Set, Relations and FunctionsPYQ Dec. 21Question 1944 of 217
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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
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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 \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.

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