**Set Theory Statements and Analysis**1. **True or False Analysis for Set Operations:** - **Statement:** For all sets \(A \cap \varnothing = A\) - **Truth Value:** True - **Analysis:** The intersection of any set \(A\) with the empty set \(\varnothing\) is indeed \(A\).2. **Statement:** For all sets \(A\), if \(A \cap \varnothing = A\), then you are the pope. - **Truth Value:** False - **Analysis:** This statement is humorous in nature and is logically false since there’s no correlation between set operation results and the identity of a person.3. **Statement:** For all sets \(A, B, \) and \(C\), \(A \cup (B \cup C) = (A \cup B) \cup C\) - **Truth Value:** True - **Analysis:** This represents the associative property of union for sets.4. **Statement:** For all sets \(A, B, \) and \(C\), \(A \cup (B \cup C) = (B \cup A) \cup C\) - **Truth Value:** True - **Analysis:** This is also valid due to the commutative and associative properties of union.5. **Statement:** If \(x \in (A \cup B)^c\) then \(x \notin A\) or \(x \notin B\). - **Truth Value:** True - **Analysis:** If \(x\) is in the complement of \(A \cup B\), then it is not in either \(A\) or \(B\).6. **Statement:** If \(x \in (A \cap B)^c\) then \(x \notin A\) or \(x \notin B\). - **Truth Value:** False - **Analysis:** If \(x\) is in the complement of \(A \cap B\), then \(x\) is not in both \(A\) and \(B\) at the same time, not necessarily 'or'.7. **Statement:** \(\varphi^c = \text{the universal set.}\) - **Truth Value:** True - **Analysis:** The complement of the empty set \(\

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T T T T T T F F F F F F F For all sets AN Q = A For all sets A, if AN q = A, then you are the pope. For all sets A, B, and C, A U (B U C) = (A U B) U C For all sets A, B, and C, A U (BU C) = (BU A) UC If x ¤ (AUB)º then x & A or x & B. If x = (ANB)c then x & A or x & B. = the universal set. φε