rioritized while awarding marks. If \( \mathrm{U}=\{1,2,3, \ldots \ldots ., 16\}, \mathrm{P}=\{\mathrm{x}: \mathrm{x} \) is a composite number, \( \mathrm{x} \leq 13\} \), \( \mathrm{Q}=\{2,3,6,9,12,15\} \) and \( \mathrm{R}=\{2,4,6,12,14,16\} \) then answer the following questions: a. Write the set P in listing method. b. Find \( \mathrm{P} \cap \mathrm{Q} R \mathrm{R} \). c. Show the given information in a Venn-Diagram. d. By using constructed Venn-diagram, verify the relation: \( \mathrm{n}(\mathrm{P})=\mathrm{n}_{0}(\mathrm{P})+\mathrm{n}_{0}(\mathrm{P} \cap \mathrm{Q})+\mathrm{n}_{0}(\mathrm{P} \cap \mathrm{R})+\mathrm{n}(\mathrm{P} \cap \mathrm{Q} \cap \mathrm{R}) \)

Let's go through each part of the problem step by step. ### a. Write the set \( P \) in listing method. The set \( U = \{1, 2, 3, \ldots, 16\} \) contains the numbers from 1 to 16. The set \( P \) is defined as the set of composite numbers \( x \) such that \( x \leq 13 \). The composite numbers less than or equal to 13 are: - 4 (2 × 2) - 6 (2 × 3) - 8 (2 × 4) - 9 (3 × 3) - 10 (2 × 5) - 12 (3 × 4) - 14 (2 × 7) (not included since it is greater than 13) - 15 (3 × 5) (not included since it is greater than 13) Thus, the set \( P \) in listing method is: \[ P = \{4, 6, 8, 9, 10, 12\} \] ### b. Find \( P \cap Q \cap R \). We have: - \( P = \{4, 6, 8, 9, 10, 12\} \) - \( Q = \{2, 3, 6, 9, 12, 15\} \) - \( R = \{2, 4, 6, 12, 14, 16\} \) Now, we will find the intersections: 1. **Find \( P \cap Q \)**: \[ P \cap Q = \{4, 6, 8, 9, 10, 12\} \cap \{2, 3, 6, 9, 12, 15\} = \{6, 9, 12\} \] 2. **Find \( P \cap R \)**: \[ P \cap R = \{4, 6, 8, 9, 10, 12\} \cap \{2, 4, 6, 12, 14, 16\} = \{4, 6, 12\} \] 3. **Find \( P \cap Q \cap R \)**: \[ P \cap Q \cap R = \{6, 9, 12\} \cap \{4, 6, 12\} = \{6, 12\} \] Thus, the intersection \( P \cap Q \cap R \) is: \[ P \cap Q \cap R = \{6, 12\} \] ### c. Show the given information in a Venn-Diagram. To create a Venn diagram, we will represent the three sets \( P \), \( Q \), and \( R \) as overlapping circles. 1. **Circle for \( P \)** will contain: \( 4, 6, 8, 9, 10, 12 \) 2. **Circle for \( Q \)** will contain: \( 2, 3, 6, 9, 12, 15 \) 3. **Circle for \( R \)** will contain: \( 2, 4, 6, 12, 14, 16 \) **Overlapping areas**: - \( P \cap Q \): \( 6, 9, 12 \) - \( P \cap R \): \( 4, 6, 12 \) - \( Q \cap R \): \( 2 \) (not in \( P \)) - \( P \cap Q \cap R \): \( 6, 12 \) ### d. Verify the relation: \[ n(P) = n_0(P) + n_0(P \cap Q) + n_0(P \cap R) + n(P \cap Q \cap R) \] Where: - \( n(P) \) is the number of elements in \( P \). - \( n_0(P) \) is the number of elements in \( P \) only. - \( n_0(P \cap Q) \) is the number of elements in \( P \cap Q \) only. - \( n_0(P \cap R) \) is the number of elements in \( P \cap R \) only. - \( n(P \cap Q \cap R) \) is the number of elements in \( P \cap Q \cap R \). Calculating the sizes: - \( n(P) =