The Nuts and Bolts of Proofs

An Introduction to Mathematical Proofs


  • Antonella Cupillari, Pennsylvania State Erie, Behrend College, U.S.A.

The Nuts and Bolts of Proof instructs students on the basic logic of mathematical proofs, showing how and why proofs of mathematical statements work. It provides them with techniques they can use to gain an inside view of the subject, reach other results, remember results more easily, or rederive them if the results are forgotten.A flow chart graphically demonstrates the basic steps in the construction of any proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems.
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Intended as a supplement for undergraduate courses in higher mathematics (including Linear Algebra and Geometry), discrete mathematics courses in computer science, and as a primer to assist engineering and physical science students on fundamental proof techniques


Book information

  • Published: August 2005
  • ISBN: 978-0-12-088509-1


“I really enjoyed the 'Collection of Proofs.' These exercises will really stretch a student’s imagination, and go a long way to impressing on them the standards for a believable proof and the necessity of understanding a proposition before embarking on its proof...The new material only makes a great book even greater.” -Robert Beezer, University of Puget Sound "It treats mathematical proofs, and mathematical thinking in general, as an art rather than a science, and does not descend into cookbook recipes for approaching their construction. Thus it empowers students to discover, write and analyze mathematical statements, and to think for themselves. It brings out important details to be considered in constructing proofs gradually and doesn’t overwhelm the reader..." -Andy Miller, University of Oklahoma

Table of Contents

Introduction and Basic TerminologyGeneral Suggestions Some basic Techniques Used in Proving a Theorem of the Form :If A then B”Direct ProofRelated StatementsProof by Contra positiveHow to Construct the Negation of a StatementSpecial Kinds of Theorems“If and only if” or Equivalence TheoremsUse of CounterexamplesMathematical InductionExistence TheoremsUniqueness TheoremsEquality of SetsEquality of NumbersComposite StatementsLimitsReview Exercises Exercises without Soultions Collection of ProofsSolutions of the Exercises at the End of the Sections and the Review ExercisesOther Books on the Subject of Proofs and Mathematical Writing