L5[1] - Review Recall: Proof methods for a statement p q...

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Review Recall: Proof methods for a statement Direct: Assume is true, prove is true. By Contraposition: Assume is true, prove that is true. By Contradiction: Assume is true, assume is true. If a contradiction is found, the proof is complete. pq p q q p p q
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Exercise Prove by contradiction that is irrational. (on board) 2
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Proof about Primes Prove: There are infinitely many prime numbers. Proof : (Euclid, ≈300 B.C., by contradiction ) Assume there are a finite number of primes. Then we can list them: p 1 ,p 2 ,p 3 ,…,p n . Consider the number q =p 1 p 2 p 3 ∙∙∙ p n +1. This number q is not divisible by any of the listed primes. So we conclude that q must also be prime. This contradicts our assumption that all of the primes are in the list.
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Mistakes in Proofs I Q: What is wrong with this? Prove that 1=2: “Proof” : Let a and b be two equal integers. 1) a=b 2) a 2 =ab 3) a 2 -b 2 =ab-b 2 4) (a-b)(a+b)=b(a-b) 5) a+b=b 6) 2b=b (since a=b) 7) 2=1 A: In the 4->5 transition we divide by a-b, which equals zero.
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Mistakes in Proofs II Q: What is wrong with this proof? Prove: If n 2 is even, then n is even. 1) Suppose n 2 is even. 2) Then n 2 =2k for some integer k. 3) Let n=2h, for some integer h. 4) Then n is even. A: In creating Step 3, we have assumed exactly what we are trying to prove! Note that the conditional statement is actually true here, but the method of showing it is faulty.
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Mistakes in Proofs II (cont‟d) The faulty reasoning used in the previous slide is often referred to as circular reasoning. This is also based on a fallacy called begging the question. This occurs when one or more steps of the proof are based on the truth of that which you are trying to prove.
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Proofs of Biconditional Statements Consider the statement “ p if and only if q ”. To prove a biconditional statement we must prove two things individually: 1) We must show “if p then q 2) We also must show “if q then p” Each of these two can be demonstrated by any method of proof.
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Vacuous Proofs Consider the conditional “if p then q ”.
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L5[1] - Review Recall: Proof methods for a statement p q...

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