1. **Problem Statement:** Given sets: $$A = \{1, 3, 5, 7, 9\}, B = \{1, 2, 3, 4, 5\}, C = \{2, 4, 6, 7, 8, 9\}$$ Find: (a) $A \cup B$ and $A \cap B$ (b) $A \times B$ (c) Verify De Morgan's Law: $\overline{A \cup B} = \overline{A} \cap \overline{B}$ 2. **Formulas and Rules:** - Union: $A \cup B = \{x | x \in A \text{ or } x \in B\}$ - Intersection: $A \cap B = \{x | x \in A \text{ and } x \in B\}$ - Cartesian Product: $A \times B = \{(a,b) | a \in A, b \in B\}$ - De Morgan's Law for sets: $\overline{A \cup B} = \overline{A} \cap \overline{B}$ where $\overline{X}$ is the complement of $X$ relative to a universal set. 3. **Step (a): Find $A \cup B$ and $A \cap B$** - $A \cup B = \{1, 3, 5, 7, 9\} \cup \{1, 2, 3, 4, 5\} = \{1, 2, 3, 4, 5, 7, 9\}$ - $A \cap B = \{1, 3, 5, 7, 9\} \cap \{1, 2, 3, 4, 5\} = \{1, 3, 5\}$ 4. **Step (b): Find $A \times B$** - Cartesian product pairs each element of $A$ with each element of $B$: $$A \times B = \{(1,1),(1,2),(1,3),(1,4),(1,5),(3,1),(3,2),(3,3),(3,4),(3,5),(5,1),(5,2),(5,3),(5,4),(5,5),(7,1),(7,2),(7,3),(7,4),(7,5),(9,1),(9,2),(9,3),(9,4),(9,5)\}$$ 5. **Step (c): Verify De Morgan's Law** - Assume universal set $U$ contains all elements from $A$, $B$, and $C$: $$U = \{1,2,3,4,5,6,7,8,9\}$$ - Find complements: - $\overline{A} = U - A = \{2,4,6,8\}$ - $\overline{B} = U - B = \{6,7,8,9\}$ - $A \cup B = \{1,2,3,4,5,7,9\}$ - $\overline{A \cup B} = U - (A \cup B) = \{6,8\}$ - Find $\overline{A} \cap \overline{B}$: $$\overline{A} \cap \overline{B} = \{2,4,6,8\} \cap \{6,7,8,9\} = \{6,8\}$$ - Since $\overline{A \cup B} = \overline{A} \cap \overline{B} = \{6,8\}$, De Morgan's Law is verified. **Final answers:** - (a) $A \cup B = \{1, 2, 3, 4, 5, 7, 9\}$, $A \cap B = \{1, 3, 5\}$ - (b) $A \times B$ is the set of 25 ordered pairs listed above. - (c) De Morgan's Law holds: $\overline{A \cup B} = \overline{A} \cap \overline{B} = \{6,8\}$