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Unformatted text preview: 1 Contradiction 1.1 Welcome Welcome to the ninth module in our course on proofs, Contradiction . Before going on, please do the assigned readings. 1.2 Introduction So far we have used the forwards-backwards method to prove statements. There are times when this is difficult. A proof by contradiction provides a new method. 1.3 How To Use Contradiction Suppose that we wish to prove that the statement “ A implies B ” is true. We assume that A is true. We must show that B is true. What would happen if B were true, but we assumed it was false, and continued our reasoning based on the assumption that B was false? Since a mathematical statement cannot be both true and false, it seems likely we would eventually encounter a mathematically non- sensical statement. Then we would ask ourselves “How did we arrive at this nonsense?” and the answer would have to be that our assumption that B was false was wrong and so B is, in fact, true. Proof by contradiction structures proofs in exactly this way. Proceed as follows. 1. Assume that A is true. 2. Assume that B is false, or equivalently, assume that NOT B is true. 3. Reason forward from A and NOT B to reach a contradiction. Unfortunately, it is not always clear what contradiction to find, or how to find it. What is more clear is when to use contradiction. 1.4 When To Use Contradiction The general rule of thumb is to use contradiction when the statement NOT B gives you useful information. There are typically two instances when this is useful. The first instance is when the statement B is one of only two alternatives. For example, if the conclusion B is the statement f ( x ) = 0 then the only two possibilities are f ( x ) = 0 and f ( x ) 6 = 0....
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- Spring '08
- Logic, Prime number, Proposition, Euclid