Set, Relations and FunctionsPYQ Jun 23Question 1946 of 217
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Given the relation R={(1,2),(2,3)}\displaystyle R = \{(1,2), (2,3)\} on the set A={1,2,3}\displaystyle A = \{1,2,3\}, the minimum number of ordered pairs which when added to R\displaystyle R make it equivalence relation is

Options

A5
B7
C6
D8
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Correct Answer

✅ Option b — 7

All Options:

  • A5
  • B7
  • C6
  • D8

Detailed Solution & Explanation

We are given set A={1,2,3}\displaystyle A = \{1, 2, 3\} and relation R={(1,2),(2,3)}\displaystyle R = \{(1,2), (2,3)\}.
An equivalence relation R′\displaystyle R' on A\displaystyle A must satisfy reflexivity, symmetry, and transitivity:
1. **Reflexivity**: R′\displaystyle R' must contain (1,1),(2,2),(3,3)\displaystyle (1, 1), (2, 2), (3, 3) (3 pairs).
2. **Symmetry**: Since (1,2)∈R\displaystyle (1, 2) \in R, we must add (2,1)∈R′\displaystyle (2, 1) \in R'. Since (2,3)∈R\displaystyle (2, 3) \in R, we must add (3,2)∈R′\displaystyle (3, 2) \in R' (2 pairs).
3. **Transitivity**: Since we have (1,2)∈R′\displaystyle (1, 2) \in R' and (2,3)∈R′\displaystyle (2, 3) \in R', we must add (1,3)∈R′\displaystyle (1, 3) \in R'.
By symmetry, since (1,3)∈R′\displaystyle (1, 3) \in R', we must also add (3,1)∈R′\displaystyle (3, 1) \in R' (2 pairs).
Thus, the smallest equivalence relation containing R\displaystyle R is the universal relation on A\displaystyle A, which contains 3×3=9\displaystyle 3 \times 3 = 9 ordered pairs:
R′={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2),(1,3),(3,1)}R' = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)\}
The original relation R\displaystyle R has 2 elements: {(1,2),(2,3)}\displaystyle \{(1, 2), (2, 3)\}.
Therefore, the minimum number of ordered pairs that must be added is:
Pairs to add=9−2=7\text{Pairs to add} = 9 - 2 = 7
Hence, **Option B** is the correct answer.

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