The conversation between human and AI assistant.
How can I prove that (A × B) − (C × D) = (A − C) × B ∪ A × (B − D)?
Okay, but that still doesn't prove the original equation, right?
Almost there. Now we just need to show that (A × B) − (C × D) is equal to (A × B − C × B) ∪ (A × B) − A × D, which we can do by using the fact that set difference distributes over set union. We get (A × B) − (C × D) = (A × B) − ((C × B) ∪ A × D), which shows that the left side is equal to the right side.