Which one of the following is a binary operation on N?

Options

A. Subtraction
B. Multiplication
C. Division
D. All the above

Answer

To determine which of the given operations is a binary operation on the set of natural numbers \( N \), we need to analyze each operation: subtraction, multiplication, and division.

### Step-by-Step Solution:

1. **Definition of Binary Operation**:
A binary operation on a set \( S \) is a function that combines any two elements \( a \) and \( b \) from \( S \) to produce another element \( c \) in \( S \). In mathematical terms, if \( * \) is a binary operation, then for all \( a, b \in S \), the result \( a * b \) must also be in \( S \).

2. **Set of Natural Numbers**:
The set of natural numbers \( N \) is defined as \( \{1, 2, 3, \ldots\} \).

3. **Analyzing Each Operation**:
- **Multiplication**:
- Let’s take two natural numbers, for example, \( 2 \) and \( 3 \).
- Performing multiplication: \( 2 \times 3 = 6 \).
- Since \( 6 \) is also a natural number, multiplication is a binary operation on \( N \).

- **Subtraction**:
- Using the same numbers, \( 2 \) and \( 3 \):
- Performing subtraction: \( 2 - 3 = -1 \).
- Since \( -1 \) is not a natural number, subtraction is not a binary operation on \( N \).

- **Division**:
- Again using \( 2 \) and \( 3 \):
- Performing division: \( 2 \div 3 = \frac{2}{3} \approx 0.67 \).
- Since \( \frac{2}{3} \) is not a natural number, division is not a binary operation on \( N \).

4. **Conclusion**:
Among the operations analyzed, only multiplication results in a natural number when applied to any two natural numbers. Therefore, the correct answer is that multiplication is the only binary operation on \( N \).

### Final Answer:
The correct option is **multiplication**.

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