: A, B, C, D)
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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

Detailed Solution & Explanation

Let the relation R\displaystyle R be defined on the set of integers Z\displaystyle Z\displaystyle \mathbb{Z} by:
ARB  ⟺  A−B is an even integerA R B \iff A - B  is an even integer\displaystyle \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 ×\displaystyle \times 0), $A R A$ holds for all A\displaystyle A. The relation is reflexive.
2. **Symmetric**: If $A R B$, 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 $B R A$. The relation is symmetric.
3. **Transitive**: If $A R B$ and $B R C$, 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 $A R C$. 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.

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