1. **Problem Statement:** Given the universal set $\varepsilon = \{x : x \text{ is a positive integer and } 5 \leq x \leq 40\}$, and sets: - $P = \{x : x \text{ is a multiple of } 4\}$ - $Q = \{x : x \text{ is a perfect square}\}$ - $R = \{x : x \text{ is a cube of numbers}\}$ Find: i. $P \cup Q$ ii. $P \cap R$ iii. $P \cap Q$ iv. $P' \cap R$ v. $P' \cap Q$ b. Find $n(P \cup Q)$ 2. **Step 1: List elements of $\varepsilon$** $$\varepsilon = \{5,6,7,\ldots,40\}$$ 3. **Step 2: Find elements of $P$ (multiples of 4 between 5 and 40)** Multiples of 4 in $\varepsilon$ are: $$P = \{8,12,16,20,24,28,32,36,40\}$$ 4. **Step 3: Find elements of $Q$ (perfect squares between 5 and 40)** Perfect squares in $\varepsilon$ are: $$Q = \{9,16,25,36\}$$ 5. **Step 4: Find elements of $R$ (cubes between 5 and 40)** Cubes in $\varepsilon$ are: $$R = \{8,27\}$$ 6. **Step 5: Calculate $P \cup Q$ (union of $P$ and $Q$)** Combine all unique elements from $P$ and $Q$: $$P \cup Q = \{8,9,12,16,20,24,25,28,32,36,40\}$$ 7. **Step 6: Calculate $P \cap R$ (intersection of $P$ and $R$)** Elements common to both $P$ and $R$: $$P \cap R = \{8\}$$ 8. **Step 7: Calculate $P \cap Q$ (intersection of $P$ and $Q$)** Elements common to both $P$ and $Q$: $$P \cap Q = \{16,36\}$$ 9. **Step 8: Calculate $P'$ (complement of $P$ in $\varepsilon$)** Elements in $\varepsilon$ not in $P$: $$P' = \varepsilon \setminus P = \{5,6,7,9,10,11,13,14,15,17,18,19,21,22,23,25,26,27,29,30,31,33,34,35,37,38,39\}$$ 10. **Step 9: Calculate $P' \cap R$ (elements in $P'$ and $R$)** Elements common to $P'$ and $R$: $$P' \cap R = \{27\}$$ 11. **Step 10: Calculate $P' \cap Q$ (elements in $P'$ and $Q$)** Elements common to $P'$ and $Q$: $$P' \cap Q = \{9,25\}$$ 12. **Step 11: Find $n(P \cup Q)$ (number of elements in $P \cup Q$)** Count elements in $P \cup Q$: $$n(P \cup Q) = 11$$ **Final answers:** i. $P \cup Q = \{8,9,12,16,20,24,25,28,32,36,40\}$ ii. $P \cap R = \{8\}$ iii. $P \cap Q = \{16,36\}$ iv. $P' \cap R = \{27\}$ v. $P' \cap Q = \{9,25\}$ b. $n(P \cup Q) = 11$