The number of non-empty equival ence relations on the set {1,2,3} is : (1)6 (2)7 (3)5 (4)4

An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive. Equivalence relations correspond exactly to partitions of the set. That means each equivalence relation divides the set into disjoint subsets called equivalence classes.

For the set {1, 2, 3}, the number of equivalence relations equals the number of ways to partition the set into non-empty subsets.

The number of partitions of a set with n elements is given by the Bell number B_n. For n=3, the Bell number B_3 = 5.

These partitions are:

1. { {1}, {2}, {3} }
2. { {1, 2}, {3} }
3. { {1, 3}, {2} }
4. { {2, 3}, {1} }
5. { {1, 2, 3} }

Each partition corresponds to one equivalence relation. Since the question asks for non-empty equivalence relations, and equivalence relations are never empty (they always contain at least the identity pairs), the answer is 5.

Imagine you have three friends: 1, 2, and 3. You want to group them so that each group has friends who are 'equivalent' to each other. An equivalence relation is like making groups where everyone in a group is connected in a special way. The question is: how many different ways can you make these groups without leaving the whole set empty? To solve this, we need to find all possible ways to split the set {1, 2, 3} into groups where each group is connected properly.

Step 1: Understand that equivalence relations correspond to partitions of the set.

Step 2: Find the number of partitions of the set {1, 2, 3}.

Step 3: The number of partitions of a set with 3 elements is the Bell number B_3.

Step 4: Bell numbers for small n are:

• B_1 = 1
• B_2 = 2
• B_3 = 5

Step 5: List the partitions of {1, 2, 3}:

• { {1}, {2}, {3} }
• { {1, 2}, {3} }
• { {1, 3}, {2} }
• { {2, 3}, {1} }
• { {1, 2, 3} }

Step 6: Each partition corresponds to one equivalence relation.

Step 7: Since equivalence relations are never empty, all 5 are valid.

Final answer: 5

So, the correct option is (3) 5.