1. List the elements of the following sets: (i) $P = \{1, 2, 3, \ldots, 13\}$ means all integers from 1 to 13 inclusive. So, $P = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}$. (ii) $S = \{x \mid x \in \mathbb{Z}, 4 \leq x \leq 9\}$ means all integers $x$ such that $4 \leq x \leq 9$. So, $S = \{4, 5, 6, 7, 8, 9\}$. (iii) $W = \{n \mid n \in \mathbb{Z}, 0 \leq n \leq 6\}$ means all integers $n$ such that $0 \leq n \leq 6$. So, $W = \{0, 1, 2, 3, 4, 5, 6\}$. 2. Given sets: $U = \{a, b, c, d, e, f, g\}$ $A = \{a, b, c, d, e\}$ $B = \{a, c, e, g\}$ $C = \{b, e, f, g\}$ Find: (i) $A \cup B$ (union of $A$ and $B$) is the set of elements in $A$ or $B$ or both. $A \cup B = \{a, b, c, d, e, g\}$. (ii) $A \cap B$ (intersection of $A$ and $B$) is the set of elements common to both. $A \cap B = \{a, c, e\}$. (iii) $A \cup C$ is the set of elements in $A$ or $C$ or both. $A \cup C = \{a, b, c, d, e, f, g\}$. (iv) $B \cap A$ is the same as $A \cap B$. $B \cap A = \{a, c, e\}$. (v) $A \cap (B \cap C)$ means elements in $A$ and also in both $B$ and $C$. First find $B \cap C = \{e, g\}$. Then $A \cap (B \cap C) = A \cap \{e, g\} = \{e\}$. (vi) $(A \cap A')'$ where $A'$ is the complement of $A$ in $U$. Since $A \cap A' = \emptyset$ (no element is both in $A$ and not in $A$), $(A \cap A')' = \emptyset' = U = \{a, b, c, d, e, f, g\}$. 3. For sets $A$ and $B$ with common intersections, the following are: (i) $A'$ is the complement of $A$ in $U$, i.e., elements not in $A$. $A' = U \setminus A = \{f, g\}$. (ii) $A' \cap B'$ is the intersection of complements of $A$ and $B$. $B' = U \setminus B = \{b, d, f\}$. So, $A' \cap B' = \{f, g\} \cap \{b, d, f\} = \{f\}$. (iii) $A' \cap B$ is elements not in $A$ but in $B$. $A' = \{f, g\}$ and $B = \{a, c, e, g\}$. So, $A' \cap B = \{g\}$. (iv) $(A \cap B)'$ is the complement of the intersection of $A$ and $B$. $A \cap B = \{a, c, e\}$. So, $(A \cap B)' = U \setminus \{a, c, e\} = \{b, d, f, g\}$. (v) $(A \cup B)'$ is the complement of the union of $A$ and $B$. $A \cup B = \{a, b, c, d, e, g\}$. So, $(A \cup B)' = U \setminus \{a, b, c, d, e, g\} = \{f\}$. Final answers are listed above.