Let \(A = \{1, 2, 3, \ldots, 12\}\). Consider the following equivalence relations on A.
Match List-I with List-II
| List-I (equivalence relation on A) | List-II (number of equivalence classes) | ||
| (A) | \(\{(a, b) : a - b \text{ is divisible by } 4\}\) | (I) | 1 |
| (B) | \(\{(a, b) : a - b \text{ is divisible by } 3\}\) | (II) | 3 |
| (C) | The identity relation on A | (III) | 12 |
| (D) | The universal relation \(A \times A\) | (IV) | 4 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(IV), (B)-(II), (C)-(III), (D)-(I)
(A) \(\{(a, b) : a - b \text{ is divisible by } 4\}\)
This matches (IV) 4.
Numbers are related when they leave the same remainder on division by 4:
\(\{1, 5, 9\}, \{2, 6, 10\}, \{3, 7, 11\}, \{4, 8, 12\}\)
(B) \(\{(a, b) : a - b \text{ is divisible by } 3\}\)
This matches (II) 3.
Numbers are related when they leave the same remainder on division by 3:
\(\{1, 4, 7, 10\}, \{2, 5, 8, 11\}, \{3, 6, 9, 12\}\)
(C) The identity relation on A
This matches (III) 12.
Each element is related only to itself, so each of the 12 elements forms its own class.
(D) The universal relation \(A \times A\)
This matches (I) 1.
Every element is related to every other, so the whole of A is one class.