Prove that: (A ∪ B) ∩ C = A ∩ (B ∩ C)

Steps to Solve

1. Recall Set Definitions To prove the equation, we need to understand the definitions of union ($\cup$) and intersection ($\cap$) in set theory.

   • Union ($A \cup B$): The set containing all elements from both $A$ and $B$.
   • Intersection ($A \cap B$): The set containing elements that are in both $A$ and $B$.

2. Rewrite the Left Side Start with the left side of the equation: $$(A \cup B) \cap C$$ This means we are interested in elements that are both in the union of $A$ and $B$, and also in set $C$.

3. Expand the Left Side Using Element Membership An element $x$ belongs to $(A \cup B) \cap C$ if:

   • $x \in (A \cup B)$, meaning $x \in A$ or $x \in B$, and
   • $x \in C$.

4. Analyze Membership in Component Sets Therefore, we can say:

   • If $x \in (A \cup B) \cap C$, then $x \in C$ and either $x \in A$ or $x \in B$. This leads to two cases:
   • Case 1: If $x \in A$, then $x \in (B \cap C)$.
   • Case 2: If $x \in B$, then $x \in (C)$.

5. Focus on the Right Side Now consider the right side of the equation: $$A \cap (B \cap C)$$ This means we are looking for elements that are in both $A$ and the intersection of $B$ and $C$.

6. Expand the Right Side Using Element Membership For $x$ to belong to $A \cap (B \cap C)$, it must satisfy:

   • $x \in A$ and $x \in B \cap C$, implying $x \in B$ and $x \in C$.

7. Show Both Sides Contain the Same Elements To show that $(A \cup B) \cap C = A \cap (B \cap C)$, we must prove that:

   • If $x \in (A \cup B) \cap C$, then $x \in A \cap (B \cap C)$, and vice versa.

8. Final Conclusion Since both expressions represent the same logical conditions for elements, we conclude that: $$(A \cup B) \cap C = A \cap (B \cap C)$$