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

Options

A. 5
B. `3!`
C. `2^(3)`
D. `3^(3)`

Answer

To find the number of equivalence relations that can be defined on the set {a, b, c}, we need to understand the properties of equivalence relations. An equivalence relation must satisfy three conditions: reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Understanding Equivalence Relations**:
- An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive.
- For the set {a, b, c}, we need to identify all possible ways to partition this set into equivalence classes.

2. **Identifying Partitions**:
- Each equivalence relation corresponds to a partition of the set. The number of equivalence relations on a set is equal to the number of ways to partition that set.
- For the set {a, b, c}, we can have the following partitions:
1. **Single class**: { {a, b, c} }
2. **Two classes**:
- { {a}, {b, c} }
- { {b}, {a, c} }
- { {c}, {a, b} }
3. **Three classes**: { {a}, {b}, {c} }

3. **Counting the Partitions**:
- From the above analysis, we can count the partitions:
- 1 partition with one class: { {a, b, c} }
- 3 partitions with two classes: { {a}, {b, c} }, { {b}, {a, c} }, { {c}, {a, b} }
- 1 partition with three classes: { {a}, {b}, {c} }
- Therefore, the total number of partitions is:
- 1 (one class) + 3 (two classes) + 1 (three classes) = 5.

4. **Conclusion**:
- The total number of equivalence relations that can be defined on the set {a, b, c} is 5.

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