contradiction - Thus the work is completed after finding...

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Theorem Proofs Using the Method of Contradiction As always, in one form or another, a theorem being proved boils down to the following structure: Hypotheses Conclusion: Then Statement A where A is the statement to be proven using the hypotheses and previous results. A Proof by Contradiction begins with a comment equivalent to “Assume not A” where “not A” is the logical complement of A. It is as if you are adding “not A” to the hypotheses. The goal no longer is to prove A. Rather one is to prove a statement that is the complement of one of the hypotheses, that is, show that at least one hypothesis is false. When this happens, a contradiction has occurred since, by definition of hypothesis, all the hypotheses are assumed to be true. What caused this to happen? Since the hypotheses are true, the only possibility is that the assumed hypothesis “not A” is false, that is, A is true and the theorem is proved.
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Unformatted text preview: Thus the work is completed after finding the contradiction. The proof usually terminates at this point with a comment equivalent to “This is a contradiction,” simply to emphasize to the reader that your goal has been achieved. It is natural to ask when a proof by contradiction is the best approach. This is not at all obvious. As you gain experience with proofs, you will sometimes get a feeling. At other times you may have to try various approaches in order to find one that shows your result. It never is wise to restrict yourself to a single method, trying to force all proofs into that one approach. The question becomes trickier when you are not sure whether a proposed theorem is true or not. This often happens when conducting research. In that case you may alternate between attempting to prove the theorem by any method you deem promising, including contradiction, and attempting to find a counterexample to it....
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