1. **Problem:** Given $S = \{1\}$, determine which of the following is *not* a subset of the power set $P(S)$. The options are: A) $\emptyset$ B) $\{1\}$ C) $\{\{1\}\}$ D) $\{\emptyset, \{1\}\}$ 2. **Recall:** The power set $P(S)$ is the set of all subsets of $S$. Since $S = \{1\}$, its subsets are $\emptyset$ and $\{1\}$. Therefore, $$P(S) = \{\emptyset, \{1\}\}$$ 3. **Check each option as a subset of $P(S)$:** - A) $\emptyset$ is the empty set, which is a subset of every set, so $\emptyset \subseteq P(S)$. - B) $\{1\}$ contains the element $1$, but $1$ is not an element of $P(S)$ (elements of $P(S)$ are sets, not numbers). So $\{1\} \not\subseteq P(S)$. - C) $\{\{1\}\}$ contains $\{1\}$, which is an element of $P(S)$, so $\{\{1\}\} \subseteq P(S)$. - D) $\{\emptyset, \{1\}\}$ is exactly $P(S)$, so it is a subset of itself. 4. **Answer:** The set that is *not* a subset of $P(S)$ is option B) $\{1\}$. --- 5. **True or False:** For any non-empty set $S$, $S$ is an element of $P(S)$. - By definition, $P(S)$ contains all subsets of $S$. Since $S$ is a subset of itself, $S \in P(S)$. - So the statement is **True**. --- 6. **True or False:** For any set $S$, the empty set $\emptyset$ is a subset of $P(S)$. - Since $\emptyset$ is a subset of every set, $\emptyset \subseteq P(S)$ is always true. - So the statement is **True**. --- 7. **Additional problem:** Given $S = \{a, b, \{a\}\}$, determine which statements are true: (i) $\{b\} \in S$? No, $b$ is in $S$, but $\{b\}$ is not. (ii) $\{a\} \subseteq P(S)$? $P(S)$ contains subsets of $S$. $\{a\}$ is a set containing $a$, but $a$ is an element of $S$, not a subset. So $\{a\}$ is not a subset of $P(S)$. (iii) $\{a, b\} \in P(S)$? $\{a, b\}$ is a subset of $S$, so yes. (iv) $\{a, b\} \in S$? No, $S$ contains $a$, $b$, and $\{a\}$, but not $\{a, b\}$. (v) $\{\{a\}\} \in P(S)$? $\{a\}$ is an element of $S$, so $\{\{a\}\}$ is a subset of $S$, hence in $P(S)$. (vi) $\{a, \{a\}\} \in P(S)$? Both $a$ and $\{a\}$ are elements of $S$, so $\{a, \{a\}\}$ is a subset of $S$, so yes. True statements are (iii), (v), and (vi). --- **Final answers:** - The set not a subset of $P(S)$ is $\{1\}$. - For any non-empty set $S$, $S \in P(S)$ is True. - For any set $S$, $\emptyset \subseteq P(S)$ is True. - For $S = \{a, b, \{a\}\}$, true statements are (iii), (v), and (vi).