Theorems, Corollaries, Lemmas, and Methods of Proof

A hands–on introduction to the tools needed for rigorous and theoretical mathematical reasoning
Successfully addressing the frustration many students experience as they make the transition from computational mathematics to advanced calculus and algebraic structures, Theorems, Corollaries, Lemmas, and Methods of Proof equips students with the tools needed to succeed while providing a firm foundation in the axiomatic structure of modern mathematics.
This essential book:Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofsReinforces the foundations of calculus and algebraExplores how to use both a direct and indirect proof to prove a theoremPresents the basic properties of real numbersDiscusses how to use mathematical induction to prove a theoremIdentifies the different types of theoremsExplains how to write a clear and understandable proofCovers the basic structure of modern mathematics and the key components of modern mathematics
A complete chapter is dedicated to the different methods of proof such as forward direct proofs, proof by contrapositive, proof by contradiction, mathematical induction, and existence proofs. In addition, the author has supplied many clear and detailed algorithms that outline these proofs.
Theorems, Corollaries, Lemmas, and Methods of Proof uniquely introduces scratch work as an indispensable part of the proof process, encouraging students to use scratch work and creative thinking as the first steps in their attempt to prove a theorem. Once their scratch work successfully demonstrates the truth of the theorem, the proof can be written in a clear and concise fashion. The basic structure of modern mathematics is discussed, and each of the key components of modern mathematics is defined. Numerous exercises are included in each chapter, covering a wide range of topics with varied levels o f difficulty.
Intended as a main text for mathematics courses such as Methods of Proof, Transitions to Advanced Mathematics, and Foundations of Mathematics, the book may also be used as a supplementary textbook in junior– and senior–level courses on advanced calculus, real analysis, and modern algebra.

Spis treści:
Preface.
Chapter 1. Introduction to Modern Mathematics.
Chapter 2. An Introduction to Symbolic Logic.
Chapter 3. Methods of Proof.
Chapter 4. Introduction to Number Theory.
Chapter 5. The Foundations of Calculus.
Chapter 6. Foundations of Algebra.
References.
Index.

Nota biograficzna:
RICHARD J. ROSSI, PHD, is Professor in the Department of Mathematics at Montana Tech of The University of Montana in Butte, Montana. He served as President of the Montana Chapter of the American Statistical Association in 1996 and 2001 and as an Associate Editor for Biometrics from 1997–2000. He is a member of the American Mathematical Society, the Institute of Mathematical Statistics, and the American Statistical Association. Dr. Rossi received his PhD in statistics from Oregon State University in 1988.


Okładka tylna:
A hands–on introduction to the tools needed for rigorous and theoretical mathematical reasoning
Successfully addressing the frustration many students experience as they make the transition from computational mathematics to advanced calculus and algebraic structures, Theorems, Corollaries, Lemmas, and Methods of Proof equips students with the tools needed to succeed while providing a firm foundation in the axiomatic structure of modern mathematics.
This essential book:Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofsReinforces the foundations of calculus and algebraExplores how to use both a direct and indirect proof to prove a theoremPre sents the basic properties of real numbersDiscusses how to use mathematical induction to prove a theoremIdentifies the different types of theoremsExplains how to write a clear and understandable proofCovers the basic structure of modern mathematics and the key components of modern mathematics
A complete chapter is dedicated to the different methods of proof such as forward direct proofs, proof by contrapositive, proof by contradiction, mathematical induction, and existence proofs. In addition, the author has supplied many clear and detailed algorithms that outline these proofs.
Theorems, Corollaries, Lemmas, and Methods of Proof uniquely introduces scratch work as an indispensable part of the proof process, encouraging students to use scratch work and creative thinking as the first steps in their attempt to prove a theorem. Once their scratch work successfully demonstrates the truth of the theorem, the proof can be written in a clear and concise fashion. The basic structure of modern mathematics is discussed, and each of the key components of modern mathematics is defined. Numerous exercises are included in each chapter, covering a wide range of topics with varied levels of difficulty.
Intended as a main text for mathematics courses such as Methods of Proof, Transitions to Advanced Mathematics, and Foundations of Mathematics, the book may also be used as a supplementary textbook in junior– and senior–level courses on advanced calculus, real analysis, and modern algebra.