3.	For every two sets A and B,
(A∪B) – (A∩B) = (A–B) ∪ (B–A).

Yes, the given set equation is valid. Let's prove it step by step.

### Theorem:
For any two sets \( A \) and \( B \),
\[
(A \cup B) - (A \cap B) = (A - B) \cup (B - A).
\]

### Proof:

#### Left-hand side (LHS):
\[
(A \cup B) - (A \cap B)
\]
This is the set difference between the union of \( A \) and \( B \), and the intersection of \( A \) and \( B \).

- \( A \cup B \) represents all elements in \( A \) or \( B \) (or both).
- \( A \cap B \) represents all elements common to both \( A \) and \( B \).
  
So, \( (A \cup B) - (A \cap B) \) represents all elements in \( A \cup B \) that are not in \( A \cap B \). This means we are removing all elements that are common to both sets, leaving behind the elements that are in \( A \) but not in \( B \), and elements that are in \( B \) but not in \( A \).

Thus,
\[
(A \cup B) - (A \cap B) = \{ x \in A \cup B \ | \ x \notin A \cap B \}.
\]

#### Right-hand side (RHS):
\[
(A - B) \cup (B - A)
\]
This is the union of the set differences \( A - B \) and \( B - A \).

- \( A - B \) represents all elements in \( A \) that are not in \( B \).
- \( B - A \) represents all elements in \( B \) that are not in \( A \).

Thus, the union of \( A - B \) and \( B - A \) gives us all elements that are in \( A \) but not in \( B \), or in \( B \) but not in \( A \). In other words,
\[
(A - B) \cup (B - A) = \{ x \in A \ | \ x \notin B \} \cup \{ x \in B \ | \ x \notin A \}.
\]

#### Conclusion:
Both sides of the equation describe the set of elements that are either in \( A \) or \( B \), but not in both. Therefore, we have shown that
\[
(A \cup B) - (A \cap B) = (A - B) \cup (B - A).
\]

### The theorem is proven. ✅

Would you like any further details on this proof?

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Here are 5 related questions to explore further:

1. How would the equation change if the universal set \( U \) is involved?
2. Can we extend this set identity to three sets \( A \), \( B \), and \( C \)?
3. What is the symmetric difference of two sets, and how does it relate to this identity?
4. How can this identity be visualized using a Venn diagram?
5. What are some applications of set operations in real-life problems?

**Tip:** Always verify set identities by considering examples with specific sets to gain better intuition.

For every two sets A and B, (A∪B) – (A∩B) = (A–B) ∪ (B–A).

Explore the step-by-step proof for the set identity (A∪B) – (A∩B) = (A–B) ∪ (B–A). This problem covers key concepts in set theory, including union, intersection, and difference, and is suitable for students in Grades 9-12.

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