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

To determine the number of non-empty equivalence relations on the set \(\{1,2,3\}\), we need to find the number of partitions of this set. An equivalence relation corresponds to a partition of the set, and we enumerate the possible partitions of the set \(\{1,2,3\}\) as follows:

• Partition with one subset (trivial partition): \(\{\{1,2,3\}\}\)
• Partitions with two subsets:
  • \(\{\{1\},\{2,3\}\}\)
  • \(\{\{2\},\{1,3\}\}\)
  • \(\{\{3\},\{1,2\}\}\)
• Partition with three subsets: \(\{\{1\},\{2\},\{3\}\}\)

Counting these partitions, we find there are 5 distinct ways to partition the set \(\{1,2,3\}\). Hence, the number of non-empty equivalence relations on the set \(\{1,2,3\}\) is 5.