Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Let's analyze each of the given sets step-by-step to determine which one is an empty set.
1. Set 1: [tex]\(\{ x \mid x \in U \text{ and } \frac{1}{2}x \text{ is prime} \}\)[/tex]
We are considering elements [tex]\( x \)[/tex] from the universal set [tex]\( U \)[/tex] such that [tex]\( \frac{1}{2} x \)[/tex] is a prime number. For [tex]\( \frac{1}{2} x \)[/tex] to be a prime number, [tex]\( x \)[/tex] must be an even number (since the half of an odd number isn't an integer). Hence, if [tex]\( x \)[/tex] is even, say [tex]\( x = 2k \)[/tex], then:
[tex]\[ \frac{1}{2} \cdot 2k = k \][/tex]
This means [tex]\( k \)[/tex] must be a prime number. We need to check the positive integers greater than 1 which after dividing by 2 give a prime number. We can find that the set is not empty as there exist such [tex]\( x \)[/tex] values that result in primes.
2. Set 2: [tex]\(\{ x \mid x \in U \text{ and } 2x \text{ is prime} \}\)[/tex]
Here, we look for elements [tex]\( x \)[/tex] from [tex]\( U \)[/tex] such that [tex]\( 2x \)[/tex] is a prime number. Since [tex]\( 2x \)[/tex] represents an even number greater than 2, and the only even prime number is 2 itself, there is a contradiction because for [tex]\( x \)[/tex] to be greater than 1, [tex]\( 2x \)[/tex] would be at least 4, which is not prime. Thus, this set is empty.
3. Set 3: [tex]\(\{ x \mid x \in U \text{ and } \frac{1}{2} x \text{ can be written as a fraction} \}\)[/tex]
For any positive integer [tex]\( x \)[/tex] in [tex]\( U \)[/tex], [tex]\( \frac{1}{2}x \)[/tex] is always a fraction. For example, if [tex]\( x = 4 \)[/tex], then [tex]\( \frac{1}{2} \cdot 4 = 2 \)[/tex], which is expressible as a fraction [tex]\( \frac{2}{1} \)[/tex]. Therefore, this set is not empty since every [tex]\( x \)[/tex] in [tex]\( U \)[/tex] satisfies this condition.
4. Set 4: [tex]\(\{ x \mid x \in U \text{ and } 2x \text{ can be written as a fraction} \}\)[/tex]
Any positive integer [tex]\( x \)[/tex] in [tex]\( U \)[/tex] when multiplied by 2 will result in an integer, which is inherently a fraction (e.g., 8 can be written as [tex]\( \frac{8}{1} \)[/tex]). Hence, this set is also not empty as all [tex]\( x \)[/tex] in [tex]\( U \)[/tex] will meet this condition.
Therefore, the only set that is empty is:
[tex]\[ \{ x \mid x \in U \text{ and } 2x \text{ is prime} \} \][/tex]
1. Set 1: [tex]\(\{ x \mid x \in U \text{ and } \frac{1}{2}x \text{ is prime} \}\)[/tex]
We are considering elements [tex]\( x \)[/tex] from the universal set [tex]\( U \)[/tex] such that [tex]\( \frac{1}{2} x \)[/tex] is a prime number. For [tex]\( \frac{1}{2} x \)[/tex] to be a prime number, [tex]\( x \)[/tex] must be an even number (since the half of an odd number isn't an integer). Hence, if [tex]\( x \)[/tex] is even, say [tex]\( x = 2k \)[/tex], then:
[tex]\[ \frac{1}{2} \cdot 2k = k \][/tex]
This means [tex]\( k \)[/tex] must be a prime number. We need to check the positive integers greater than 1 which after dividing by 2 give a prime number. We can find that the set is not empty as there exist such [tex]\( x \)[/tex] values that result in primes.
2. Set 2: [tex]\(\{ x \mid x \in U \text{ and } 2x \text{ is prime} \}\)[/tex]
Here, we look for elements [tex]\( x \)[/tex] from [tex]\( U \)[/tex] such that [tex]\( 2x \)[/tex] is a prime number. Since [tex]\( 2x \)[/tex] represents an even number greater than 2, and the only even prime number is 2 itself, there is a contradiction because for [tex]\( x \)[/tex] to be greater than 1, [tex]\( 2x \)[/tex] would be at least 4, which is not prime. Thus, this set is empty.
3. Set 3: [tex]\(\{ x \mid x \in U \text{ and } \frac{1}{2} x \text{ can be written as a fraction} \}\)[/tex]
For any positive integer [tex]\( x \)[/tex] in [tex]\( U \)[/tex], [tex]\( \frac{1}{2}x \)[/tex] is always a fraction. For example, if [tex]\( x = 4 \)[/tex], then [tex]\( \frac{1}{2} \cdot 4 = 2 \)[/tex], which is expressible as a fraction [tex]\( \frac{2}{1} \)[/tex]. Therefore, this set is not empty since every [tex]\( x \)[/tex] in [tex]\( U \)[/tex] satisfies this condition.
4. Set 4: [tex]\(\{ x \mid x \in U \text{ and } 2x \text{ can be written as a fraction} \}\)[/tex]
Any positive integer [tex]\( x \)[/tex] in [tex]\( U \)[/tex] when multiplied by 2 will result in an integer, which is inherently a fraction (e.g., 8 can be written as [tex]\( \frac{8}{1} \)[/tex]). Hence, this set is also not empty as all [tex]\( x \)[/tex] in [tex]\( U \)[/tex] will meet this condition.
Therefore, the only set that is empty is:
[tex]\[ \{ x \mid x \in U \text{ and } 2x \text{ is prime} \} \][/tex]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
