1. **State the problem:** We have sets $A = \{l, a, c, e\}$, $B = \{a, c, e\}$, $C = \{e, a, r\}$, and $D = \{e, c\}$. We need to determine which sets are subsets of which others. 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 option:** - A. $D \subseteq A$? $D = \{e, c\}$ and $A = \{l, a, c, e\}$. Both $e$ and $c$ are in $A$, so yes, $D \subseteq A$. - B. All of these are subsets of each other? This would mean all sets are equal, which is false because $A$ has $l$ and $C$ has $r$ which are not in all sets. - C. $B \subseteq D$? $B = \{a, c, e\}$, $D = \{e, c\}$. $a$ is in $B$ but not in $D$, so no. - D. $D \subseteq C$? $D = \{e, c\}$, $C = \{e, a, r\}$. $c$ is not in $C$, so no. - E. $B \subseteq A$? $B = \{a, c, e\}$, $A = \{l, a, c, e\}$. All elements of $B$ are in $A$, so yes. - F. None of these are subsets of each other? False, since we found some subsets. - G. $A \subseteq D$? $A$ has $l$ and $a$ not in $D$, so no. - H. $D \subseteq B$? $D = \{e, c\}$, $B = \{a, c, e\}$. Both $e$ and $c$ are in $B$, so yes. - I. $A \subseteq B$? $A$ has $l$ not in $B$, so no. - J. $A \subseteq C$? $A$ has $l$ and $c$ not in $C$, so no. - K. $B \subseteq C$? $B$ has $c$ not in $C$, so no. 4. **Summary of correct subset relations:** - $D \subseteq A$ - $B \subseteq A$ - $D \subseteq B$ 5. **Final answer:** The correct subset relations are options A, E, and H.