A△B△C = (A − (B ∪ C)) ∪ (B − (C ∪ A)) ∪ (C − (A ∪ B)) ∪ (A ∩ B ∩ C)

The expression \( A \triangle B \triangle C \) represents the **symmetric difference** between the sets \( A \), \( B \), and \( C \). Symmetric difference means the elements that are in one of the sets but not in both, extended across more sets.

To break it down:

### Symmetric difference between two sets:
For two sets, \( A \triangle B \) (symmetric difference) is defined as:
\[
A \triangle B = (A - B) \cup (B - A)
\]
This gives the elements that are in either \( A \) or \( B \), but not in both.

### Symmetric difference for three sets \( A \triangle B \triangle C \):
Now, for three sets, \( A \triangle B \triangle C \), the formula is more complex:
\[
A \triangle B \triangle C = (A - (B \cup C)) \cup (B - (C \cup A)) \cup (C - (A \cup B))
\]
Additionally, to include elements that might belong to all three sets (but without double-counting):
\[
A \triangle B \triangle C = (A - (B \cup C)) \cup (B - (C \cup A)) \cup (C - (A \cup B)) \cup (A \cap B \cap C)
\]
This is precisely the expression you have provided. It breaks down into:
- \( A - (B \cup C) \): Elements in \( A \), but not in \( B \) or \( C \).
- \( B - (C \cup A) \): Elements in \( B \), but not in \( C \) or \( A \).
- \( C - (A \cup B) \): Elements in \( C \), but not in \( A \) or \( B \).
- \( A \cap B \cap C \): Elements that are in all three sets.

### Key Takeaway:
The formula combines the pairwise symmetric differences with the common intersection of all three sets, ensuring all non-overlapping and overlapping portions are covered.

Let me know if you need further details or clarifications!

### 5 Related Questions:
1. How would the symmetric difference between two sets \( A \triangle B \) be represented graphically using Venn diagrams?
2. How does the formula for symmetric difference generalize for more than three sets?
3. What are the properties of the symmetric difference, such as commutativity and associativity?
4. How can symmetric differences be useful in computer science or data analysis?
5. What is the difference between symmetric difference and union or intersection?

### Tip:
The symmetric difference operation is associative and commutative, meaning the order of the sets doesn't matter when computing it.

Explore how to compute the symmetric difference for three sets with the formula A△B△C = (A − (B ∪ C)) ∪ (B − (C ∪ A)) ∪ (C − (A ∪ B)) ∪ (A ∩ B ∩ C). This guide covers key set theory concepts and formulas, perfect for students in Grades 10-12.

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