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Unformatted text preview: Proofs Crash Course Winter 2011 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 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 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. 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. 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 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). 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. 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. 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. 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. 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. O Girls don’t date jerks. Deductive Proof Example How would you prove that Jerry won’t get a family if he’s a jerk?...
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This note was uploaded on 02/09/2012 for the course MATH 112 taught by Professor Jarvis during the Winter '08 term at BYU.
 Winter '08
 JARVIS
 Calculus

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