The set of netural numbers is closed under the binary operations of :

Options

A. addition , subtraction, multiplication and division
B. addition, subtraction , multiplication but not division
C. addition and multiplication but not subtraction and division
D. addtion and substraction but not multiplication and division

Answer

To determine whether the set of natural numbers is closed under the given binary operations, we need to analyze each operation one by one. The operations we will consider are addition, subtraction, multiplication, and division.

### Step-by-Step Solution:

1. **Understand Closure**:
- A set is said to be closed under a binary operation if performing that operation on members of the set always produces a member of the same set.

2. **Addition**:
- Take any two natural numbers, for example, \( a = 5 \) and \( b = 6 \).
- When we add them: \( a + b = 5 + 6 = 11 \).
- Since 11 is also a natural number, the set of natural numbers is closed under addition.

3. **Subtraction**:
- Now, consider the subtraction of two natural numbers, for example, \( a = 5 \) and \( b = 10 \).
- When we subtract them: \( a - b = 5 - 10 = -5 \).
- Since -5 is not a natural number, the set of natural numbers is not closed under subtraction.

4. **Multiplication**:
- Next, take two natural numbers, for example, \( a = 3 \) and \( b = 4 \).
- When we multiply them: \( a \times b = 3 \times 4 = 12 \).
- Since 12 is a natural number, the set of natural numbers is closed under multiplication.

5. **Division**:
- Finally, consider the division of two natural numbers, for example, \( a = 5 \) and \( b = 2 \).
- When we divide them: \( a \div b = 5 \div 2 = 2.5 \).
- Since 2.5 is not a natural number, the set of natural numbers is not closed under division.

### Conclusion:
- The set of natural numbers is closed under addition and multiplication, but not under subtraction and division. Therefore, the correct answer is that the set of natural numbers is closed under the binary operations of addition and multiplication.