Question

If \(A=\{1,2,3\}\), then the number of equivalence relations on \(A\) containing \((1,2)\) is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Count partitions of \(\{1,2,3\}\) where 1 and 2 are in the same block.

Answer 1

The Correct Option is B

Step 1: Direct counting via block sizes:
Since 1 and 2 must be equivalent, treat \(\{1,2\}\) as one merged unit; the remaining element 3 is either in this block or its own block.

Step 2: Case 1:
3 joins \(\{1,2\}\): gives the single block \(\{1,2,3\}\).

Step 3: Case 2:
3 stays alone: gives blocks \(\{1,2\},\{3\}\).

Final Answer:
Exactly \(\boxed{2}\) partitions arise, so 2 equivalence relations contain \((1,2)\).