To prove that $(A - B) - C = (A - C) - (B - C)$ for arbitrary sets $A$, $B$, and $C$, we will demonstrate that each side of the equation contains the same elements. We proceed by showing that any element in $(A - B) - C$ is also in $(A - C) - (B - C)$, and vice versa.

1. Prove that $(A - B) - C \subseteq (A - C) - (B - C)$

Let $x \in (A - B) - C$.

This means:

• $x \in A - B$, which implies $x \in A$ and $x \notin B$.
• Additionally, $x \notin C$.

We need to show that $x \in (A - C) - (B - C)$.

• Since $x \in A$ and $x \notin C$, we have $x \in A - C$.
• Since $x \notin B$ and $x \notin C$, it follows that $x \notin B - C$.

Therefore, $x \in (A - C) - (B - C)$.

2. Prove that $(A - C) - (B - C) \subseteq (A - B) - C$

Let $x \in (A - C) - (B - C)$.

This means:

• $x \in A - C$, which implies $x \in A$ and $x \notin C$.
• Additionally, $x \notin B - C$, which implies $x \notin B$ or $x \in C$.

We need to show that $x \in (A - B) - C$.

• Since $x \in A$ and $x \notin B$, we have $x \in A - B$.
• Since $x \notin C$, we have $x \notin (A - B) - C$.

Therefore, $x \in (A - B) - C$.

Conclusion

Since we have shown both $(A - B) - C \subseteq (A - C) - (B - C)$ and $(A - C) - (B - C) \subseteq (A - B) - C$, it follows that $(A - B) - C = (A - C) - (B - C)$.

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Would you like more details on this proof or have any other questions?

Here are 5 related questions to consider:

1. What are the properties of set difference that were used in this proof?
2. Can the equality $(A - B) - C = (A - C) - (B - C)$ be generalized to more than three sets?
3. How does the symmetric difference of two sets relate to set difference?
4. How would the proof change if $B$ or $C$ were empty sets?
5. What are some real-world applications of set difference in computer science?

Tip: When working with set operations, drawing Venn diagrams can be a helpful way to visualize the relationships between the sets.