The maximum number of equivalence relations on the set `A = {phi , {phi}, {{phi}}}` are

Options

A. 1
B. 2
C. 3
D. 5

Explanation

To find the maximum number of equivalence relations on the set \( A = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\} \), we need to understand the concept of equivalence relations and partitions.

### Step-by-Step Solution:

1. **Identify the Elements of the Set**:
The set \( A \) contains three distinct elements:
- \( \emptyset \) (the empty set)
- \( \{\emptyset\} \) (a set containing the empty set)
- \( \{\{\emptyset\}\} \) (a set containing a set that contains the empty set)

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

3. **Counting Partitions**:
The maximum 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 \( B_n \).

4. **Calculate the Bell Number for \( n = 3 \)**:
The Bell number \( B_3 \) can be calculated or looked up. The Bell numbers are:
- \( B_0 = 1 \)
- \( B_1 = 1 \)
- \( B_2 = 2 \)
- \( B_3 = 5 \)

Therefore, \( B_3 = 5 \).

5. **List the Partitions**:
The partitions of the set \( A \) are:
- \( \{\{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}\} \) (all elements in one subset)
- \( \{\{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset\}\} \) (two elements in one subset and one in another)
- \( \{\{\emptyset\}, \{\emptyset\}, \{\{\emptyset\}\}\} \) (two elements in one subset and one in another)
- \( \{\{\emptyset\}, \{\{\emptyset\}\}\}, \{\emptyset\} \) (two subsets)
- \( \{\{\emptyset\}\}, \{\{\emptyset\}\}, \{\{\emptyset\}\} \) (each element in its own subset)

6. **Conclusion**:
The maximum number of equivalence relations on the set \( A \) is \( 5 \).

### Final Answer:
The maximum number of equivalence relations on the set \( A = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\} \) is \( 5 \).