Let's start by defining the sets given in the problem:
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Set \( x \): Whole numbers less than 24 \[ x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} \]
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Set \( P \): Prime numbers less than 24 \[ P = {2, 3, 5, 7, 11, 13, 17, 19, 23} \]
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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. \( (P \cup Q)' = {1, 8, 9, 15, 21} \)
1 answer