To illustrate and prove the distributive law $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ using Venn diagrams, we can follow these steps:

Step 1: Define the Sets

Let’s define three sets $A$, $B$, and $C$ within a universal set. We will use Venn diagrams to visualize the interactions between these sets.

Step 2: Create Venn Diagrams

1. Diagram for $A \cap (B \cup C)$:

   • Shade the area representing $B \cup C$ (the union of sets $B$ and $C$).
   • Then, shade the intersection of $A$ with the shaded area (i.e., $A \cap (B \cup C)$).

2. Diagram for $(A \cap B) \cup (A \cap C)$:

   • First, shade the intersection $A \cap B$.
   • Next, shade the intersection $A \cap C$.
   • Finally, combine these two shaded areas to illustrate the union $(A \cap B) \cup (A \cap C)$.

Step 3: Proving the Law

To prove the law holds for all sets $A$, $B$, and $C$:

1. Choose Elements: Consider any element $x$ in the universal set.

2. Case Analysis:

   • If $x \in A$: Then we check if $x$ is also in $B$ or $C$:

     • If $x \in B$, then $x \in A \cap B$.
     • If $x \in C$, then $x \in A \cap C$.
     • In both cases, $x$ belongs to $(A \cap B) \cup (A \cap C)$.

   • If $x \notin A$: Then $x$ cannot be in $A \cap (B \cup C)$ since $A$ is the limiting factor. Hence, it’s not in the union either.

3. Conclusion: Since any element $x$ that is in $A \cap (B \cup C)$ is also in $(A \cap B) \cup (A \cap C)$ and vice versa, we conclude that: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

Summary

This proof shows that the distributive law holds true for all sets $A$, $B$, and $C$. The Venn diagrams effectively illustrate how the elements relate under the intersection and union operations.

Questions to Consider

1. What are other examples of distributive laws in set theory?
2. How can we visualize the intersection and union of more than three sets?
3. What happens if we change the operations in the distributive law (e.g., using difference)?
4. Can the distributive law be applied in probability theory? If so, how?
5. What are some practical applications of Venn diagrams in data analysis?

Tip

When working with Venn diagrams, always ensure that the universal set is clearly defined to avoid confusion in intersections and unions.