1. **Problem 1: Membership in $\mathbb{Z}$** We need to determine if each object is an integer ($\in \mathbb{Z}$). - $\{5\}$ is a set containing 5, not an integer itself, so $\{5\} \notin \mathbb{Z}$. - $\{3, -1\}$ is a set, not an integer, so $\{3, -1\} \notin \mathbb{Z}$. - $7.12$ is a decimal number, not an integer, so $7.12 \notin \mathbb{Z}$. - $\sqrt{5}$ is irrational, not an integer, so $\sqrt{5} \notin \mathbb{Z}$. - $a$ = the 200th decimal digit of $\pi$ is a digit (0-9), which is an integer, so $a \in \mathbb{Z}$. 2. **Problem 2: Sets of digits $A$ and $B$ from $\frac{41}{333}$ and $\frac{44}{333}$** - Compute decimal expansions: $$\frac{41}{333} = 0.123123123...$$ (repeating 123) $$\frac{44}{333} = 0.132132132...$$ (repeating 132) - Set $A$ digits: $\{1, 2, 3\}$ - Set $B$ digits: $\{1, 3, 2\}$ Since sets ignore order, $A = B = \{1, 2, 3\}$. 3. **Problem 3: Set $C$ of digits in $\frac{4036363637}{33300000000}$ equals $A$?** - Approximate decimal expansion of $\frac{4036363637}{33300000000}$ is about 0.1212... - Digits appearing include 0,1,2,3,6,7,4. - Set $C$ digits differ from $A = \{1,2,3\}$. Therefore, $C \neq A$. 4. **Problem 4: Membership table for $a_i$ in sets $A$ to $F$** Recall sets: $A=\{1, \{4\}, \{2\}, 3, 4, 5\}$ $B=\{\{\{1,4,5,3,1\}\}\}$ $C=\{1, \{3\}, 2, 1\}$ $D=\{1,1,3\}$ (duplicates ignored) $E=\{1,4, \{5\}, \{3\}\}$ $F=\{1,8, \{1,2,3,4\}\}$ Elements: $a_1=1$ $a_2=\{2\}$ $a_3=\{2,1\}$ $a_4=\{2,1,3,4\}$ $a_5=\{3,1,5\}$ Membership: - $a_1=1$ in $A$ (yes), $B$ (no), $C$ (yes), $D$ (yes), $E$ (yes), $F$ (yes) - $a_2=\{2\}$ in $A$ (yes), $B$ (no), $C$ (no), $D$ (no), $E$ (no), $F$ (no) - $a_3=\{2,1\}$ in none of $A$ to $F$ (no), since sets contain only singletons or different sets - $a_4=\{2,1,3,4\}$ in none (no) - $a_5=\{3,1,5\}$ in none (no) | | A | B | C | D | E | F | |-----|---|---|---|---|---|---| | a1 | ∈ | | ∈ | ∈ | ∈ | ∈ | | a2 | ∈ | | | | | | | a3 | | | | | | | | a4 | | | | | | | | a5 | | | | | | | 5. **Problem 5: Calculate sets** - $A \cap C = \{1,3,4,5,2\} \cap \{1,2,3\} = \{1,2,3\}$ - $B \cap F = \{\{\{1,4,5,3,1\}\}\} \cap \{1,8,\{1,2,3,4\}\} = \emptyset$ - $D \cup C = \{1,3\} \cup \{1,2,3\} = \{1,2,3\}$ - $C \cap E = \{1,2,3\} \cap \{1,4,\{5\},\{3\}\} = \{1\}$ - $D \cap F = \{1,3\} \cap \{1,8,\{1,2,3,4\}\} = \{1\}$ - $C \cup (D \cap F) = \{1,2,3\} \cup \{1\} = \{1,2,3\}$ - $A \cap E = \{1,\{4\},\{2\},3,4,5\} \cap \{1,4,\{5\},\{3\}\} = \{1,4\}$ **Final answers:** $A \cap C = \{1,2,3\}$ $B \cap F = \emptyset$ $D \cup C = \{1,2,3\}$ $C \cap E = \{1\}$ $C \cup (D \cap F) = \{1,2,3\}$ $A \cap E = \{1,4\}$