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hw3-sol - CS 173 Discrete Structures Fall 2011 Homework 3...

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CS 173: Discrete Structures, Fall 2011 Homework 3 - Solution This homework contains 4 problems worth a total of 40 points. It is due on Friday, September 16th at 4pm. When writing your proofs, be sure to use the definitions of key concepts (e.g. divisible) as presented in class. Also one goal of this problem set is to practice certain proof techniques. So be sure to use the proof technique specified by the problem instructions, even if there might be other ways to prove the claim. 1. Proof by cases [8 points] Problem 5 from Homework 2 defined the “extended real numbers.” An extended real number has the form a + , where ǫ is a special new positive number whose square is zero. To compare the size of two extended real numbers, we use the definition: a + bǫ < c + whenever either a < c , or a = c and b < d . Using this definition and proof by cases, prove the following claim: Claim: For any extended real numbers a + , c + , and p + , if a + bǫ < c + and c + dǫ < p + , then a + bǫ < p + . Solution: Suppose that a + , c + and p + are extended real numbers such that a + bǫ < c + and c + dǫ < p + .
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