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

The count of non-empty equivalence relations on the set \(\{1,2,3\}\) is equivalent to the number of partitions of this set. Each equivalence relation uniquely defines a partition. The partitions of \(\{1,2,3\}\) are enumerated as follows:

• Single subset partition: \(\{\{1,2,3\}\}\)
• Two subset partitions:
  • \(\{\{1\},\{2,3\}\}\)
  • \(\{\{2\},\{1,3\}\}\)
  • \(\{\{3\},\{1,2\}\}\)
• Three subset partition: \(\{\{1\},\{2\},\{3\}\}\)

In total, there are 5 unique partitions for the set \(\{1,2,3\}\). Therefore, there are 5 non-empty equivalence relations on the set \(\{1,2,3\}\).