The maximum number of equivalence relations on the set A = {1, 2, 3} are

Options

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

Explanation

To find the maximum number of equivalence relations on the set \( A = \{1, 2, 3\} \), we need to understand the concept of equivalence relations and how they relate to partitions of a set.

### Step-by-Step Solution:

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

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

3. **Calculating Bell Number for \( n = 3 \)**:
The Bell number \( B_3 \) can be calculated as follows:
- The partitions of the set \( A = \{1, 2, 3\} \) are:
1. \( \{\{1\}, \{2\}, \{3\}\} \) (each element in its own set)
2. \( \{\{1, 2\}, \{3\}\} \) (1 and 2 together, 3 alone)
3. \( \{\{1, 3\}, \{2\}\} \) (1 and 3 together, 2 alone)
4. \( \{\{2, 3\}, \{1\}\} \) (2 and 3 together, 1 alone)
5. \( \{\{1, 2, 3\}\} \) (all elements together)

4. **Counting the Partitions**:
From the above, we can see that there are 5 distinct partitions of the set \( A = \{1, 2, 3\} \).

5. **Conclusion**:
Therefore, the maximum number of equivalence relations on the set \( A = \{1, 2, 3\} \) is \( 5 \).

### Final Answer:
The maximum number of equivalence relations on the set \( A = \{1, 2, 3\} \) is \( 5 \).

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