Question

Let \( U = \{1, 2, \dots, 15\} \). Let \( P \subseteq U \) consist of all prime numbers, \( Q \subseteq U \) consist of all even numbers and \( R \subseteq U \) consist of all multiples of 3. Let \( T = P - Q \). Then, which of the following is/are CORRECT?

• A. \( |T| = 5 \) and \( |T \cup R| = 9 \)
• B. \( |T| = 6 \) and \( |T \cup R| = 9 \)
• C. \( |T| = 5 \) and \( |T \cap R| = 1 \)
• D. \( |T| = 6 \) and \( |T \cap R| = 1 \)

Hint:

When performing set operations like \( T = P - Q \), carefully identify the elements in each set and apply the appropriate operations (union, intersection, or difference).

Answer 1

The Correct Option is A, C

Step 1: Identifying the sets.
- \( P = \{2, 3, 5, 7, 11, 13\} \) (prime numbers) - \( Q = \{2, 4, 6, 8, 10, 12, 14\} \) (even numbers) - \( R = \{3, 6, 9, 12, 15\} \) (multiples of 3)
\( T = P - Q \), so \( T = \{3, 5, 7, 11, 13\} \), which is the set of prime numbers excluding even numbers. This gives \( |T| = 5 \).
Step 2: Calculating \( |T \cup R| \).
\( T \cup R = \{3, 5, 7, 11, 13, 6, 9, 12, 15\} \), so \( |T \cup R| = 9 \).
Step 3: Calculating \( |T \cap R| \).
\( T \cap R = \{3\} \), so \( |T \cap R| = 1 \).
Step 4: Conclusion.
The correct answer is (C) \( |T| = 5 \) and \( |T \cap R| = 1 \).