Simplify using set identities [B \cup A^c \ (B \cap A^c)]^c

We are tasked with simplifying the following expression using set identities:

\[
[B \cup A^c \ (B \cap A^c)]^c
\]

### Step-by-Step Simplification:

1. **Break down the initial expression:**
   The expression inside the brackets can be written as:
   \[
   B \cup (A^c \setminus (B \cap A^c))
   \]
   Here, \( A^c \setminus (B \cap A^c) \) represents the difference of sets, i.e., elements in \( A^c \) but not in \( B \cap A^c \).

2. **Simplify the difference of sets:**
   Using the identity \( A \setminus B = A \cap B^c \), we get:
   \[
   A^c \setminus (B \cap A^c) = A^c \cap (B \cap A^c)^c
   \]
   Simplify \( (B \cap A^c)^c \) using De Morgan's Law:
   \[
   (B \cap A^c)^c = B^c \cup A
   \]
   Therefore:
   \[
   A^c \setminus (B \cap A^c) = A^c \cap (B^c \cup A)
   \]
   Now, distribute the intersection:
   \[
   A^c \cap (B^c \cup A) = (A^c \cap B^c) \cup (A^c \cap A)
   \]
   Since \( A^c \cap A = \emptyset \), we are left with:
   \[
   A^c \setminus (B \cap A^c) = A^c \cap B^c
   \]

3. **Substitute back into the main expression:**
   Now we substitute \( A^c \cap B^c \) back into the original expression:
   \[
   B \cup (A^c \cap B^c)
   \]
   Using the distributive property:
   \[
   B \cup (A^c \cap B^c) = (B \cup A^c) \cap (B \cup B^c)
   \]
   Since \( B \cup B^c = U \) (where \( U \) is the universal set), we get:
   \[
   B \cup (A^c \cap B^c) = B \cup A^c
   \]

4. **Apply the complement:**
   Now apply the complement to the simplified expression:
   \[
   (B \cup A^c)^c
   \]
   Using De Morgan's Law again:
   \[
   (B \cup A^c)^c = B^c \cap A
   \]

### Final Answer:
The simplified expression is:

\[
B^c \cap A
\]

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

Here are 5 related questions to deepen understanding:
1. How does De Morgan's Law help in simplifying set expressions?
2. What is the significance of the complement in set theory?
3. Can you prove the identity \( A \setminus B = A \cap B^c \) with an example?
4. How does the distributive property apply in set operations?
5. What is the result of \( (A^c)^c \), and how does it simplify?

**Tip**: When simplifying set expressions, always try applying De Morgan's Laws and distributive properties early—they often lead to major simplifications!

Simplify using set identities [B ∪ A^c \ (B ∩ A^c)]^c

Learn how to simplify the set expression [B ∪ A^c \ (B ∩ A^c)]^c using set identities. This problem involves core concepts like De Morgan's Law, set difference, and set complements. Suitable for advanced high school or early college students.

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