Question

consider \\(u = \\{x | x \text{ is a positive integer greater than } 1\\}\\).

which is an empty set?

\\(\bigcirc\\) \\(\\{x | x \in u \text{ and } \frac{1}{2}x \text{ is prime}\\}\\)
\\(\bigcirc\\) \\(\\{x | x \in u \text{ and } 2x \text{ is prime}\\}\\)
\\(\bigcirc\\) \\(\\{x | x \in u \text{ and } \frac{1}{2}x \text{ can be written as a fraction}\\}\\)
\\(\bigcirc\\) \\(\\{x | x \in u \text{ and } 2x \text{ can be written as a fraction}\\}\\)

Explanation:

Define the universal set

$$ U = \{2, 3, 4, 5, \dots\} $$

Analyze each option

• For \(\{x \mid x \in U \text{ and } \frac{1}{2}x \text{ is prime}\}\):

If \(x = 4 \in U\), then \(\frac{1}{2}(4) = 2\), which is prime. This set is not empty.

• For \(\{x \mid x \in U \text{ and } 2x \text{ is prime}\}\):

Since \(x \ge 2\), \(2x \ge 4\). Any number \(2x\) has at least factors \(1\), \(2\), and \(2x\), so \(2x\) is composite. Thus, no \(x \in U\) makes \(2x\) prime. This set is empty.

• For \(\{x \mid x \in U \text{ and } \frac{1}{2}x \text{ can be written as a fraction}\}\):

If \(x = 2 \in U\), then \(\frac{1}{2}(2) = 1 = \frac{1}{1}\). This set is not empty.

• For \(\{x \mid x \in U \text{ and } 2x \text{ can be written as a fraction}\}\):

If \(x = 2 \in U\), then \(2(2) = 4 = \frac{4}{1}\). This set is not empty.

Identify the empty set

$$ \{x \mid x \in U \text{ and } 2x \text{ is prime}\} $$

Answer:

• (A) \(\{x|x \in \text{U and } \frac{1}{2}x \text{ is prime}\}\)
• (B) \(\{x|x \in \text{U and } 2x \text{ is prime}\}\) (Correct answer)
• (C) \(\{x|x \in \text{U and } \frac{1}{2}x \text{ can be written as a fraction}\}\)
• (D) \(\{x|x \in \text{U and } 2x \text{ can be written as a fraction}\}\)