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.