To simplify the given formula:

The formula is:

$(A \setminus (B \cup C)) \setminus (\overline{A} \cup (C \setminus B))$

Step-by-Step Simplification:

1. Rewrite using set operations:
   The formula can be broken into parts. The operator \setminus means "set difference." So, for two sets $X \setminus Y$, it means $X \cap \overline{Y}$, where $\overline{Y}$ is the complement of $Y$. Let's simplify each part:

   $A \setminus (B \cup C) = A \cap \overline{(B \cup C)} = A \cap (\overline{B} \cap \overline{C})$

   So the first part becomes: $A \cap \overline{B} \cap \overline{C}.$

2. Simplify the second part:
   The second part of the formula is: $\overline{A} \cup (C \setminus B).$

   Here, $C \setminus B = C \cap \overline{B}$, so: $\overline{A} \cup (C \cap \overline{B}).$

3. Combine the two parts:
   Now the formula becomes: $(A \cap \overline{B} \cap \overline{C}) \setminus (\overline{A} \cup (C \cap \overline{B})).$

   Using the set difference property, this becomes: $(A \cap \overline{B} \cap \overline{C}) \cap \overline{(\overline{A} \cup (C \cap \overline{B}))}.$

4. Simplify the complement:
   The complement of $\overline{A} \cup (C \cap \overline{B})$ is: $\overline{\overline{A}} \cap \overline{(C \cap \overline{B})}.$

   Using De Morgan's laws: $\overline{\overline{A}} = A, \quad \text{and} \quad \overline{(C \cap \overline{B})} = \overline{C} \cup B.$

   So: $\overline{(\overline{A} \cup (C \cap \overline{B}))} = A \cap (\overline{C} \cup B).$

5. Combine everything:
   Substituting back, the formula becomes: $(A \cap \overline{B} \cap \overline{C}) \cap \left( A \cap (\overline{C} \cup B) \right).$

   Distribute the intersection: $(A \cap \overline{B} \cap \overline{C}) \cap A \cap (\overline{C} \cup B).$

   Since $A \cap A = A$, we have: $A \cap \overline{B} \cap \overline{C} \cap (\overline{C} \cup B).$

6. Simplify further:
   Use the distributive property: $(\overline{C} \cap \overline{C}) \cup (\overline{C} \cap B) = \overline{C} \cup (\overline{C} \cap B).$

   Combining terms: $A \cap \overline{B} \cap \overline{C}.$

Final Simplified Form:

$A \cap \overline{B} \cap \overline{C}.$

This is the simplified result.

Would you like me to break down any specific step further or solve a related example?

Related Questions:

1. What are De Morgan's laws and how are they applied to set complements?
2. How does the set difference operator $X \setminus Y$ translate into intersections and complements?
3. What is the role of distributive properties in simplifying set expressions?
4. How do complements interact with unions and intersections in set theory?
5. Can we visualize $A \cap \overline{B} \cap \overline{C}$ using Venn diagrams?

Tip:

Always translate set differences into complements and intersections to simplify complex set operations step-by-step!