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Unformatted text preview: Math 280 Strategies of Proof Fall 2007 Class Notes 3.8 Proof by Contradiction Proof by contradiction is a method of indirect proof based upon the fact that a statement p must be either true or false but cannot be both. If we can show that assuming the statement p is false leads to a logical contradiction, a statement known to be false, then p must in fact be true. The logical basis for a proof by contradiction is the implication [ p ( r r )] p, contained in Theorem 1.3.1(h). Since this is a tautology, if we can show that the hypothesis p ( r r ) is true, then the conclusion p of the implication must be true. In order to show that the hypothesis is true, we suppose p is true or equivalently, the statement p is false. We then show that this leads to a contradiction r r for some statement r . We can then conclude that the statement p is true. This method of proof is also known as reductio ad absurdum , reduction to an absurdity, or reductio ad impossibile , reduction to an impossibility, the point being that our assumption that the original statement is false leads us to deduce an absurdity or impossibility. The exact form of a proof by contradiction will depend upon the specific form of the statement to be proved, but all such proofs follow the following basic outline. Proof by Contradiction of a Statement p 1. Suppose the statement p is false or, equivalently, the negation p is true. The statement p will typically involve quantifiers and logical connectives, so we usually deter- mine the negation p , and then suppose the negation is true. 2. Show that this assumption leads to a contradiction. We need to deduce a contradiction r r from the supposition that p is false. Note that the statement r is not specified at the beginning of the proof but is usually determined from the supposition. Either r or r may in fact be part of the supposition. 3. Conclude that the statement p is true. A contradiction arising out of our supposition that p is false enables us to conclude that p must be true. The proof is then complete once we obtain the contradiction. With any proof, its important to keep the reader informed about the procedure being used. Its particularly important with a proof by contradiction since you are starting by assuming that the statement to be proved is false. One of the classic proofs by contradiction dating back to the fourth century b.c. is the proof that 2 is irrational. The version of the proof in the following example is similar to that in the tenth book of Euclids Elements of Geometry . Euclids proof uses the fact that any fraction a/b of two integers can be written in lowest terms, that is, in such a way that the integers a and b have no common divisors. The reason this is true is that if the numerator a and the denominator b have a common divisor, then they must have a greatest common divisor d (since the set of all common divisors is finite). We may then cancel...
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This note was uploaded on 10/20/2009 for the course MATH 280 taught by Professor Solheid during the Spring '08 term at CSU Fullerton.
- Spring '08
- Indirect Proof