If A = {1, b}, then the number of binary operations that can be defined on A is

Options

A. 4
B. 2
C. 16
D. 1

Answer

To find the number of binary operations that can be defined on the set \( A = \{1, b\} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of elements in the set \( A \)**:
- The set \( A \) has 2 elements: \( 1 \) and \( b \).
- Therefore, \( n = 2 \).

2. **Understand the concept of binary operations**:
- A binary operation on a set \( A \) is a function that takes two elements from \( A \) and combines them to produce another element from \( A \).
- The number of binary operations that can be defined on a set with \( n \) elements is given by the formula \( m^{n^2} \), where \( m \) is the number of elements in the set.

3. **Apply the formula**:
- Here, \( m = 2 \) (the number of elements in set \( A \)).
- We need to calculate \( 2^{n^2} \).
- Since \( n = 2 \), we have \( n^2 = 2^2 = 4 \).

4. **Calculate the number of binary operations**:
- Now substituting into the formula gives us:
\[ 2^{n^2} = 2^4 = 16 \]

5. **Conclusion**:
- Therefore, the number of binary operations that can be defined on the set \( A = \{1, b\} \) is \( 16 \).

### Final Answer:
The number of binary operations that can be defined on \( A \) is \( 16 \).

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