Let's answer each part step by step with explanations. --- ### **1. Show that \( F = \{ \emptyset, E, E^c, R \} \) is the σ-algebra of subsets of \( R \) generated by \( \{E\} \).** #### **Step 1: Show \( F \) is a σ-algebra.** - **Explanation:** A σ-algebra over \( R \) must contain \( R \), be closed under complements, and closed under countable unions. - \( R \in F \) (given). - \( \emptyset \in F \) (given). - \( E \in F \Rightarrow E^c \in F \). - \( E^c \in F \Rightarrow (E^c)^c = E \in F \). - The union of any two elements in \( F \) is still in \( F \) (since all possible unions are either \( \emptyset, E, E^c, R \)). - Countable unions are also within \( F \) since \( F \) is finite. #### **Step 2: Show \( F \) is the smallest σ-algebra containing \( E \).** - **Explanation:** Any σ-algebra containing \( E \) must also contain \( E^c \) (since σ-algebras are closed under complements), \( R \), and \( \emptyset \). The only σ-algebra containing exactly these elements is \( F \), so \( F \) is the σ-algebra generated by \( \{E\} \). #### **Final Answer for 1:** The σ-algebra generated by \( \{E\} \) is \( F = \{\emptyset, E, E^c, R\} \). --- ### **2. Let \( x = \{1,2,3,4\} \) and \( C = \{\{1\}, \{2,3\}\} \subset P(x) \). Determine \( \sigma(C) \), the σ-algebra generated by \( C \).** #### **Step 1: Find all sets that must be in \( \sigma(C) \).** - **Explanation:** σ-algebras are closed under complements and unions. - \( x \) must be in \( \sigma(C) \). - \( \emptyset \) must be in \( \sigma(C) \). - \( \{1\} \in \sigma(C) \Rightarrow x \setminus \{1\} = \{2,3,4\} \in \sigma(C) \). - \( \{2,3\} \in \sigma(C) \Rightarrow x \setminus \{2,3\} = \{1,4\} \in \sigma(C) \). - Union: \( \{1\} \cup \{2,3\} = \{1,2,3\} \in \sigma(C) \). - Complement: \( x \setminus \{1,2,3\} = \{4\} \in \sigma(C) \). #### **Step 2: List all elements explicitly.** - **Explanation:** Collect all subsets generated by these operations: - \( \emptyset \) - \( \{1\} \) - \( \{2,3\} \) - \( \{1,2,3\} \) - \( \{4\} \) - \( \{1,4\} \) - \( \{2,3,4\} \) - \( x = \{1,2,3,4\} \) These 8 sets form a σ-algebra (the partition is \(\{1\},\{2,3\},\{4\}\)), confirming closure under complements and unions. #### **Final Answer for 2:** The σ-algebra generated by \( C \) is: \[ \sigma(C) = \{\emptyset, \{1\}, \{2,3\}, \{1,2,3\}, \{4\}, \{1,4\}, \{2,3,4\}, \{1,2,3,4\}\} \]