1. **Problem:** Identify which statement about three sets $A$, $B$, and $C$ is NOT true. 2. **Statements:** A) $B - A \subset A'$ B) $\{ A \subset B, B \subset C, C \subset A \} \Rightarrow A = C$ C) $B - A' = B \cap A$ D) $A \subset B \Rightarrow B' \subset A'$ --- 3. **Recall definitions and rules:** - $A'$ is the complement of $A$. - $B - A = B \cap A'$. - If $A \subset B$, then $B' \subset A'$ is generally false. --- 4. **Check each statement:** - A) $B - A = B \cap A'$, so $B - A \subset A'$ is true because $B - A$ is a subset of $A'$. - B) If $A \subset B$, $B \subset C$, and $C \subset A$, then $A = B = C$ by antisymmetry of subset relation, so $A = C$ is true. - C) $B - A' = B \cap (A')' = B \cap A$, since complement of complement is the set itself. So this is true. - D) $A \subset B$ does NOT imply $B' \subset A'$. The inclusion reverses for complements: $A \subset B$ implies $B' \subset A'$ is false. --- 5. **Answer for question 1:** Statement D is NOT true. --- 6. **Problem 2:** Given $A = \{a, \{a, b\}, \emptyset\}$, identify which statement is NOT true. Statements: A) $\{a, b\} \in A$ B) $\{\{a, b\}\} \subseteq A$ C) $\{\{a, b\}\} \subseteq A$ (same as B, likely a typo, treat as same) D) $\{\emptyset\} \subset A$ --- 7. **Check membership and subset:** - $A$ contains $a$, the set $\{a,b\}$, and the empty set $\emptyset$. - A) $\{a,b\} \in A$ is true because $\{a,b\}$ is an element of $A$. - B) $\{\{a,b\}\} \subseteq A$ means the set containing $\{a,b\}$ is a subset of $A$. Since $\{a,b\} \in A$, this is true. - C) Same as B, true. - D) $\{\emptyset\} \subset A$ means the set containing $\emptyset$ is a proper subset of $A$. Since $\emptyset \in A$, this is true. --- 8. **Answer for question 2:** All statements are true, but since B and C are identical, if one is considered a duplicate, no false statement is present. If the user intended a different option, the only possible false would be if $\{a,b\} \in A$ was false, but it is true. --- **Final answers:** - For question 1, the NOT true statement is D. - For question 2, all given statements are true (assuming B and C are duplicates).