A B by definition of set difference In either case x A Band so x A B Therefore

# A b by definition of set difference in either case x

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A-Bby definition of set difference. In either case,x /A-Band soxA-B. Therefore,ABA-B.Putting both statementsA-BABandABA-Btogether we have provedA-B=AB.ii.Using Laws of Sets:(AB)(AB) =A(BB)Distributive Law=A∩ UComplement Law=AUsing Subset Containment:Proof.We first prove(AB)(AB)A.Letxbe a specific but arbitrarily-chosen element of(AB)(AB). ThenxABorxAB. Consider two cases:1.xAB: By definition we havexA.2.xAB: By definition we havexA. Homework 3: Introductory Set Theory Solutions (c)Proof.Suppose by way of contradiction thatBAandA6⊂B. Therefore there existsxAsuch thatx /B. However this meansxB. By hypothesis,Bis a subset ofAsoxAas well. But thenx /AandxAis a contradiction.(d) The statement “It follows thatx /Aorx /Bby definition of complement, and sox /ABby definitionof union” is logically incorrect. Imagine the statement • • • 