1. **Problem Statement:** We are given three sets: - $M$: students studying Mathematics - $A$: students studying Additional Mathematics - $U$: all students in the school We need to illustrate two relationships using Venn diagrams: i. All students who study Additional Mathematics also study Mathematics, but some students who study Mathematics do not study Additional Mathematics. ii. Not all students study Additional Mathematics, but every student studying Mathematics studies Additional Mathematics. 2. **Understanding the relationships:** - For (i): $A \subseteq M$ but $M \not\subseteq A$. This means the set $A$ is entirely inside $M$, but $M$ has elements outside $A$. - For (ii): $M \subseteq A$ but $A \neq U$. This means the set $M$ is entirely inside $A$, and $A$ is a proper subset of $U$ (not all students study Additional Mathematics). 3. **Explanation:** - In (i), the Venn diagram shows $A$ as a smaller circle inside $M$, and $M$ partially outside $A$. - In (ii), the Venn diagram shows $M$ as a smaller circle inside $A$, and $A$ as a smaller circle inside $U$ but not equal to $U$. 4. **Summary:** - (i) $A \subseteq M$ and $M \not\subseteq A$ - (ii) $M \subseteq A \subset U$ These relationships can be visualized with nested circles representing the sets.