Show that (A-B)-C=(A-C)-(B-C)Let A,B,C be arbitrary sets 

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.

Show that (A-B)-C=(A-C)-(B-C). Let A, B, C be arbitrary sets.

Explore how to prove the set equality (A-B)-C = (A-C)-(B-C) using fundamental concepts of set theory. This problem demonstrates the properties of set difference and subset relations, suitable for undergraduate students.

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