Let \(A = \{1, 2, 3, 4, 5\}\).
Match List-I with List-II
| List-I (relation on A) | List-II (number of ordered pairs) | ||
| (A) | The identity relation on A | (I) | 10 |
| (B) | The universal relation on A | (II) | 25 |
| (C) | The smallest equivalence relation on A containing (1, 2) and (2, 3) | (III) | 5 |
| (D) | \(\{(a, b) : a \lt b\}\) | (IV) | 11 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(III), (B)-(II), (C)-(IV), (D)-(I)
(A) The identity relation on A
This matches (III) 5.
It is \(\{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\}\), one pair for each element.
(B) The universal relation on A
This matches (II) 25.
It is \(A \times A\), with \(5 \times 5 = 25\) pairs.
(C) The smallest equivalence relation on A containing (1, 2) and (2, 3)
This matches (IV) 11.
1 is related to 2 and 2 to 3, so by symmetry and transitivity 1, 2, 3 must all be related to one another. 4 and 5 are related only to themselves. The equivalence classes are {1, 2, 3}, {4} and {5}, so the number of pairs is
\(3 \times 3 + 1 + 1 = 11\)
(D) \(\{(a, b) : a \lt b\}\)
This matches (I) 10.
Each pair of two different elements gives exactly one pair with the smaller one first:
\(4 + 3 + 2 + 1 = 10\)