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Section 3.1: Direct Proof and Counterexample 1

Section : Direct Proof and Counterexample 1In this chapter, we introduce the notion of Proof in mathematics. Amathematical Proof is valid logical argument in mathematics whichshows that a given conclusion is true under the assumption that thepremisses are true. All major mathematical results you haveconsideredsince you first started studying mathematics have all been derived inthis way Pythagoras Theorem, Fundamental Theorem of Calculus,Fundamental Theorem of Algebra. Most of these proofs are long andcomplicated and will be considered in further mathematics this course, we shall consider more elementary proofs, mainly innumber theory, to start and strengthen our Proof writing stated at the beginning of the course, one of the most importantparts of mathematical Proof is knowing and understanding the defini-tions of what you are trying to prove things about. In this classandall future classesif you do not learn and understand the definitionsyou will notbe able to prove things.

Section 3.1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. ... is rational, then x is rational”. Disprove this statement by giving a counter example. ... (ii) Start the proof by supposing that x is a particular, but arbi-trary chosen element of D for which P(x) is true.

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