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mid1prac - 18.100B and 18.100C Fall 2011 First midterm...

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18.100B and 18.100C Fall 2011 First midterm review sheet The first midterm covers all the material from chapters one and two, except for the appendix to chapter one. On the midterm you will be asked to give the proof, or a part of the proof, of one or two of these theorems: 1.11, 1.20, 2.12, 2.14, 2.19, 2.23, 2.24, 2.28, 2.34, 2.35, 2.36, 2.37. You will also be asked to solve several of the following problems. You may use results from chapters 1 and 2 of Rudin without proof. You may not use results from problem sets or from the exercises at the ends of the chapters. Note that these problems are not arranged in order of difficulty! 1. Problem 9 from page 22. 2. Give an alternative proof that | x · y | ≤ | x || y | for any x, y R n and for any n N by using the fact that the dot product of x + ty with itself is nonnegative for any t R . ( Hint: Complete the square.) 3. Let A be a nonempty set, and let P ( A ) be the set of subsets of A . Prove that the cardinality of P ( A ) is strictly greater than that of A using the following two steps:
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