Let’s tackle the problem step-by-step by defining the sets [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( R \)[/tex] and identifying their elements.
1. Define Set [tex]\( P \)[/tex]:
[tex]\( P = \{x \in \mathbb{N} \mid 50 < x < 70\} \)[/tex].
This means [tex]\( P \)[/tex] contains all natural numbers greater than 50 and less than 70. Hence,
[tex]\[
P = \{51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69\}
\][/tex]
2. Define Set [tex]\( Q \)[/tex]:
[tex]\( Q = \{x \in P \mid x \text{ is odd}\} \)[/tex].
This means [tex]\( Q \)[/tex] contains all the odd numbers in set [tex]\( P \)[/tex]. Checking the aforementioned elements in [tex]\( P \)[/tex]:
[tex]\[
Q = \{51, 53, 55, 57, 59, 61, 63, 65, 67, 69\}
\][/tex]
3. Define Set [tex]\( R \)[/tex]:
[tex]\( R = \{x \in P \mid x \text{ is prime}\} \)[/tex].
This means [tex]\( R \)[/tex] contains all the prime numbers in set [tex]\( P \)[/tex]. We need to check each element in [tex]\( P \)[/tex] to see if it is prime:
- 51 is not a prime (divisible by 3 and 17).
- 52 is not a prime (divisible by 2).
- 53 is a prime.
- 54 is not a prime (divisible by 2 and 3).
- 55 is not a prime (divisible by 5 and 11).
- 56 is not a prime (divisible by 2 and 7).
- 57 is not a prime (divisible by 3 and 19).
- 58 is not a prime (divisible by 2).
- 59 is a prime.
- 60 is not a prime (divisible by 2, 3, and 5).
- 61 is a prime.
- 62 is not a prime (divisible by 2).
- 63 is not a prime (divisible by 3 and 7).
- 64 is not a prime (divisible by 2).
- 65 is not a prime (divisible by 5 and 13).
- 66 is not a prime (divisible by 2 and 3).
- 67 is a prime.
- 68 is not a prime (divisible by 2).
- 69 is not a prime (divisible by 3).
Therefore,
[tex]\[
R = \{53, 59, 61, 67\}
\][/tex]
Summarizing the results, we have:
- Elements of [tex]\( P \)[/tex]:
[tex]\[
P = \{51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69\}
\][/tex]
- Elements of [tex]\( Q \)[/tex]:
[tex]\[
Q = \{51, 53, 55, 57, 59, 61, 63, 65, 67, 69\}
\][/tex]
- Elements of [tex]\( R \)[/tex]:
[tex]\[
R = \{53, 59, 61, 67\}
\][/tex]
So, the elements of sets [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( R \)[/tex] are given as above.
