1. Problem: Find the elements and cardinality of $A \cap B$. Step 1: Identify $A = \{a,b,c,e,f\}$ and $B = \{b,c,d,e,f\}$. Step 2: Intersection $A \cap B$ contains elements in both $A$ and $B$. $$A \cap B = \{b,c,e,f\}$$ Step 3: Cardinality $|A \cap B| = 4$. 2. Problem: Find $A' \cup D$ where $A'$ is the complement of $A$ in $U$. Step 1: Universal set $U = \{a,b,c,d,e,f,g,h,i,j,k\}$. Step 2: $A' = U - A = \{d,g,h,i,j,k\}$. Step 3: $D = \{a,e,g,i\}$. Step 4: Union $A' \cup D = \{a,d,e,g,h,i,j,k\}$. Step 5: Cardinality $|A' \cup D| = 8$. 3. Problem: Find $(C \cup B)'$. Step 1: $B = \{b,c,d,e,f\}$, $C = \{f,g,h,k\}$. Step 2: $C \cup B = \{b,c,d,e,f,g,h,k\}$. Step 3: Complement $(C \cup B)' = U - (C \cup B) = \{a,i,j\}$. Step 4: Cardinality $|(C \cup B)'| = 3$. 4. Problem: Find $C \oplus D$ (symmetric difference). Step 1: $C = \{f,g,h,k\}$, $D = \{a,e,g,i\}$. Step 2: Symmetric difference is elements in either set but not both. $$C \oplus D = (C - D) \cup (D - C) = \{f,h,k\} \cup \{a,e,i\} = \{a,e,f,h,i,k\}$$ Step 3: Cardinality $|C \oplus D| = 6$. 5. Problem: Find $A \sim B$ (set difference $A - B$). Step 1: $A = \{a,b,c,e,f\}$, $B = \{b,c,d,e,f\}$. Step 2: $A \sim B = \{a\}$ (elements in $A$ not in $B$). Step 3: Cardinality $|A \sim B| = 1$. 6. Problem: Find $A \cap C'$. Step 1: $C = \{f,g,h,k\}$, so $C' = U - C = \{a,b,c,d,e,i,j\}$. Step 2: $A = \{a,b,c,e,f\}$. Step 3: Intersection $A \cap C' = \{a,b,c,e\}$. Step 4: Cardinality $|A \cap C'| = 4$. 7. Problem: Find $A \sim U$. Step 1: $U$ is the universal set, so $A \sim U = A - U = \emptyset$. Step 2: Cardinality $|A \sim U| = 0$. 8. Problem: Find $(A \oplus B)'$. Step 1: $A = \{a,b,c,e,f\}$, $B = \{b,c,d,e,f\}$. Step 2: $A \oplus B = (A - B) \cup (B - A) = \{a\} \cup \{d\} = \{a,d\}$. Step 3: Complement $(A \oplus B)' = U - \{a,d\} = \{b,c,e,f,g,h,i,j,k\}$. Step 4: Cardinality $|(A \oplus B)'| = 9$. 9. Problem: Find $(A \cap C) \sim B$. Step 1: $A = \{a,b,c,e,f\}$, $C = \{f,g,h,k\}$, $B = \{b,c,d,e,f\}$. Step 2: $A \cap C = \{f\}$. Step 3: $(A \cap C) \sim B = \{f\} - \{b,c,d,e,f\} = \emptyset$. Step 4: Cardinality $| (A \cap C) \sim B | = 0$. 10. Problem: Find $(A \sim B) \cap C$. Step 1: $A \sim B = \{a\}$. Step 2: $C = \{f,g,h,k\}$. Step 3: Intersection $\{a\} \cap \{f,g,h,k\} = \emptyset$. Step 4: Cardinality $| (A \sim B) \cap C | = 0$. Final answers: 1. $\{b,c,e,f\}, 4$ 2. $\{a,d,e,g,h,i,j,k\}, 8$ 3. $\{a,i,j\}, 3$ 4. $\{a,e,f,h,i,k\}, 6$ 5. $\{a\}, 1$ 6. $\{a,b,c,e\}, 4$ 7. $\emptyset, 0$ 8. $\{b,c,e,f,g,h,i,j,k\}, 9$ 9. $\emptyset, 0$ 10. $\emptyset, 0$