Question

The number of non-empty equivalence relations on the set \(\{1,2,3\}\) is :

• A. 6
• B. 7
• C. 5
• D. 4

Hint:

To count the number of equivalence relations, find all the possible partitions of the set.

Answer 1

The Correct Option is C

Step 1: Understanding equivalence relations. An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive. For a set of three elements, \( \{1, 2, 3\} \), we need to count all possible non-empty equivalence relations. 
Step 2: List all possible partitions of the set. The number of equivalence relations corresponds to the number of partitions of the set \( \{1, 2, 3\} \). The possible partitions are: \[ \{ \{1\}, \{2\}, \{3\} \}, \quad \{ \{1, 2\}, \{3\} \}, \quad \{ \{1, 3\}, \{2\} \}, \quad \{ \{2, 3\}, \{1\} \}, \quad \{ \{1, 2, 3\} \}. \] Step 3: Count the number of partitions. From the above, there are 5 possible partitions, and hence, 5 possible equivalence relations. Thus, the number of non-empty equivalence relations on the set is \( \boxed{5} \).