1. **Problem:** For any three sets $A$, $B$, and $C$, determine which of the following statements is NOT true: A) $B - A \subseteq A^c$ B) $[A \subset B, B \subset C, C \subset A] \Rightarrow A = C$ C) $B - A^c = B \cap A$ D) $A \subset B \Rightarrow B^c \subset A^c$ --- 2. **Step 1: Understand the notation and definitions** - $B - A$ means elements in $B$ but not in $A$. - $A^c$ is the complement of $A$. - $\subset$ means proper subset. - $\subseteq$ means subset (possibly equal). --- 3. **Analyze each statement:** **A)** $B - A \subseteq A^c$ means elements in $B$ but not in $A$ are in the complement of $A$. This is true by definition because if $x \in B$ and $x \notin A$, then $x \in A^c$. **B)** $[A \subset B, B \subset C, C \subset A] \Rightarrow A = C$. - This means $A$ is a proper subset of $B$, $B$ is a proper subset of $C$, and $C$ is a proper subset of $A$. - This is a cycle of strict subsets, which is impossible because strict subset relation is transitive and antisymmetric. - Therefore, this statement is NOT true. **C)** $B - A^c = B \cap A$. - $B - A^c$ means elements in $B$ but not in $A^c$, which is $B \cap A$. - This is true. **D)** $A \subset B \Rightarrow B^c \subset A^c$. - If $A$ is a proper subset of $B$, then the complement of $B$ is a subset of the complement of $A$. - This is true. --- 4. **Conclusion:** The statement that is NOT true is **B**. --- **Final answer:** $$\boxed{\text{B) } [A \subset B, B \subset C, C \subset A] \Rightarrow A = C \text{ is NOT true}}$$