Which of the following is a null set?

Options

A. `{x:x in N, 2x-1=3}`
B. `{x:xin N, x^(2)lt20}`
C. `{x:x "is even prime greater than "2}`
D. `x:x in I, 3x+7=1}`

Answer

To determine which of the given options is a null set, we will analyze each option step by step.

### Step 1: Understand the Definition of a Null Set
A null set (or empty set) is a set that contains no elements.

### Step 2: Analyze Each Option

#### Option 1: \( x \) where \( x \) belongs to any natural number and \( 2x - 1 = 3 \)
1. Solve the equation \( 2x - 1 = 3 \).
- Add 1 to both sides: \( 2x = 4 \)
- Divide by 2: \( x = 2 \)
2. Since \( 2 \) is a natural number, the set is \( \{2\} \).
3. Conclusion: This is **not a null set**.

#### Option 2: \( x \) where \( x \) is any natural number and \( x^2 < 20 \)
1. Find the natural numbers \( x \) such that \( x^2 < 20 \).
- \( x = 1 \): \( 1^2 = 1 < 20 \) (True)
- \( x = 2 \): \( 2^2 = 4 < 20 \) (True)
- \( x = 3 \): \( 3^2 = 9 < 20 \) (True)
- \( x = 4 \): \( 4^2 = 16 < 20 \) (True)
- \( x = 5 \): \( 5^2 = 25 \not< 20 \) (False)
2. The valid values are \( 1, 2, 3, 4 \). Thus, the set is \( \{1, 2, 3, 4\} \).
3. Conclusion: This is **not a null set**.

#### Option 3: \( x \) where \( x \) is an even prime number greater than 2
1. The only even prime number is \( 2 \).
2. There are no even prime numbers greater than \( 2 \).
3. Therefore, the set is empty: \( \{\} \).
4. Conclusion: This **is a null set**.

#### Option 4: \( x \) where \( x \) belongs to any integer and \( 3x + 7 = 1 \)
1. Solve the equation \( 3x + 7 = 1 \).
- Subtract 7 from both sides: \( 3x = -6 \)
- Divide by 3: \( x = -2 \)
2. The set is \( \{-2\} \).
3. Conclusion: This is **not a null set**.

### Final Conclusion
The only option that is a null set is **Option 3: \( x \) where \( x \) is an even prime number greater than 2**.

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