1. **Problem:** Compute the union and intersection of given sets and other set operations. Given sets: $$U = \{a, b, c, d, e, f, g, h, k\}$$ $$A = \{a, b, c, g\}$$ $$B = \{d, e, f, g\}$$ $$C = \{a, c, f\}$$ $$D = \{f, h, k\}$$ 2. **Formulas and rules:** - Union: $$A \cup B = \{x | x \in A \text{ or } x \in B\}$$ - Intersection: $$A \cap B = \{x | x \in A \text{ and } x \in B\}$$ - Set difference: $$A - B = \{x | x \in A \text{ and } x \notin B\}$$ - Complement: $$\overline{A} = U - A$$ 3. **Calculations:** **a) $$A \cup B$$** $$A \cup B = \{a, b, c, g\} \cup \{d, e, f, g\} = \{a, b, c, d, e, f, g\}$$ **b) $$B \cup C$$** $$B \cup C = \{d, e, f, g\} \cup \{a, c, f\} = \{a, c, d, e, f, g\}$$ **c) $$B \cap D$$** $$B \cap D = \{d, e, f, g\} \cap \{f, h, k\} = \{f\}$$ **d) $$A - B$$** $$A - B = \{a, b, c, g\} - \{d, e, f, g\} = \{a, b, c\}$$ **e) $$\overline{A}$$** $$\overline{A} = U - A = \{a, b, c, d, e, f, g, h, k\} - \{a, b, c, g\} = \{d, e, f, h, k\}$$ **Final answers:** - a) $$\{a, b, c, d, e, f, g\}$$ - b) $$\{a, c, d, e, f, g\}$$ - c) $$\{f\}$$ - d) $$\{a, b, c\}$$ - e) $$\{d, e, f, h, k\}$$