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Discrete Math1Proof Techniques2Last updated: Tuesday 16thFebruary, 2016 15:2834Axiomatic Proof5The notion of proof dated back to Euclid and6Archimedes1. The word axiom means something7that we assume to be true.You could however8argue whether the axiom applied to the problem9we are trying to solve or not. If it does then that10is good, but if it does not you pick the wrong set11of axiom.12The axiom usually contains trivial stuff like13you can commute addition (a+b=b+a) or if14a=bandc=dthena+c=b+detc. Another15example would be that the area is the same for16every observer no matter who observe it or no17matter how we move or rotate it.2.18The idea of proof is to show that the state-19ment is true/false beyond reasonable doubt. Let20us work on our first proof21Theorem:Given a right triangle22abc23Then, the length of all three sides are related by24c2=a2+b2Proof:First, we pieces the tringle in a useful25way326abcabc27We can calculate the area of this trapezoid in28two ways29A1=12(b+a)×(b+a)or we can add up the area of all triangles30A2=12ab+12ab+12c2Since they are the same area,31A1=A2(1)12(b2+a2+ 2ab)=12ab+12ab+12c2(2)a2+b2=c2(3)3233You can see from the above proof that the34idea of the proof is that we start from knowl-35edge which we know to be true: how to find the36area.Then we build the idea in a small step37making sure each step is reasonably obvious to38the reader.Then after enough steps we should39reach the final goal.Think of this as trying to40explain to your friends in this class. Make sure41that if your friends read this proof then he/she42should be convinced what you are trying to say.431The story of his death is quite intriguing. The city got invaded soldier comes in to his house. He refused to surrendersince he is working on math.2This is only true in flat space.3This proof is credited to President Garfield.Discrete Math: Week 11of 7Proof Techniques
For a given proposition there are usually44many ways to write it.Writing a good proof45is a work of art. In this chapter, you will see a46couple of common proof techniques.47Let us do a couple more examples.48Def:An integeraiseveniffnIsuch that 2n=aDef:An integeraisoddiffnIsuch that 2n+ 1 =aFor example, 6 is even since ifn= 3 then we49get 2×3 = 6. 7 is odd sincen= 3 then we get502×3 + 1 = 7.51Let us use these definitions to show that52Theorem:Assume thatxis even andyis odd53thenxyis even.54Proof:Sincexis even this means, by definition55above, that56nIsuch that 2n=xLet us call the numbernxI572nx=xSimilarly, forysince it is odd, by definition,58we can find an integernysuch that592ny+ 1 =yConsider the product of the two60xy= 2nx×(2ny+ 1)= 2(nx×(2ny+ 1))= 2mSincemis a product of integers,mis an integer.

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Term
Winter
Professor
Boonsit Yimwadsana
Tags
Number Theory, Prime number, xy, Mathematical proof

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