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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.
 Fall '08
 STAFF
 Math, Calculus

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