1. **State the problem:** We want to show through a Venn diagram and logical reasoning that if $A \subset B$ then $B' \subset A'$, and conversely, if $B' \subset A'$, then $A \subset B$. 2. **Recall definitions:** - $A \subset B$ means every element of $A$ is also an element of $B$. - $B'$ is the complement of $B$, i.e., all elements not in $B$. - $A'$ is the complement of $A$, i.e., all elements not in $A$. 3. **Using Venn diagrams:** - Assume $A \subset B$. This means circle $A$ is completely inside circle $B$. - Then, the area outside $B$ ($B'$) is larger than the area outside $A$ ($A'$), so every element outside $B$ is necessarily outside $A$. - Therefore, $B' \subset A'$. 4. **Verify the converse:** - Suppose $B' \subset A'$. - This means every element not in $B$ is also not in $A$. - Hence, if an element is in $A$, it cannot be outside $B$, so that element must be inside $B$. - Therefore, $A \subset B$. 5. **Final conclusion:** The relations $A \subset B$ and $B' \subset A'$ are logically equivalent. This completes the proof by using set definitions and reasoning with Venn diagrams.