To simplify the expression \( (¬(A ∪ ¬B) ∖ A) ∩ ¬(B ∪ C) \), let's break it down step by step.

### 1. Simplify \( ¬(A ∪ ¬B) \):
Using De Morgan's law:
\[
¬(A ∪ ¬B) = ¬A ∩ B
\]

### 2. Simplify \( ¬(B ∪ C) \):
Again, using De Morgan's law:
\[
¬(B ∪ C) = ¬B ∩ ¬C
\]

### 3. Subtract \( A \) from \( ¬(A ∪ ¬B) \):
The difference operation \( X ∖ Y \) is equivalent to \( X ∩ ¬Y \), so:
\[
¬(A ∪ ¬B) ∖ A = (¬A ∩ B) ∩ ¬A
\]
Simplifying this:
\[
(¬A ∩ B) ∩ ¬A = ¬A ∩ B ∩ ¬A = ¬A ∩ B
\]

### 4. Combine with \( ¬(B ∪ C) \):
Now, substitute into the original expression:
\[
(¬A ∩ B) ∩ (¬B ∩ ¬C)
\]
Simplify the intersection:
\[
¬A ∩ B ∩ ¬B ∩ ¬C
\]
Note that \( B ∩ ¬B = \emptyset \), so the entire expression simplifies to:
\[
\emptyset
\]

### Final Answer:
The expression simplifies to the **empty set** \( \emptyset \).

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Would you like more details on any step, or do you have any other questions?

Here are 5 related questions for practice:
1. How would you simplify \( (A ∪ B) ∩ ¬(A ∪ C) \)?
2. What is the result of \( (A ∩ ¬B) ∪ (¬A ∩ B) \)?
3. Can you simplify \( (¬A ∪ B) ∩ (A ∪ ¬B) \)?
4. How does De Morgan's law apply to \( ¬(A ∩ B) \)?
5. What is the result of \( A ∖ (A ∩ B) \)?

**Tip:** Always remember that the intersection of a set with its complement (e.g., \( B ∩ ¬B \)) results in the empty set.

Explore the step-by-step solution to simplify the set expression (¬(A ∪ ¬B) ∖ A) ∩ ¬(B ∪ C). Learn about De Morgan's laws, set difference, and set intersection operations. Suitable for high school students in Grades 11-12.

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