The number of equivalence relations that can be defined on set {a, b, c}, is

Options

A. 3
B. 5
C. 7
D. 8

Explanation

To find the number of equivalence relations that can be defined on the set \( S = \{a, b, c\} \), we need to understand the concept of equivalence relations and how they correspond to partitions of the set.

### Step-by-Step Solution:

1. **Understanding Equivalence Relations**:
An equivalence relation on a set is a relation that satisfies three properties: reflexive, symmetric, and transitive. Each equivalence relation corresponds to a partition of the set.

2. **Counting Partitions**:
The number of equivalence relations on a set is equal to the number of ways to partition that set. For a set with \( n \) elements, the number of partitions is given by the Bell number \( B_n \).

3. **Finding the Bell Number**:
For our set \( S \) with 3 elements, we need to find \( B_3 \). The Bell numbers for small values of \( n \) are:
- \( B_0 = 1 \)
- \( B_1 = 1 \)
- \( B_2 = 2 \)
- \( B_3 = 5 \)

4. **Listing the Partitions**:
We can explicitly list the partitions of the set \( S = \{a, b, c\} \):
- **1 partition**: \( \{\{a, b, c\}\} \)
- **3 partitions**: \( \{\{a\}, \{b, c\}\}, \{\{b\}, \{a, c\}\}, \{\{c\}, \{a, b\}\} \)
- **1 partition**: \( \{\{a\}, \{b\}, \{c\}\} \)

Thus, the partitions are:
1. \( \{\{a, b, c\}\} \) (1 subset)
2. \( \{\{a\}, \{b, c\}\} \)
3. \( \{\{b\}, \{a, c\}\} \)
4. \( \{\{c\}, \{a, b\}\} \)
5. \( \{\{a\}, \{b\}, \{c\}\} \) (3 subsets)

5. **Conclusion**:
Therefore, the total number of equivalence relations that can be defined on the set \( S = \{a, b, c\} \) is \( 5 \).

### Final Answer:
The number of equivalence relations that can be defined on the set \( \{a, b, c\} \) is **5**.