1. **State the problem:** Determine which subset relationships among the sets $A=\{3,5,7\}$, $B=\{3,7\}$, $C=\{5,7\}$, and $D=\{5,7,9\}$ are true. 2. **Recall the definition of subset:** A set $X$ is a subset of set $Y$ (written $X \subseteq Y$) if every element of $X$ is also an element of $Y$. 3. **Check each subset relationship:** - $B \subseteq C$? $B=\{3,7\}$, $C=\{5,7\}$. Element 3 is in $B$ but not in $C$, so $B \not\subseteq C$. - $A \subseteq D$? $A=\{3,5,7\}$, $D=\{5,7,9\}$. Element 3 is in $A$ but not in $D$, so $A \not\subseteq D$. - $B \subseteq D$? $B=\{3,7\}$, $D=\{5,7,9\}$. Element 3 is in $B$ but not in $D$, so $B \not\subseteq D$. - $B \subseteq A$? $B=\{3,7\}$, $A=\{3,5,7\}$. All elements of $B$ (3 and 7) are in $A$, so $B \subseteq A$ is true. - $D \subseteq B$? $D=\{5,7,9\}$, $B=\{3,7\}$. Element 5 and 9 are in $D$ but not in $B$, so $D \not\subseteq B$. - $C \subseteq D$? $C=\{5,7\}$, $D=\{5,7,9\}$. All elements of $C$ are in $D$, so $C \subseteq D$ is true. - $C \subseteq A$? $C=\{5,7\}$, $A=\{3,5,7\}$. All elements of $C$ are in $A$, so $C \subseteq A$ is true. 4. **Final answers:** - $B \subseteq C$: false - $A \subseteq D$: false - $B \subseteq D$: false - $B \subseteq A$: true - $D \subseteq B$: false - $C \subseteq D$: true - $C \subseteq A$: true