1. **Stating the problem:** We want to prove De Morgan's law for sets, which states that the complement of the union of two sets $A$ and $B$ is equal to the intersection of their complements: $$(A \cup B)' = A' \cap B'$$ 2. **Formula and explanation:** De Morgan's laws relate the complement of unions and intersections of sets. The complement of a set $X$, denoted $X'$, contains all elements not in $X$. The union $A \cup B$ contains elements in $A$ or $B$ or both. The intersection $A' \cap B'$ contains elements not in $A$ and not in $B$. 3. **Using a Venn diagram:** - Draw two overlapping circles labeled $A$ and $B$ inside a rectangle representing the universal set. - The union $A \cup B$ is the area covered by both circles. - The complement $(A \cup B)'$ is everything outside both circles. - The complements $A'$ and $B'$ are the areas outside each circle. - The intersection $A' \cap B'$ is the area outside both circles, which matches $(A \cup B)'$. 4. **Conclusion:** The shaded regions for $(A \cup B)'$ and $A' \cap B'$ are identical in the Venn diagram, proving the law visually. Thus, De Morgan's law $$(A \cup B)' = A' \cap B'$$ is true.