If `A={1,2,3}` then the maximum number of equivalence relations on A is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the maximum number of equivalence relations on the set \( A = \{1, 2, 3\} \), we can follow these steps:

### Step 1: Understand Equivalence Relations
An equivalence relation on a set must satisfy three properties:
1. **Reflexivity**: Every element must be related to itself. For example, \( (1, 1), (2, 2), (3, 3) \) must be included.
2. **Symmetry**: If one element is related to another, then the second must be related to the first. For example, if \( (1, 2) \) is included, then \( (2, 1) \) must also be included.
3. **Transitivity**: If one element is related to a second, and the second is related to a third, then the first must be related to the third. For example, if \( (1, 2) \) and \( (2, 3) \) are included, then \( (1, 3) \) must also be included.

### Step 2: Identify Partitions of Set A
Equivalence relations correspond to partitions of the set. We need to find all possible ways to partition the set \( A \).

1. **Single partition**: All elements in one group:
- \( \{ \{1, 2, 3\} \} \)

2. **Two partitions**: One group has two elements, and the other has one:
- \( \{ \{1, 2\}, \{3\} \} \)
- \( \{ \{1, 3\}, \{2\} \} \)
- \( \{ \{2, 3\}, \{1\} \} \)

3. **Three partitions**: Each element in its own group:
- \( \{ \{1\}, \{2\}, \{3\} \} \)

### Step 3: Count the Partitions
Now, we count the partitions we identified:
- 1 partition with all elements together: \( \{ \{1, 2, 3\} \} \)
- 3 partitions with one element separate: \( \{ \{1, 2\}, \{3\} \}, \{ \{1, 3\}, \{2\} \}, \{ \{2, 3\}, \{1\} \} \)
- 1 partition with all elements separate: \( \{ \{1\}, \{2\}, \{3\} \} \)

### Step 4: Total Number of Equivalence Relations
Adding these together gives us:
- 1 (all together) + 3 (two together, one separate) + 1 (all separate) = 5

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

### Final Answer
The maximum number of equivalence relations on \( A \) is **5**.

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