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: Understanding the Concept:
An equivalence relation on a set corresponds exactly to a partition of that set into disjoint blocks (equivalence classes). We need those partitions of \(\{1,2,3\}\) in which 1 and 2 land in the same block.

Step 2: Listing the partitions of a 3-element set:
The partitions of \(\{1,2,3\}\) are: \(\{1\}\{2\}\{3\}\), \(\{1,2\}\{3\}\), \(\{1,3\}\{2\}\), \(\{2,3\}\{1\}\), and \(\{1,2,3\}\).

Step 3: Keeping only those with 1 and 2 together:
Only \(\{1,2\}\{3\}\) and \(\{1,2,3\}\) place 1 and 2 in the same block, so their relations contain the pair \((1,2)\).

Final Answer:
There are exactly \(\boxed{2}\) such equivalence relations.