State True or False
Rational numbers are closed under addition and multiplication but not under subtraction.

Answer

To determine whether the statement "Rational numbers are closed under addition and multiplication but not under subtraction" is true or false, we will analyze the closure properties of rational numbers under addition, subtraction, and multiplication.

### Step 1: Understand Rational Numbers
Rational numbers are numbers that can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

**Hint:** Rational numbers include fractions, integers, and terminating or repeating decimals.

### Step 2: Check Closure Under Addition
Let's take two rational numbers, for example, \( \frac{1}{2} \) and \( \frac{3}{2} \).

- Adding these two rational numbers:
\[ \frac{1}{2} + \frac{3}{2} = \frac{1 + 3}{2} = \frac{4}{2} = 2 \]
The result, \( 2 \), is a rational number.

**Hint:** When you add two rational numbers, the result should also be a rational number.

### Step 3: Check Closure Under Subtraction
Now, let's subtract the same two rational numbers.

- Subtracting these two rational numbers:
\[ \frac{1}{2} - \frac{3}{2} = \frac{1 - 3}{2} = \frac{-2}{2} = -1 \]
The result, \( -1 \), is also a rational number.

**Hint:** When you subtract one rational number from another, the result should also be a rational number.

### Step 4: Check Closure Under Multiplication
Next, we will multiply the two rational numbers.

- Multiplying these two rational numbers:
\[ \frac{1}{2} \times \frac{3}{2} = \frac{1 \times 3}{2 \times 2} = \frac{3}{4} \]
The result, \( \frac{3}{4} \), is a rational number.

**Hint:** When you multiply two rational numbers, the result should also be a rational number.

### Conclusion
Since we have shown that rational numbers are closed under addition, subtraction, and multiplication, the original statement "Rational numbers are closed under addition and multiplication but not under subtraction" is **False**.

**Final Answer:** False