unit 1 introduction to proofs
TRANSCRIPT
![Page 1: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/1.jpg)
Introduction to proofs
![Page 2: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/2.jpg)
terminology
• Theorem/facts/results:• A statement of some importance that can be shown to be
true• Propositions:• Less important statements which can be shown to be true
• Proof• A valid argument that establishes the truth of the theorem
![Page 3: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/3.jpg)
• Lemma• A less important theorem that is helpful in the proof of other results• Corollary• Theorem that can be established directly from a theorem that has
been proved• Conjecture• A statement that is being proposed to be true statement on basis of
partial evidence, intuition of an expert
![Page 4: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/4.jpg)
Methods of proving theorems
• Direct proof• Proofs by contraposition• Vacuous and trivial proofs• Proofs of equivalence• counterexamples
![Page 5: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/5.jpg)
Direct proof
• A direct proof of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed u• A direct proof shows that a conditional statement p → q
is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs. Using rules of inference, with the final step showing that q must also be true
![Page 6: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/6.jpg)
Definition of even and odd integers
• The integer n is even if there exists an integer k such that n = 2k, and n is odd if there exists an integer k such that n = 2k + 1
![Page 7: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/7.jpg)
Same parity
• Two integers have the same parity when both are even or both are odd; they have opposite parity when one is even and the other is odd.
![Page 8: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/8.jpg)
example
• Give a direct proof of the theorem “If n is an odd integer, then n2 is odd.”
![Page 9: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/9.jpg)
example
• Give a direct proof that if m and n are both perfect squares, then nm is also a perfect square.
![Page 10: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/10.jpg)
Proof by contraposition
![Page 11: Unit 1 introduction to proofs](https://reader038.vdocuments.us/reader038/viewer/2022102720/587e58e01a28abeb1a8b75f5/html5/thumbnails/11.jpg)
Mistakes in proofs