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 proofs
* Reinforces the foundations of calculus and algebra
* Explores how to use both a direct and indirect proof to prove a
theorem
* Presents the basic properties of real numbers
* Discusses how to use mathematical induction to prove a
theorem
* Identifies the different types of theorems
* Explains how to write a clear and understandable proof
* Covers 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.
General
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