104-final-s - Math 104 - Lecture 1 - Final ExamSolutions 1....

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Unformatted text preview: Math 104 - Lecture 1 - Final ExamSolutions 1. Given a set of real numbers X (given as cuts), give a definition of inf X in terms of cuts. Prove that your definition is correct. Assume that X is not empty and bounded below. Let A = R X Left ( ) . Then put inf X = ( A \ { max A } ) | rest of Q . Since X is nonempty, the right side is nonempty. Since X is bounded below, the left side is nonempty. The left side has no maximal ele- ment since we threw it out. Everything in Q is in one side or the other by definition. The left side is less than or equal to the left side of everything in X , so it is a lower bound. Given any lower bound , Left ( ) must be a subset of the left here, so it is the greatest lower bound. 2. Prove or disprove: The set of all nondecreasing functions from Q to { , 1 } is countable. ( f is nondecreasing if x < y = f ( x ) f ( y ).) par False. Call this set of functions X . Then the map f : R X defined by f ( x )( q ) = ( q < x 1 q x is an injection. You can think of the elements ofis an injection....
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104-final-s - Math 104 - Lecture 1 - Final ExamSolutions 1....

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