Let \(A = \{1, 2, 3, \ldots, 9\}\) and let R be the equivalence relation on A given by \(R = \{(a, b) : a^{2} - b^{2} \text{ is divisible by } 3\}\). The number of elements in the equivalence class [2] is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 6
The class [2] contains every b in A with \(2^{2} - b^{2} = 4 - b^{2}\) divisible by 3. So it depends only on the remainder of \(b^{2}\) on division by 3.
If b is a multiple of 3, then \(b^{2}\) leaves remainder 0. Otherwise, b is \(3k + 1\) or \(3k + 2\), and
\((3k + 1)^{2} = 9k^{2} + 6k + 1\), remainder 1
\((3k + 2)^{2} = 9k^{2} + 12k + 4\), remainder 1
Since \(4 = 2^{2}\) leaves remainder 1, the class of 2 is made of the numbers of A that are not multiples of 3:
\([2] = \{1, 2, 4, 5, 7, 8\}\)
So [2] has 6 elements. The other class is {3, 6, 9}.