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 can follow these steps:

### Step 1: Understand the concept of 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.

### Step 2: Identify the number of elements in the set
The set \( A \) has 3 distinct elements: \( 1, 2, \) and \( 3 \). Thus, we have \( m = 3 \).

### Step 3: Use the formula for the number of equivalence relations
The maximum number of equivalence relations on a set with \( m \) distinct elements is given by the formula:
\[ 2^{m - 1} \]
This formula arises because each element can either be in a separate equivalence class or combined with others.

### Step 4: Substitute the value of \( m \)
Now, substituting \( m = 3 \) into the formula:
\[ 2^{3 - 1} = 2^2 = 4 \]

### Step 5: Count the partitions
To find the maximum number of equivalence relations, we need to count the partitions of the set \( A \):
1. Each element in its own class: \( \{ \{1\}, \{2\}, \{3\} \} \)
2. One class with two elements and one separate: \( \{ \{1, 2\}, \{3\} \} \), \( \{ \{1, 3\}, \{2\} \} \), \( \{ \{2, 3\}, \{1\} \} \)
3. All elements in one class: \( \{ \{1, 2, 3\} \} \)

This gives us the following partitions:
1. \( \{ \{1\}, \{2\}, \{3\} \} \)
2. \( \{ \{1, 2\}, \{3\} \} \)
3. \( \{ \{1, 3\}, \{2\} \} \)
4. \( \{ \{2, 3\}, \{1\} \} \)
5. \( \{ \{1, 2, 3\} \} \)

### Step 6: Count the total
Counting all the distinct partitions, we find:
- 1 partition where all elements are separate
- 3 partitions where two elements are together
- 1 partition where all elements are together

Thus, the total number of equivalence relations is \( 5 \).

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

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