1. **State the problem:** Given sets: - $A = \{x \in \mathbb{Z} \mid 3 \leq x < 36, x \text{ divisible by } 3\}$ - $B = \{x \in \mathbb{Z} \mid 4 < x \leq 48, x \text{ divisible by } 4\}$ - $C = \{x \in \mathbb{Z} \mid 2 \leq x < 50, x \text{ divisible by } 5\}$ Find elements of $A$, $B$, and $C$ and perform: (i) $(A \cap C) \cup (A \cap B)$ (ii) $A - (B \cup C)$ (iii) $(B \cup C) \cap (A \cup C)$ (iv) $C - (B \cap A)$ 2. **List elements:** - $A$: multiples of 3 from 3 to 33 (since $x<36$): $\{3,6,9,12,15,18,21,24,27,30,33\}$ - $B$: multiples of 4 greater than 4 up to 48: $\{8,12,16,20,24,28,32,36,40,44,48\}$ - $C$: multiples of 5 from 2 to 49: $\{5,10,15,20,25,30,35,40,45\}$ 3. **Calculate intersections and unions:** - $A \cap C$: elements in both $A$ and $C$: - $A = \{3,6,9,12,15,18,21,24,27,30,33\}$ - $C = \{5,10,15,20,25,30,35,40,45\}$ - Common: $\{15,30\}$ - $A \cap B$: - $A = \{3,6,9,12,15,18,21,24,27,30,33\}$ - $B = \{8,12,16,20,24,28,32,36,40,44,48\}$ - Common: $\{12,24\}$ - $(A \cap C) \cup (A \cap B) = \{15,30\} \cup \{12,24\} = \{12,15,24,30\}$ 4. **Calculate $B \cup C$:** $B \cup C = \{5,8,10,12,15,16,20,24,25,28,30,32,35,36,40,44,45,48\}$ 5. **Calculate $A - (B \cup C)$:** Elements in $A$ not in $B \cup C$: $A = \{3,6,9,12,15,18,21,24,27,30,33\}$ Remove $\{5,8,10,12,15,16,20,24,25,28,30,32,35,36,40,44,45,48\}$ Remaining: $\{3,6,9,18,21,27,33\}$ 6. **Calculate $(B \cup C) \cap (A \cup C)$:** - $A \cup C = \{3,5,6,9,10,12,15,18,20,21,24,25,27,30,33,35,40,45\}$ - Intersection with $B \cup C$: $B \cup C = \{5,8,10,12,15,16,20,24,25,28,30,32,35,36,40,44,45,48\}$ Common elements: $\{5,10,12,15,20,24,25,30,35,40,45\}$ 7. **Calculate $B \cap A$:** $B \cap A = \{12,24\}$ 8. **Calculate $C - (B \cap A)$:** $C = \{5,10,15,20,25,30,35,40,45\}$ Remove $\{12,24\}$ (none in $C$), so $C - (B \cap A) = C = \{5,10,15,20,25,30,35,40,45\}$ **Final answers:** (i) $\{12,15,24,30\}$ (ii) $\{3,6,9,18,21,27,33\}$ (iii) $\{5,10,12,15,20,24,25,30,35,40,45\}$ (iv) $\{5,10,15,20,25,30,35,40,45\}$