104-exam1-s - Math 104 - Lecture 1 - Exam 1 Solutions...

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Math 104 - Lecture 1 - Exam 1 Solutions Monday, July 12, 2010 1. Recall that for any set S , the power set of S , denoted P ( S ), is the collection of all sub- sets of S . That is P ( S ) = { X S } . Prove that if A and B have the same cardinality, then P ( A ) and P ( B ) have the same cardinality. Let f : A B be a bijection. Define g : P ( A ) → P ( B ) by g ( X ) = f [ X ] = { f ( x ) : x X } . Since f is a bijection, f inverse exists. Then the map h ( Y ) = f - 1 [ Y ] is an inverse of g because h ( g ( X )) = f - 1 h f [ X ] ± = X and g ( h ( Y )) = f h f - 1 [ Y ] i = Y . 2. Prove that if a < b are real numbers, then there is a rational number q so that a < q < b . (Do not use the theorem in the book which includes this fact.) Let a be the cut A | A 0 and b be the cut B | B 0 . Since a < b , A ( B . Pick q 0 B \ A . Since B is a the left side of a cut, pick some q B q > q 0 . Then a < q < b as desired. 3. Prove or disprove:
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104-exam1-s - Math 104 - Lecture 1 - Exam 1 Solutions...

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