Use the element method for proving that a set equals the empty set to prove the following statement. (Assume that all sets are subsets of a universal set U.) If U denotes a universal set, then UC = ø....

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Extracted text: Use the element method for proving that a set equals the empty set to prove the following statement. (Assume that all sets are subsets of a universal set U.) If U denotes a universal set, then UC = ø. Proof by contradiction: Consider the sentences in the following scrambled list. But, by definition of a universal set, U contains all elements under discussion, and so x E U. So, by definition of complement x E U. Let U be a universal set and suppose Uº = Ø. Let U be a universal set and suppose UC ± Ø. Then there exists an element x in UC. Thus x EU and x € U, which is a contradiction. So, by definition of complement x ¢ U. But, by definition of a universal set, UC contains no elements. We construct the proof by selecting appropriate sentences from the list and putting them in the correct order. 1. Let U be a universal set and suppose UC = Ø. 2. Let U be a universal set and suppose UC = Ø. 3. ---Select--- 4. ---Select--- 5. Thus x e U and x ¢ U, which is a contradiction. 6. Hence the supposition is false, and so UC = ø.