Logo
Jeeify
MCQMediumJEE 2023Relations

JEE Mathematics 2023 Question with Solution

The minimum number of elements that must be added to the relation R={(a,b),(b,c)}R = \{(a, b), (b, c)\} on the set {a,b,c}\{a, b, c\} so that it becomes symmetric and transitive is:

  • A

    44

  • B

    77

  • C

    55

  • D

    33

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: R={(a,b),(b,c)}R = \{(a,b),(b,c)\} on the set {a,b,c}\{a,b,c\}.

Find: The minimum number of ordered pairs to be added so that the relation becomes symmetric and transitive.

For symmetry, from

(a,b),(b,c)∈R(a,b),(b,c) \in R

we must also have

(b,a),(c,b)∈R(b,a),(c,b) \in R

For transitivity, from

(a,b),(b,c)∈R(a,b),(b,c) \in R

we must have

(a,c)∈R(a,c) \in R

Now, since (a,c)∈R(a,c) \in R and the relation must be symmetric, we also need

(c,a)∈R(c,a) \in R

Next, using transitivity:

(a,b),(b,a)∈R⇒(a,a)∈R(a,b),(b,a) \in R \Rightarrow (a,a) \in R

and

(b,c),(c,b)∈R⇒(b,b),(c,c)∈R(b,c),(c,b) \in R \Rightarrow (b,b),(c,c) \in R

Therefore, the elements to be added are

{(b,a),(c,b),(a,c),(c,a),(a,a),(b,b),(c,c)}\{(b,a),(c,b),(a,c),(c,a),(a,a),(b,b),(c,c)\}

Hence, the number of elements to be added is 77.

Therefore, the correct option is B.

Common mistakes

  • Adding only (b,a)(b,a) and (c,b)(c,b) for symmetry is incomplete. Symmetry alone is not enough; transitivity must also hold. After adding reverse pairs, check all possible compositions again.

  • Adding (a,c)(a,c) for transitivity but forgetting (c,a)(c,a) is incorrect. Once (a,c)(a,c) is included, symmetry forces its reverse pair as well.

  • Ignoring reflexive-looking pairs (a,a),(b,b),(c,c)(a,a), (b,b), (c,c) is a common error. They are not added because reflexivity is asked, but because transitivity of the newly added symmetric pairs forces them.

Practice more Relations questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trialMore on Relations

Related questions