For any two sets A and B Prove the following Identity A-(A∩B) = A-B

To prove the identity \( A - (A \cap B) = A - B \) for any two sets \( A \) and \( B \), we will show that each side of the equation is a subset of the other.

### Proof:

#### Step 1: Show that \( A - (A \cap B) \subseteq A - B \)

Let \( x \in A - (A \cap B) \). 

This means:
- \( x \in A \) (because \( x \) is in the set \( A - (A \cap B) \))
- \( x \notin A \cap B \) (because \( x \) is not in the intersection)

Since \( x \notin A \cap B \), \( x \) is not in both \( A \) and \( B \). However, since \( x \in A \), the only possibility is that \( x \notin B \).

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

Thus, \( A - (A \cap B) \subseteq A - B \).

#### Step 2: Show that \( A - B \subseteq A - (A \cap B) \)

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

This means:
- \( x \in A \) (because \( x \) is in \( A - B \))
- \( x \notin B \) (because \( x \) is not in \( B \))

Since \( x \notin B \), it cannot be in \( A \cap B \) because the intersection \( A \cap B \) consists of elements that are in both \( A \) and \( B \).

Therefore, \( x \in A - (A \cap B) \).

Thus, \( A - B \subseteq A - (A \cap B) \).

#### Conclusion:
Since we have shown both:
\[ A - (A \cap B) \subseteq A - B \quad \text{and} \quad A - B \subseteq A - (A \cap B), \]

we can conclude that:
\[ A - (A \cap B) = A - B. \]

This completes the proof.

Would you like further details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How does the identity \( A - (A \cup B) \) compare to \( A - B \)?
2. What is the relationship between the complement of \( A \) and \( A - B \)?
3. How can De Morgan's laws be applied to the identity \( A - (A \cap B) = A - B \)?
4. Can you prove the identity \( A - (B \cap C) = (A - B) \cup (A - C) \)?
5. What happens to the identity \( A - (A \cap B) = A - B \) when \( A \) is a subset of \( B \)?

**Tip:** When working with set identities, using Venn diagrams can often provide an intuitive understanding of why the identities hold.

Explore the proof of the set identity A - (A ∩ B) = A - B for any two sets A and B. This detailed explanation shows how each side of the equation is a subset of the other, demonstrating fundamental concepts in set theory and set operations.

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