1. **State the problem:** Determine which of the given sets are subsets of the set $$A = \{1, \{2, 3, 4\}, \{5, \{6\}\}, 7, 8\}$$. 2. **Recall the definition of subset:** A set $$B$$ is a subset of $$A$$, written $$B \subseteq A$$, if every element of $$B$$ is also an element of $$A$$. 3. **Analyze each candidate subset:** - $$\emptyset \subseteq A$$: The empty set is a subset of every set by definition. - $$\{2, 3, 4\} \subseteq A$$: Check if elements 2, 3, and 4 are in $$A$$. They are elements inside the set $$\{2, 3, 4\}$$ which is an element of $$A$$, but 2, 3, and 4 themselves are not elements of $$A$$. So $$\{2, 3, 4\} \not\subseteq A$$. - $$\{1, 7, 8\} \subseteq A$$: Elements 1, 7, and 8 are all elements of $$A$$, so this is a subset. - $$\{\{2, 3, 4\}\} \subseteq A$$: The set $$\{2, 3, 4\}$$ is an element of $$A$$, so this singleton set is a subset. - $$\{5, \{6\}\} \subseteq A$$: Check if 5 and $$\{6\}$$ are elements of $$A$$. The element $$\{5, \{6\}\}$$ is in $$A$$, but 5 alone is not an element of $$A$$, nor is $$\{6\}$$. So this is not a subset. - $$\{\{5, \{6\}\}, 7, 8\} \subseteq A$$: The elements $$\{5, \{6\}\}$$, 7, and 8 are all elements of $$A$$, so this is a subset. 4. **Final answers:** - $$\emptyset \subseteq A$$: True - $$\{2, 3, 4\} \subseteq A$$: False - $$\{1, 7, 8\} \subseteq A$$: True - $$\{\{2, 3, 4\}\} \subseteq A$$: True - $$\{5, \{6\}\} \subseteq A$$: False - $$\{\{5, \{6\}\}, 7, 8\} \subseteq A$$: True