1. **State the problem:** We are given sets $A = \{3, 5, 7\}$, $B = \{3, 7\}$, $C = \{5, 7\}$, and $D = \{5, 7, 9\}$. We need to determine which of the subset relations are true: $$ C \subseteq D, \quad D \subseteq B, \quad C \subseteq A, \quad B \subseteq C, \quad B \subseteq A, \quad A \subseteq D, \quad B \subseteq D $$ 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 relation:** - $C \subseteq D$? $C = \{5,7\}$ and $D = \{5,7,9\}$. Every element of $C$ is in $D$, so **true**. - $D \subseteq B$? $D = \{5,7,9\}$ and $B = \{3,7\}$. Element $5$ in $D$ is not in $B$, so **false**. - $C \subseteq A$? $C = \{5,7\}$ and $A = \{3,5,7\}$. Every element of $C$ is in $A$, so **true**. - $B \subseteq C$? $B = \{3,7\}$ and $C = \{5,7\}$. Element $3$ in $B$ is not in $C$, so **false**. - $B \subseteq A$? $B = \{3,7\}$ and $A = \{3,5,7\}$. Every element of $B$ is in $A$, so **true**. - $A \subseteq D$? $A = \{3,5,7\}$ and $D = \{5,7,9\}$. Element $3$ in $A$ is not in $D$, so **false**. - $B \subseteq D$? $B = \{3,7\}$ and $D = \{5,7,9\}$. Element $3$ in $B$ is not in $D$, so **false**. 4. **Summary of truth values:** $$ C \subseteq D: \text{true} $$ $$ D \subseteq B: \text{false} $$ $$ C \subseteq A: \text{true} $$ $$ B \subseteq C: \text{false} $$ $$ B \subseteq A: \text{true} $$ $$ A \subseteq D: \text{false} $$ $$ B \subseteq D: \text{false} $$ This completes the evaluation of all subset relations.