\( \{ [ ( A - B ) \cap B ] \cup [ ( A \cup B ) \cap C ] \} ^ { \prime } \)

To simplify the expression \( \{ [ ( A - B ) \cap B ] \cup [ ( A \cup B ) \cap C ] \} ^ { \prime } \), we will use set theory operations and De Morgan's laws. 1. **Understanding the components**: - \( A - B \) is the set of elements in \( A \) that are not in \( B \). - \( (A - B) \cap B \) is the intersection of \( A - B \) and \( B \). This set is empty because there are no elements that can be in both \( A - B \) and \( B \) at the same time. - Therefore, \( (A - B) \cap B = \emptyset \). 2. **Simplifying the union**: - Now we have: \[ \{ [ ( A - B ) \cap B ] \cup [ ( A \cup B ) \cap C ] \} = \{ \emptyset \cup [ ( A \cup B ) \cap C ] \} = ( A \cup B ) \cap C \] 3. **Applying the complement**: - Now we need to find the complement of \( ( A \cup B ) \cap C \): \[ \{ ( A \cup B ) \cap C \} ^ { \prime } \] - By De Morgan's laws, the complement of an intersection is the union of the complements: \[ ( ( A \cup B ) \cap C ) ^ { \prime } = ( A \cup B ) ^ { \prime } \cup C ^ { \prime } \] 4. **Finding the complements**: - The complement of \( A \cup B \) is: \[ ( A \cup B ) ^ { \prime } = A ^ { \prime } \cap B ^ { \prime } \] - Therefore, we have: \[ ( A \cup B ) ^ { \prime } \cup C ^ { \prime } = ( A ^ { \prime } \cap B ^ { \prime }) \cup C ^ { \prime } \] 5. **Final result**: - The final simplified expression is: \[ ( A ^ { \prime } \cap B ^ { \prime }) \cup C ^ { \prime } \] Thus, the simplified form of the original expression is: \[ ( A ^ { \prime } \cap B ^ { \prime }) \cup C ^ { \prime } \]