chap mathematical preliminaries thought processwhich in turn clarifies your explanations secondif you use one of the standard proof structures such as proof by contradiction or an induction proofthen both you and your reader are working from shared understanding of that structure that makes for less complexity to your reader to understand your proofbecause the reader need not decode the structure of your argument from scratch this section briefly introduces three commonly used proof techniques(ideductionor direct proof(iiproof by contradictionand (iiiproof by mathematical induction direct proof in generala direct proof is just "logical explanation direct proof is sometimes referred to as an argument by deduction this is simply an argument in terms of logic often written in english with words such as "if then,it could also be written with logic notation such as " even if we don' wish to use symbolic logic notationwe can still take advantage of fundamental theorems of logic to structure our arguments for exampleif we want to prove that and are equivalentwe can first prove and then prove in some domainsproofs are essentially series of state changes from start state to an end state formal predicate logic can be viewed in this waywith the various "rules of logicbeing used to make the changes from one formula or combining couple of formulas to make new formula on the route to the destination symbolic manipulations to solve integration problems in introductory calculus classes are similar in spiritas are high school geometry proofs proof by contradiction the simplest way to disprove theorem or statement is to find counterexample to the theorem unfortunatelyno number of examples supporting theorem is sufficient to prove that the theorem is correct howeverthere is an approach that is vaguely similar to disproving by counterexamplecalled proof by contradiction to prove theorem by contradictionwe first assume that the theorem is false we then find logical contradiction stemming from this assumption if the logic used to find the contradiction is correctthen the only way to resolve the contradiction is to recognize that the assumption that the theorem is false must be incorrect that iswe conclude that the theorem must be true example here is simple proof by contradiction theorem there is no largest integer