1. **State the problem:** We have two sets $A = \{x, y, z, w\}$ and $B = \{x, y\}$. We need to determine which of the following statements are true: - $B \not\subseteq A$ - $A \not\subseteq B$ - $B \subseteq A$ - $A \subseteq B$ 2. **Recall definitions:** - $X \subseteq Y$ means every element of set $X$ is also in set $Y$ (subset). - $X \not\subseteq Y$ means there is at least one element in $X$ not in $Y$. 3. **Check each statement:** - $B \not\subseteq A$? Since $B = \{x, y\}$ and $A = \{x, y, z, w\}$, all elements of $B$ are in $A$. So $B \subseteq A$ is true, thus $B \not\subseteq A$ is false. - $A \not\subseteq B$? $A$ has elements $z$ and $w$ which are not in $B$. So $A \subseteq B$ is false, hence $A \not\subseteq B$ is true. - $B \subseteq A$? As above, all elements of $B$ are in $A$, so this is true. - $A \subseteq B$? Since $z$ and $w$ are in $A$ but not in $B$, this is false. 4. **Final answers:** - $B \not\subseteq A$: False - $A \not\subseteq B$: True - $B \subseteq A$: True - $A \subseteq B$: False Therefore, the true statements are $A \not\subseteq B$ and $B \subseteq A$.