- The set \(U\) consists of all integers \(x > 1\).
- Analyze each set to see if any \(x \in U\) satisfies the given condition.
- The set \(\{x \mid x \in U \text{ and } 2x \text{ is prime}\}\) is the empty set because \(2x\) is always a composite number for any \(x \ge 2\).
- The empty set is \(\boxed{\{x \mid x \in U \text{ and } 2x \text{ is prime}\}}\).
Explanation
- Definition of Universal Set and Empty Set
The problem requires identifying which of the four given sets is an empty set. The universal set \(U\) is defined as the set of all integers greater than 1, so \(U = \{2, 3, 4, \dots\}\). An empty set is a set that contains no elements, meaning no value of \(x \in U\) can satisfy the condition given in the set-builder notation.
- Evaluating the First Set
The first set is \(S_1 = \{x \mid x \in U \text{ and } \frac{1}{2}x \text{ is prime}\}\). We check if there exists an \(x \in U\) such that \(\frac{x}{2}\) is a prime number. If we let \(x=4\), which is in \(U\), then \(\frac{x}{2} = \frac{4}{2} = 2\). Since \(2\) is a prime number, \(x=4\) satisfies the condition. Therefore, \(S_1\) is not empty (it contains at least 4).
- Evaluating the Second Set
The second set is \(S_2 = \{x \mid x \in U \text{ and } 2x \text{ is prime}\}\). We check if there exists an \(x \in U\) such that \(2x\) is a prime number. Since \(x \ge 2\), the smallest value for \(2x\) is \(2(2) = 4\). Any value \(2x\) for \(x \in U\) is an even number greater than or equal to 4. All even numbers greater than or equal to 4 are composite because they are divisible by 1, 2, and themselves (and potentially other numbers). Therefore, no value of \(x \in U\) can make \(2x\) prime. Consequently, \(S_2\) is an empty set.
- Evaluating the Third and Fourth Sets
The third set is \(S_3 = \{x \mid x \in U \text{ and } \frac{1}{2}x \text{ can be written as a fraction}\}\). Every rational number can be written as a fraction \(\frac{p}{q}\) where \(p, q\) are integers and \(q
eq 0\). For any \(x \in U\), \(\frac{x}{2}\) is a rational number. For example, if \(x=3\), \(\frac{3}{2}\) is a fraction. Since every \(x \in U\) satisfies this, the set is not empty. Similarly, the fourth set \(S_4 = \{x \mid x \in U \text{ and } 2x \text{ can be written as a fraction}\}\) contains all \(x \in U\), as any integer \(2x\) can be written as the fraction \(\frac{2x}{1}\). Therefore, these sets are not empty.
- Conclusion
Comparing the findings, we see that \(S_1\), \(S_3\), and \(S_4\) are non-empty, whereas \(S_2\) contains no elements from \(U\) that satisfy its condition. Thus, the empty set is \(\{x \mid x \in U \text{ and } 2x \text{ is prime}\}\). The final answer is \(\boxed{\{x \mid x \in U \text{ and } 2x \text{ is prime}\}}\).
Examples
Understanding empty sets is crucial in fields like database management or programming, where you might query data based on certain conditions. For instance, if you are looking for all products in a inventory database that satisfy a very restrictive condition (like being "both perishable and non-perishable"), the result set might be an empty set, telling you that no such items exist. This helps in validating data integrity and ensuring that your logical filters are working correctly. In mathematics, defining these empty conditions helps us classify numbers and identify impossible scenarios.