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proofs_crash_course - Proofs Crash Course Winter 2011...

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Proofs Crash Course Winter 2011
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Today’s Topics O Why are Proofs so Hard? O Proof by Deduction O Proof by Contrapositive O Proof by Contradiction O Proof by Induction
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Why are Proofs so Hard? “If it is a miracle, any sort of evidence will answer, but if it is a fact, proof is necessary -Mark Twain
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Why are Proofs so Hard? O Proofs are very different from the math problems that you’re used to in High School. O Proofs are problems that require a whole different kind of thinking. O Most proofs will not give you all of the information you need to complete them.
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Before we go further… O Understanding the purpose of proofs is fundamental to understanding how to solve them. O Doing proofs is like making a map O The goal is to get from point A to B using paths, roads, and highways. O Proofs show us how two statements logically connect to each other through theorems, definitions, and laws.
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Proof by Deduction “The two operations of our understanding, intuition and deduction, on which alone we have said we must rely in the acquisition of knowledge .” -Rene Descartes
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Proof by Deduction O This is the most basic proof technique. O By using laws, definitions, and theorems you can get from A to B by starting at A and progressively moving towards B. O You start by assuming the conditional (the “if” part) and showing the logical flow to the conclusion (the “then” part).
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Deductive Proof Example Suppose you know the following: if A then B if B then C if C then D Show that if A then D.
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Deductive Proof Example Remember that deductive proofs start at the beginning and proceed towards the conclusion Proof: Assume A is true. Therefore B must be true. Since B is true, C is true. Because C is true, D is true. Hence, D.
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Deductive Proof Analysis O Notice that the path taken to get from A to D was very direct and linear. O We started by assuming that A was true. O Then we used the given “laws” to show that D was true.
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Deductive Proof Example Prove the following statement: If Jerry is a jerk, Jerry won’t get a family. Note: Many of you likely can prove this using some form of intuition. However, in order to definitively prove something, there need to be some agreed upon guidelines.
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Deductive Proof Example If Jerry is a jerk, Jerry won’t get a family. Let’s also suppose that we have some guidelines: O If somebody doesn’t date, they won’t get married. O If you don’t get married, you won’t get a family.
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