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.