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ma8_wk1_wednesday_notes_2010

# ma8_wk1_wednesday_notes_2010 - MATH 8 SECTION 1 WEEK 1...

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2 TA: PADRAIC BARTLETT 3. Proofs by Contradiction, Part II Last time, we described proofs by contradiction as the following process: suppose we want to prove that some statement P is true. How can we do this? Well, there are only two possiblities: either P is true, or ¬ P is. So: if we show that ¬ P is impossible – in other words, that assuming ¬ P leads to a contradiction – then by process of elimination we have tht P must hold! On Monday, we gave one example of a proof by contradiction; today, we have two more proofs to further illustrate the method. Lemma 3.1. (Euclid) There are infinitely many prime numbers. Proof. (N.b.: for the entertainment of those who’ve already encountered Euclid’s proof about prime numbers, the proof below is written in the “way of prime” – i.e. all of the words below contain prime numbers of letters. Silly, yes. But fun!) Suppose not: in other words, suppose n prime numbers exist, for n N . Label these prime numbers p 1 , p 2 , . . . p n , and examine the product p 1 · p 2 · . . . · p n .
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ma8_wk1_wednesday_notes_2010 - MATH 8 SECTION 1 WEEK 1...

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