1. **Stating the problem:** We need to prove that the complement of the union of two sets $A$ and $B$, denoted as $(A \cup B)^c$, is the region outside both $A$ and $B$ inside the universal set $U$. 2. **Recall the definition of union and complement:** - The union $A \cup B$ contains all elements that are in $A$, or in $B$, or in both. - The complement $(A \cup B)^c$ contains all elements in $U$ that are *not* in $A \cup B$. 3. **Express the complement of the union:** $$ (A \cup B)^c = \{ x \in U : x \notin A \cup B \} $$ This means $x$ is neither in $A$ nor in $B$. 4. **Rewrite using logical equivalences:** $$ x \notin A \cup B \iff (x \notin A) \text{ and } (x \notin B) $$ So, $$ (A \cup B)^c = \{ x \in U : x \notin A \text{ and } x \notin B \} = A^c \cap B^c $$ 5. **Interpretation:** The complement of the union is the intersection of the complements. This means the shaded region outside both circles $A$ and $B$ inside the rectangle $U$ is exactly $(A \cup B)^c$. 6. **Conclusion:** The hatching marks outside the circles $A$ and $B$ inside $U$ correctly represent $(A \cup B)^c$, proving the statement. This is a direct application of De Morgan's law for sets.