To prove these equalities using Venn diagrams, we can break down each statement visually. Here’s the approach and reasoning for each:

(i) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

1. Left Side ( $A \cap (B \cup C)$ ):

   • First, draw the sets $A$, $B$, and $C$.
   • Shade the region representing $B \cup C$ (the union of $B$ and $C$).
   • Then, intersect this shaded area with $A$, showing only the parts where $A$ overlaps with $B \cup C$.

2. Right Side ( $(A \cap B) \cup (A \cap C)$ ):

   • Shade the intersection $A \cap B$, showing only the part where $A$ overlaps with $B$.
   • Then, shade the intersection $A \cap C$, showing only the part where $A$ overlaps with $C$.
   • The union of these two shaded areas will illustrate $(A \cap B) \cup (A \cap C)$.

3. Conclusion:

   • The regions obtained from both sides should match, proving that $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

(ii) $(A \cup B)^c = A^c \cap B^c$

1. Left Side ( $(A \cup B)^c$ ):

   • Draw sets $A$ and $B$ and find their union $A \cup B$.
   • The complement $(A \cup B)^c$ represents everything outside this union, shading only the area not covered by $A$ or $B$.

2. Right Side ( $A^c \cap B^c$ ):

   • Shade $A^c$, which is everything outside of $A$.
   • Then, shade $B^c$, which is everything outside of $B$.
   • The intersection $A^c \cap B^c$ is the area that lies outside both $A$ and $B$, matching the region of $(A \cup B)^c$.

3. Conclusion:

   • The regions obtained from both sides should be the same, thus proving that $(A \cup B)^c = A^c \cap B^c$.

Would you like further visual details or any additional clarifications on these proofs?

Here are five related questions that expand upon this topic:

1. How can you use De Morgan’s laws in other Venn diagram proofs?
2. Can you explain the difference between the union and intersection in set theory?
3. What would happen if we introduced a fourth set, $D$, in a Venn diagram?
4. How does the complement of a set change when working with multiple sets?
5. Can these laws be applied in real-world scenarios, such as probability or logic?

Tip: When working with Venn diagrams, always start with shading individual sets to ensure you clearly see the intersections and unions.