Which of the following options is incorrect?

Options

A. Addition and multiplication are commutative for rational numbers.
B. Subtraction and division are not associative for rational numbers.
C. Addition is associative for rational numbers.
D. Rational numbers are closed under division.

Answer

To determine which of the given options is incorrect, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Check the First Statement:**
- **Statement:** Addition and multiplication are commutative for rational numbers.
- **Explanation:** The commutative property states that the order of the numbers does not affect the sum or product. For example, if we take two rational numbers \( \frac{2}{3} \) and \( \frac{1}{3} \):
- Addition: \( \frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1 \) and \( \frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1 \).
- Multiplication: \( \frac{2}{3} \times \frac{1}{3} = \frac{2}{9} \) and \( \frac{1}{3} \times \frac{2}{3} = \frac{2}{9} \).
- **Conclusion:** This statement is **correct**.

2. **Check the Second Statement:**
- **Statement:** Subtraction and division are not associative for rational numbers.
- **Explanation:** The associative property states that the way in which numbers are grouped does not affect the result.
- For subtraction: \( x - (y - z) \neq (x - y) - z \). For example, let \( x = \frac{2}{3}, y = \frac{1}{3}, z = \frac{5}{3} \):
- Left side: \( \frac{2}{3} - \left(\frac{1}{3} - \frac{5}{3}\right) = \frac{2}{3} - (-\frac{4}{3}) = \frac{2}{3} + \frac{4}{3} = 2 \).
- Right side: \( \left(\frac{2}{3} - \frac{1}{3}\right) - \frac{5}{3} = \frac{1}{3} - \frac{5}{3} = -\frac{4}{3} \).
- For division: \( x \div (y \div z) \neq (x \div y) \div z \).
- **Conclusion:** This statement is **correct**.

3. **Check the Third Statement:**
- **Statement:** Addition is associative for rational numbers.
- **Explanation:** The associative property for addition states that \( (x + y) + z = x + (y + z) \). Using the same rational numbers:
- Left side: \( \left(\frac{2}{3} + \frac{1}{3}\right) + \frac{5}{3} = 1 + \frac{5}{3} = \frac{8}{3} \).
- Right side: \( \frac{2}{3} + \left(\frac{1}{3} + \frac{5}{3}\right) = \frac{2}{3} + 2 = \frac{8}{3} \).
- **Conclusion:** This statement is **correct**.

4. **Check the Fourth Statement:**
- **Statement:** Rational numbers are closed under division.
- **Explanation:** Closure under an operation means that performing the operation on two elements of the set results in an element that is also in the set. For rational numbers, if we take \( \frac{2}{3} \) and \( \frac{1}{3} \):
- \( \frac{2}{3} \div \frac{1}{3} = \frac{2}{3} \times \frac{3}{1} = 2 \), which is a rational number.
- However, if we take \( \frac{2}{3} \) and \( 0 \), \( \frac{2}{3} \div 0 \) is undefined.
- **Conclusion:** This statement is **incorrect**.

### Final Answer:
The incorrect option is **Option 4: Rational numbers are closed under division**.