1. The problem involves finding the symmetric difference $A \Delta B$ for given sets $A$ and $B$. The symmetric difference is defined as the set of elements in either $A$ or $B$ but not in their intersection. 2. The formula for symmetric difference is: $$A \Delta B = (A \cup B) \setminus (A \cap B)$$ This means we take the union of $A$ and $B$ and remove the elements that are in both. 3. For example, if $A = \{1,2,5\}$ and $B = \{2,3,5\}$: - $A \cup B = \{1,2,3,5\}$ - $A \cap B = \{2,5\}$ - So, $A \Delta B = \{1,2,3,5\} \setminus \{2,5\} = \{1,3\}$ 4. This matches option d. $\{1,3\}$. 5. Similarly, for other questions: - Number of subsets of $\{x, \{x,y\}, x, y\}$: Since $x$ repeats, the set is $\{x, \{x,y\}, y\}$ with 3 distinct elements, so $2^3=8$ subsets. - Intersection $A \cap B$ for $A=\{2,4,6\}$ and $B=\{1,2,3,4\}$ is $\{2,4\}$. - $A \cup (B \cap C)$ for $A=\{1,2\}$, $B=\{2,3\}$, $C=\{3,4\}$: - $B \cap C = \{3\}$ - $A \cup \{3\} = \{1,2,3\}$ - Set of prime numbers less than 10 is $\{2,3,5,7\}$. - If $A$ and $B$ are disjoint, $A \Delta B = A \cup B$. - For $A=\{1,2,3\}$ and $B=\{3,4,5\}$, $A \Delta B = \{1,2,4,5\}$. - Symmetric difference is commutative: $A \Delta B = B \Delta A$. - Subsets of $\{a,b,c\}$ include $\{a,b\}$. - Complement of $A$ in universal set $U$ is $U \setminus A$. - Symmetric difference is associative. - For $A=\{x,y,z\}$ and $B=\{z,w\}$, $A \Delta B = \{x,y,w\}$. - Symmetric difference of identical sets is the empty set. - Union $A \cup B$ for $A=\{1,2,3\}$ and $B=\{3,4,5\}$ is $\{1,2,3,4,5\}$. - Symmetric difference does not satisfy distributive property. Final answers for the multiple choice questions are: 1. d 2. d 3. a 4. c 5. c 6. d 7. a 8. a 9. c 10. b 11. d 12. b 13. a 14. c 15. a