Constructing Proofs CS 1050
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This 0 page Class Notes was uploaded by Alayna Veum on Monday November 2, 2015. The Class Notes belongs to CS 1050 at Georgia Institute of Technology - Main Campus taught by Richard Lipton in Fall. Since its upload, it has received 9 views. For similar materials see /class/234014/cs-1050-georgia-institute-of-technology-main-campus in ComputerScienence at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
CS 1050 A Class Notes Lecture by Prof Merrick Furst and Prof Dick Lipton 7 Jan 2004 1 Some notes about basic components of theorems de nitions lemmas etc In math there are many ways of conveying ideas and for now we will focus on things called Theorems De nitions Lemmas and so forth Each is a way of stating some thing that we believe to be true Theorems relay ideas about some mathematical concept or about some mathematical object De nitions do just what their name means they de ne some mathematical object or relation Now lets take a look at a sample de nition were not concentrating on what this de nition actually means De nition Following Some reference we wil say that two nite relational structures 1 and 2 are VARk equivilents described as 1 E VARMIZ if for every sentence 7 E F0k thenll We want to focus on a few things here First o note how it is presented clearly and neatly which is a key to communicating thoughts and thus your proofs Next note the phrase for every This is important and is what allows us to create theorems and proofs that are actually useful This phrase means that some theorem or de nition or lemme applies to everything of a certain characteristic ad in nitum and not just for some limited cases In fact if we dont have the phrase for every then we are reduced to proving something by complete exhaustion for every possible situation or by exhaustive case analysis well there are really only 72 dz erent types of situations here and then exhaustively proving each case Now lets talk about the symbol What this symbol means in plain terms is that if we have A gt B then B is true if and only if A is true That is if A is false then B is false and if A is true then B is true and if B is false then A is also false and if B is true then A is also true see why we abbreviate it This can and will also be abbreviated as i Lets take a look at another example but rst lets make some de nitions that although informal we7ll typeset them formally so you can get used to seeing them formally De nition A number n is considered a Natural Number if and only if n E 0 1 2 3 1
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