https://file.aiquickdraw.com/mathbot/file/1727416456993pgsglq4m.jpg

The expression in the image is:

\[
(A - B) \cup \left[(B \cap C') \cap \left((B' \cup A) - B\right)\right]
\]

This is a set-theoretic expression involving the following operations:
1. **Difference** (\(A - B\)): The set of elements in \(A\) but not in \(B\).
2. **Union** (\(\cup\)): The set containing elements from either of the involved sets.
3. **Intersection** (\(\cap\)): The set containing only the elements common to both sets.
4. **Complement** (\(C'\)): The set of elements not in \(C\).
5. **Complement** (\(B'\)): The set of elements not in \(B\)).

To simplify this expression step by step:
1. **Start with the inner expression** \((B' \cup A) - B\):
   - Find the union \(B' \cup A\).
   - Subtract \(B\) from the result of \(B' \cup A\), which leaves elements in \(B' \cup A\) but not in \(B\).

2. **Now focus on** \((B \cap C')\):
   - Find the intersection between \(B\) and the complement of \(C\) (i.e., elements in \(B\) but not in \(C\)).

3. **Then compute** \((B \cap C') \cap ((B' \cup A) - B)\):
   - Find the intersection of the two sets calculated above.

4. **Finally, compute the union** \((A - B) \cup \left[\text{previous result}\right]\):
   - This gives the final set, which includes elements from \(A - B\) and the result from the intersection of \((B \cap C')\) and \(((B' \cup A) - B)\).

Would you like to go through a detailed breakdown of each step, or ask about specific parts of the expression?

Here are some related questions:
1. How do you find the complement of a set?
2. What is the result of the difference between two sets?
3. What properties of union and intersection can simplify this expression?
4. Can De Morgan's laws be applied to the expression?
5. How would you interpret the set operations geometrically using a Venn diagram?

**Tip:** Visualizing set operations with Venn diagrams can make it easier to understand and simplify complex set expressions.

Simplify the set-theoretic expression: (A - B) ∪ [(B ∩ C') ∩ ((B' ∪ A) - B)]

Explore the step-by-step process to simplify the set-theoretic expression (A - B) ∪ [(B ∩ C') ∩ ((B' ∪ A) - B)], covering key operations like difference, union, intersection, and complement. Suitable for advanced high school or undergraduate students.

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