Proofs involving sets

• Read carefully pages 108 through 111 (section 4.4)
• Some key questions to answer (try these without looking at the book, but after you’ve read the book):
  1. How do we prove two sets are equal?
  2. Prove that for sets \(A\), \(B\) we have \(A\setminus B = A\cap \bar B\).
  3. Prove that for sets \(A\), \(B\) we have \((A\cup B)\setminus(A\cap B) = (A\setminus B) \cup (B\setminus A)\).
  4. Prove that for sets \(A\), \(B\), \(C\) if \(A\subseteq B\) and \(B\subseteq C\) then \(A\subseteq C\).
  5. Prove that for sets \(A\), \(B\) we have \(A\cup B = A\) if and only if \(B\subseteq A\). Also devise a different proof of the forward direction, by using that \(B\subseteq A\cup B\).
• Practice problems from section 4.4 (page 116): 4.42, 4.43, 4.45, 4.46, 4.49