1. **Problem Statement:** Find the union, intersection, difference of sets A, B, C and verify De Morgan's law using Venn diagrams. 2. **Given Sets:** $A = \{1,2,3,4,5\}$, $B = \{3,4,5,6,7\}$, $C = \{5,6,7,8\}$ 3. **Formulas and Rules:** - Union: $A \cup B = \{x | x \in A \text{ or } x \in B\}$ - Intersection: $A \cap B = \{x | x \in A \text{ and } x \in B\}$ - Difference: $A - B = \{x | x \in A \text{ and } x \notin B\}$ - Complement: $A' = U - A$ where $U$ is the universal set - De Morgan's Law: $(A \cup B)' = A' \cap B'$ 4. **Step-by-step Solutions:** **i) Find $A \cup B$:** Combine all unique elements from $A$ and $B$: $$A \cup B = \{1,2,3,4,5,6,7\}$$ **ii) Find $A \cap B$:** Find common elements in $A$ and $B$: $$A \cap B = \{3,4,5\}$$ **iii) Find $(A \cup B) \cap C$:** First find $A \cup B$ as above, then intersect with $C$: $$ (A \cup B) \cap C = \{5,6,7\} $$ **iv) Find $A - B$:** Elements in $A$ not in $B$: $$ A - B = \{1,2\} $$ 5. **Verify De Morgan's Law $(A \cup B)' = A' \cap B'$:** - Universal set $U = \{1,2,3,4,5,6,7,8\}$ - $A' = U - A = \{6,7,8\}$ - $B' = U - B = \{1,2,8\}$ - $A' \cap B' = \{8\}$ - $A \cup B = \{1,2,3,4,5,6,7\}$ - $(A \cup B)' = U - (A \cup B) = \{8\}$ Since $(A \cup B)' = A' \cap B' = \{8\}$, De Morgan's law is verified. 6. **Find $A' \cap (A \cup B)$:** - $A' = U - A = \{2,4,6,7,8\}$ - $A \cup B = \{1,2,3,4,5,6,7\}$ - Intersection: $$ A' \cap (A \cup B) = \{2,4,6,7\} $$ --- **Summary:** - $A \cup B = \{1,2,3,4,5,6,7\}$ - $A \cap B = \{3,4,5\}$ - $(A \cup B) \cap C = \{5,6,7\}$ - $A - B = \{1,2\}$ - De Morgan's law verified: $(A \cup B)' = A' \cap B' = \{8\}$ - $A' \cap (A \cup B) = \{2,4,6,7\}$