1873_solutions

3 strictly speaking there are two cases to consider

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Unformatted text preview: s into two cases and compete the proof in each of these cases. To obtain a contradiction we assume that the condition 0  x  2 is false. That means that either x 0 or x 2 and so we must consider two cases. The case x 0: In this case, since x 2 2x  x x 2 and since x 0 and x 2  0 we have x 2 x 0, contradicting the assumption that x 2 x  0. The case x 2: In this case, since x 2 2x  x x 2 and since x  0 and x 2 0 we have x 2 x 0, contradicting the assumption that x 2 x  0. Since each of the two cases leads to a contradiction our proof is complete. 4. Suppose that f is a given function defined on the interval 0, 1 and suppose that we wish to prove that this function f has the property that there exists a number   0 such that whenever t and x belong to the interval 0, 1 and |t x |  , we have |f t f x |  1. Write down the first line of a proof of this assertion that uses the method of proof by contradiction. Solution: To obtain a contradiction, suppose that it is impossible to find a number   0 such that whenever t and x belong to the interval 0, 1 and |t x |   we have |f t f x |  1. Alternative Solution: To obtain a contradiction, suppose that for every number   0 there exist numbers t and x in the interval 0, 1 such that |t x |  1 and |f t f x | 1. 5. Suppose that x 1 , x 2 , x n is a subset of a vector space V and that we wish to prove that the set x 1 , x 2 , x n is linearly independent. Write down the first line of a proof of this assertion that uses the method of proof by contradiction. (Try to be specific. Don’t just suppose that the set is linearly dependent.) The assertion that the set x 1 , x 2 , x n is linearly dependent says that there exist numbers c 1 , c 2 , , c n that are not all zero such that n cjxj  0 j1 where 0 is the “origin" in the vector space. A proof by contradiction that x 1 , x 2 , x n is linearly independed could begin as follows: To obtain a contradiction, assume that x 1 , x 2 , x n is linearly dependent. Choose numbers c 1 , c 2 , , c n that are not all zero such that n cjxj  0 j1 and, using the fact that not all of the numbers c 1 , c 2 , , c n are zero, choose j such that c j Some Additional Exercises In each of the following exercises, decide whether the statement is true or false and then write a carefully 23 0. worded proof to justify your assertion. 1. For every number x 0, 1 there exists a positive integer N such that for every number integer n N we have nx . 1  n2x2  0 and every Solution: This statement is false. The denial of the statement says that there exists a number 0, 1 such that for every positive integer N there exists a number  0 and an integer n N such that nx . 1  n2x2 To prove this denial we shall give an example of a number x 0, 1 such that for every positive integer N there exists a number  0 and an integer n N such that nx . 1  n2x2 x We define x  1 2 . We now want to prove that something happens for every positive integer N and so we continue: Suppose t...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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