The conversation between human and AI assistant.
How do you prove that for all sets A, B and C, (A∪B) \C = (A\C) ∪ (B \C)?
That would be great!
Okay, here's the proof: Let x be an arbitrary element in (A∪B) \C. This means that x is in either A or B, but not in C. Therefore, x must either be in A\C or B\C, since it cannot be in C. Thus, x is in (A\C) ∪ (B\C). Conversely, let x be an arbitrary element in (A\C) ∪ (B\C). This means that x is either in A\C or B\C. Therefore, x must be in either A or B, but not in C. Thus, x is in (A∪B)\C. Therefore, we have shown that (A∪B) \C = (A\C) ∪ (B\C) for all sets A, B, and C.