Distributing difference over intersection: $(A - B) \cap (C - D) = A \cap C - B \cap C - A \cap D$

Question

Was reading Topology by Munkres Chapter 1 today, and came across the following:

$$(A - C) \times (B - D) = A \times B - C \times B - A \times D$$

The proof made me think whether $$(A - B) \cap (C - D) = A \cap C - B \cap C - A \cap D$$ was correct, here is my proof:

$(A - B) \cap (C - D) = ((A - B) \cap C) - ((A - B) \cap D) = A \cap C - B \cap C - (A \cap D - B \cap D) = (A \cap C - B \cap C - A \cap D) \cup ((A \cap C - B \cap C) \cap (B \cap D)) = (A \cap C - B \cap C - A \cap D) \cup ((A - B) \cap C \cap B \cap D) = A \cap C - B \cap C - A \cap D$

where it uses the facts:

• $(A - B)\times C = A\times C - B\times C$
• $A-(B-C) = (A-B)\cup(A\cap C)$

Is the statement/my proof correct? Couldn't find anything on google, only the first point.

Answer 1

Yes, both the statement and your proof of it are correct.