If `A={1,3,5}`, then the number of equivalence relations on A containing (1,3) is

Options

A. 1
B. 2
C. 4
D. 5

Answer

To find the number of equivalence relations on the set \( A = \{1, 3, 5\} \) that contain the pair \( (1, 3) \), we need to follow these steps:

### Step 1: Understand the properties of equivalence relations
An equivalence relation must satisfy three properties:
1. **Reflexivity**: Every element must be related to itself. Therefore, \( (1, 1), (3, 3), (5, 5) \) must be included.
2. **Symmetry**: If \( (a, b) \) is in the relation, then \( (b, a) \) must also be included. Since \( (1, 3) \) is included, \( (3, 1) \) must also be included.
3. **Transitivity**: If \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be included.

### Step 2: Start with the required pairs
Since we must include \( (1, 3) \) and \( (3, 1) \), we can start building our equivalence relation:
- Include \( (1, 1), (3, 3), (5, 5) \) for reflexivity.
- Include \( (1, 3) \) and \( (3, 1) \) for symmetry.

### Step 3: Determine how to include \( 5 \)
Now we need to consider how to include the element \( 5 \):
- We can either relate \( 5 \) to itself only, which gives us one equivalence class: \( \{1, 3\} \) and \( \{5\} \).
- Or we can relate \( 5 \) to \( 1 \) and \( 3 \), which gives us a single equivalence class: \( \{1, 3, 5\} \).

### Step 4: List the equivalence relations
1. **Relation 1**: \( \{(1, 1), (3, 3), (5, 5), (1, 3), (3, 1)\} \) - This corresponds to the equivalence classes \( \{1, 3\} \) and \( \{5\} \).
2. **Relation 2**: \( \{(1, 1), (3, 3), (5, 5), (1, 3), (3, 1), (1, 5), (5, 1), (3, 5), (5, 3)\} \) - This corresponds to the equivalence class \( \{1, 3, 5\} \).

### Step 5: Count the equivalence relations
Thus, we have identified two equivalence relations on the set \( A \) that contain the pair \( (1, 3) \).

### Final Answer
The number of equivalence relations on \( A \) containing \( (1, 3) \) is **2**.

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