The problem you provided asks to show that for all sets $A$, $B$, and $C$, the following identity holds:

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

This is a well-known set theory identity that we can prove by showing that each side of the equation is a subset of the other.

Proof:

1. Show that $A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C)$:

Let $x \in A \cup (B \cap C)$. This means that $x$ is either in $A$ or in both $B$ and $C$. We consider two cases:

• If $x \in A$, then clearly $x \in A \cup B$ and $x \in A \cup C$, so $x \in (A \cup B) \cap (A \cup C)$.
• If $x \in B \cap C$, then $x \in B$ and $x \in C$. Hence, $x \in A \cup B$ and $x \in A \cup C$, so $x \in (A \cup B) \cap (A \cup C)$.

Thus, in both cases, $x \in (A \cup B) \cap (A \cup C)$. Therefore, we have shown that:

$A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C)$

2. Show that $(A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C)$:

Let $x \in (A \cup B) \cap (A \cup C)$. This means that $x \in A \cup B$ and $x \in A \cup C$. We consider two cases:

• If $x \in A$, then clearly $x \in A \cup (B \cap C)$ because $x \in A$.
• If $x \notin A$, then $x$ must be in both $B$ and $C$ (since $x \in A \cup B$ implies $x \in B$, and $x \in A \cup C$ implies $x \in C$). Therefore, $x \in B \cap C$, and so $x \in A \cup (B \cap C)$.

Thus, in both cases, $x \in A \cup (B \cap C)$. Therefore, we have shown that:

$(A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C)$

Conclusion:

Since we have shown both $A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C)$ and $(A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C)$, we conclude that:

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

This completes the proof.

Do you have any questions or would you like more details on any step of this proof?

Here are five related questions you might find interesting:

1. How do you prove other set theory identities, like De Morgan's laws?
2. What are the key differences between union and intersection in set theory?
3. Can you extend this result to more than three sets? How would that change the proof?
4. How do Venn diagrams help in visualizing set theory proofs like this one?
5. How is this identity used in probability theory?

Tip: When working with sets, always break down the proof into cases to carefully handle each part of the set operations (e.g., union, intersection). This ensures clarity in your reasoning.