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f2006lecNotesWeek04 - 11 11.1 Proof in Mathematics(cont...

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11 Proof in Mathematics (cont.) — Mathematical Induction 11.1 Proof by contradiction • § 3.6 in Epp here, to prove that p is true, we show that assuming p leads to a contradiction classic example: prove that 2 is irrational 11.2 Comparing methods of proof many possible approaches to proving theorems no guarantee that any will work sometimes many approaches work an example prove that x R , if x 2 - 5 x + 4 < 0 then x > 0 a direct proof if x 2 - 5 x + 4 < 0 then x 2 + 4 < 5 x 5 x > x 2 + 4 > 0 5 x > 0 x > 0 an indirect proof — using contrapositive show that x R , if x 0 , then x 2 - 5 x + 4 0 if x 0 then x - 1 0 and x - 4 0 ( x - 1)( x - 4) 0 x 2 - 5 x + 4 0 a proof by contradiction suppose x 2 - 5 x + 4 < 0 and x 0 then x 2 < 5 x - 4 but if x 0 5 x 0 5 x - 4 < 0 so x 2 < 0 but x R , x 2 0 we have a contradiction 11.3 Existence proofs • § 3.1 in Epp useful for proving theorems of the form: x, P ( x ) constructive existence proofs explain e.g. there is at least one positive integer whose value is the sum of its positive divisors (other than itself) 6 = 1 + 2 + 3 25
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non-constructive existence proofs explain e.g. show that there exist irrational numbers x and y such that x y is rational consider x = 2 and y = 2 recall division into cases: ( p q ) ( p r ) ( q r ) r 11.4 Proof by Counterexample useful for proving theorems of the form: ∼ ∀ x, P ( x ) these theorems can be rephrased in the equivalent form: x, P ( x ) we only need find one x for which P ( x ) is false such an x is called a counterexample e.g. Prove that the expression n 2 - n + 41 where n Z + is not always prime. show that it is prime for many values n = 41, however, serves as a counterexample 11.5 Mathematical induction mathematical induction is another proof technique very useful for proving theorems of the form: n Z
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