1. **Problem statement:** Given sets $A = \{a, b, c\}$, $B = \{c, d, e\}$, and universal set $U = \{a, b, c, d, e, f, g\}$, find the following sets: 2. **Union $A \cup B$:** The union of two sets contains all elements that are in $A$, or $B$, or both. $$A \cup B = \{a, b, c\} \cup \{c, d, e\} = \{a, b, c, d, e\}$$ 3. **Intersection $A \cap B$:** The intersection contains elements common to both $A$ and $B$. $$A \cap B = \{c\}$$ 4. **Complement $A'$:** The complement of $A$ relative to $U$ contains all elements in $U$ not in $A$. $$A' = U \setminus A = \{a, b, c, d, e, f, g\} \setminus \{a, b, c\} = \{d, e, f, g\}$$ 5. **Set difference $A \sim B$:** Elements in $A$ but not in $B$. $$A \sim B = A \setminus B = \{a, b, c\} \setminus \{c, d, e\} = \{a, b\}$$ 6. **Symmetric difference $A \oplus B$:** Elements in either $A$ or $B$ but not in both. $$A \oplus B = (A \setminus B) \cup (B \setminus A) = \{a, b\} \cup \{d, e\} = \{a, b, d, e\}$$ Each result corresponds to the respective Venn diagram region: union covers all areas in $A$ or $B$, intersection is the overlap, complement is outside $A$, difference is the part of $A$ excluding overlap, and symmetric difference excludes the overlap but includes the rest. **Final answers:** - $A \cup B = \{a, b, c, d, e\}$ - $A \cap B = \{c\}$ - $A' = \{d, e, f, g\}$ - $A \sim B = \{a, b\}$ - $A \oplus B = \{a, b, d, e\}$