Question

Given that x= {whole numbers less then 24}
P={prime numbers less than 24}
Q={Even numbers less than 24}
Find:A. P intersection Q
B. P union Q
C. P' intersection Q
D. (P union Q)'

Answer 1

Let's start by defining the sets given in the problem:

1. **Set \( x \)**: Whole numbers less than 24
\[
x = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\}
\]

2. **Set \( P \)**: Prime numbers less than 24
\[
P = \{2, 3, 5, 7, 11, 13, 17, 19, 23\}
\]

3. **Set \( Q \)**: Even numbers less than 24
\[
Q = \{0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22\}
\]

Now, let's find the sets as per the requests.

### A. \( P \cap Q \)
The intersection of sets \( P \) and \( Q \) includes elements that are both prime and even.
Since the only even prime number is 2:
\[
P \cap Q = \{2\}
\]

### B. \( P \cup Q \)
The union of sets \( P \) and \( Q \) includes all elements that are in either set.
\[
P \cup Q = \{0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23\}
\]

### C. \( P' \cap Q \)
The complement \( P' \) consists of all elements in \( x \) that are not in \( P \).
\[
P' = x - P = \{0, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22\}
\]
Now, we find the intersection of \( P' \) and \( Q \):
\[
P' \cap Q = \{0, 4, 6, 10, 12, 14, 16, 18, 20, 22\}
\]

### D. \( (P \cup Q)' \)
The complement of \( (P \cup Q) \) consists of all elements in \( x \) that are not in \( P \cup Q \).
\[
P \cup Q = \{0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23\}
\]
The complement of \( (P \cup Q) \) is:
\[
(P \cup Q)' = x - (P \cup Q) = \{1, 8, 9, 15, 21\}
\]

### Final Answers:
- A. \( P \cap Q = \{2\} \)
- B. \( P \cup Q = \{0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23\} \)
- C. \( P' \cap Q = \{0, 4, 6, 10, 12, 14, 16, 18, 20, 22\} \)
- D. \( (P \cup Q)' = \{1, 8, 9, 15, 21\} \)