1. **Problem statement:** Given sets $U=\{a,b,c,d,e\}$, $A=\{a,b,c\}$, and $B=\{b,c,d,e\}$, verify: (i) $ (A \cup B)' = A' \cap B' $ (ii) $ (A \cap B)' = A' \cup B' $ 2. **Formula and rules:** - The complement of a union is the intersection of complements: $ (A \cup B)' = A' \cap B' $ - The complement of an intersection is the union of complements: $ (A \cap B)' = A' \cup B' $ 3. **Step-by-step verification:** (i) Calculate $A \cup B$: $$ A \cup B = \{a,b,c\} \cup \{b,c,d,e\} = \{a,b,c,d,e\} $$ Calculate complement $(A \cup B)'$ relative to $U$: $$ (A \cup B)' = U - (A \cup B) = \{a,b,c,d,e\} - \{a,b,c,d,e\} = \emptyset $$ Calculate complements $A'$ and $B'$: $$ A' = U - A = \{d,e\} $$ $$ B' = U - B = \{a\} $$ Calculate $A' \cap B'$: $$ A' \cap B' = \{d,e\} \cap \{a\} = \emptyset $$ Since $(A \cup B)' = \emptyset$ and $A' \cap B' = \emptyset$, they are equal. (ii) Calculate $A \cap B$: $$ A \cap B = \{a,b,c\} \cap \{b,c,d,e\} = \{b,c\} $$ Calculate complement $(A \cap B)'$ relative to $U$: $$ (A \cap B)' = U - (A \cap B) = \{a,b,c,d,e\} - \{b,c\} = \{a,d,e\} $$ Calculate $A' \cup B'$: $$ A' \cup B' = \{d,e\} \cup \{a\} = \{a,d,e\} $$ Since $(A \cap B)' = \{a,d,e\}$ and $A' \cup B' = \{a,d,e\}$, they are equal. **Final answer:** Both equalities are verified.