Define a relation ∼ on the set 1,2,3,4,5,6,7,8,9,10 by a∼b if a-2b is divisible by 3. Then:

C. ∼ is symmetric but neither transitive nor reflexive. Step 1: a-2b≡0 (mod 3). Since -2≡1 (mod 3), this is the same as a+b≡0 (mod 3). Step 2 (reflexive?): a∼a needs 2a≡0 (mod 3), true only for multiples of 3. 1∼1 fails ( 1-2=-1). Not reflexive. Step 3 (symmetric?): a+b≡0 (mod 3) is the same condition as b+a≡0 (mod 3). Symmetric. Step 4 (transitive?): 1∼2 ( 1-4=-3) and 2∼1 ( 2-2=0), but 1∼1 is false. Not transitive. Hence symmetric only.

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Define a relation $\sim$ on the set $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ by $a \sim b$ if $a - 2 b$ is divisible by 3. Then:

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Answer: Option C — $\sim$ is symmetric but neither transitive nor reflexive

Detailed Solution

$a - 2 b \equiv 0 (m o d 3)$ . Since $- 2 \equiv 1 (m o d 3)$ , this is the same as $a + b \equiv 0 (m o d 3)$ .

Step 2 (reflexive?): $a \sim a$ needs $2 a \equiv 0 (m o d 3)$ , true only for multiples of 3. $1 \sim 1$ fails ( $1 - 2 = - 1$ ). Not reflexive.

Step 3 (symmetric?): $a + b \equiv 0 (m o d 3)$ is the same condition as $b + a \equiv 0 (m o d 3)$ . Symmetric.

Step 4 (transitive?): $1 \sim 2$ ( $1 - 4 = - 3$ ) and $2 \sim 1$ ( $2 - 2 = 0$ ), but $1 \sim 1$ is false. Not transitive.

Hence symmetric only.

Quick tip

$a - 2 b \equiv 0 (mod 3) \Leftrightarrow a + b \equiv 0 (mod 3)$

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Relations & Mappings — Preparation

This question tests your understanding of Relations & Mappings, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A relation from set A to set B is a subset of the Cartesian product A x B. A mapping, or function, is a special relation in which every element of A is related to exactly one element of B. This topic covers types of relations (reflexive, symmetric, transitive and equivalence relations), domain, codomain and range, and types of functions such as one-one (injective), onto (surjective) and bijective functions, along with composition and inverse functions. Exam questions test identifying the type of a given relation or function, counting the number of relations or functions between two sets, and finding the domain or range of a function. Useful counting facts: if A has m elements and B has n elements, there are 2^(mn) relations and n^m functions from A to B. Practise the relations and mappings questions below, each with a step-by-step solution.

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Define a relation ∼ on the set 1,2,3,4,5,6,7,8,9,10 by a∼b if a−2b is divisible by 3. Then:

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Relations & Mappings — Preparation

Define a relation $\sim$ on the set $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ by $a \sim b$ if $a - 2 b$ is divisible by 3. Then: