06 Intro to Proofs

06 Intro to Proofs - CS103A HO #6 Slides-Intro to Proofs...

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CS103A HO #6 Slides--Intro to Proofs 1/14/08 1 What is a proof? Science : "prove" a hypothesis using inductive reasoning, i.e., the scientific method. We gather bits of specific information together and use our own knowledge and experience in order to make an observation about what must be true. Does not produce mathematical certainty. Everyday example: Observation: John came to class late this morning. Observation: John’s hair was uncombed. Prior experience: John is very fussy about his hair. Conclusion: John overslept 1 = 1 1 + 3 = 2 1 + 3 + 5 = 3 1 + 3 + 5 + 7 = 4 Do we ever use inductive reasoning in mathematics? Yes, we use it to form hypotheses. 2 2 2 2 What is a proof? Law : convince one set of 12 people, or a judge, none of whom may be experts in the subject matter, that your explanation is valid. Can use deductive and inductive reasoning, precedent, psychological tricks, etc. What is a proof? Math : we do proofs with deductive reasoning and the rules of logic. A proof is a step-by-step demonstration that a conclusion follows from some premises. Each step is a logical consequence of the ones that come before it. All proofs must be rigorous . That is, each step in a proof must provide definitive evidence that the intermediate conclusion follows from things already established. (a) If my glasses are on the kitchen table, then I saw them at breakfast.
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06 Intro to Proofs - CS103A HO #6 Slides-Intro to Proofs...

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