The problem asks which of the provided set expressions is correct. Let's go over each option one by one:

1. $A \times (B \cap C) = (A \cap B) \cup (A \times C)$:

   • This is the Cartesian product on the left-hand side and a union of sets on the right-hand side. Cartesian products and intersections generally don't follow this pattern, so this expression is incorrect.

2. $(B \setminus A) \cap (C \setminus A) = (B \cap C) \setminus A$:

   • This appears to involve the distributive property of set difference and intersection. We can verify this as correct since distributing the intersection over the set difference would hold in this case. Thus, this is a valid identity.

3. $A \cup (B \times C) = (A \times B) \cup (A \times C)$:

   • On the left-hand side, we have a union of set $A$ with a Cartesian product of sets $B \times C$, and on the right-hand side, we have two Cartesian products. The two sides are not generally equal, so this is incorrect.

4. $(A \cap B) \cup (C \setminus \bar{B}) = (A \cup C) \setminus B$:

   • This expression mixes intersections, unions, and complements, but does not follow any set-theoretic identity. Thus, it is incorrect.

Given this analysis, the second option is the correct one: $(B \setminus A) \cap (C \setminus A) = (B \cap C) \setminus A$

Would you like a more detailed breakdown of any specific step?

5 Questions for Further Thought:

1. What are the key properties of Cartesian products in set theory?
2. How does the distributive property work for intersections and set differences?
3. Why are unions and Cartesian products typically not distributive?
4. How can we prove set-theoretic identities using Venn diagrams?
5. What are common mistakes to avoid when manipulating set expressions?

Tip: When dealing with set expressions, always try to apply basic properties like distribution, complement, and De Morgan’s laws to simplify complex expressions.