Mathematical Proofs : A Transition to Advanced Mathematics

by Chartrand, Gary; Polimeni, Albert D.; Zhang, Ping

• ISBN13: 9780321390530
• ISBN10: 0321390539
• Edition: 2nd
• Format: Hardcover
• Copyright: 2007-10-03
• Publisher: Pearson

Summary

Mathematical Proofs: A Transition to Advanced Mathematics, 2/e,prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.KEY TOPICS: Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory.MARKET: For all readers interested in advanced mathematics and logic.

Author Biography

Gary Chartrand is Professor Emeritus of Mathematics at Western Michigan University. He received his Ph.D. in mathematics from Michigan State University. His research is in the area of graph theory. Professor Chartrand has authored or co-authored more than 275 research papers and a number of textbooks in discrete mathematics and graph theory as well as the textbook on mathematical proofs. He has given over 100 lectures at regional, national and international conferences and has been a co-director of many conferences. He has supervised 22 doctoral students and numerous undergraduate research projects and has taught a wide range of subjects in undergraduate and graduate mathematics. He is the recipient of the University Distinguished Faculty Scholar Award and the Alumni Association Teaching Award from Western Michigan University and the Distinguished Faculty Award from the State of Michigan. He was the first managing editor of the Journal of Graph Theory. He is a member of the Institute of Combinatorics and Its Applications, the American Mathematical Society, the Mathematical Association of America and the editorial boards of the Journal of Graph Theory and Discrete Mathematics.

Albert D. Polimeni is an Emeritus Professor of Mathematics at the State University of New York at Fredonia. He received his Ph.D. degree in mathematics from Michigan State University. During his tenure at Fredonia he taught a full range of undergraduate courses in mathematics and graduate mathematics. In addition to the textbook on mathematical proofs, he co-authored a textbook in discrete mathematics. His research interests are in the area of finite group theory and graph theory, having published several papers in both areas. He has given addresses in mathematics to regional, national and international conferences. He served as chairperson of the Department of Mathematics for nine years.

Ping Zhang is Professor of Mathematics at Western Michigan University. She received her Ph.D. in mathematics from Michigan State University. Her research is in the area of graph theory and algebraic combinatorics. Professor Zhang has authored or co-authored more than 200 research papers and four textbooks in discrete mathematics and graph theory as well as the textbook on mathematical proofs. She serves as an editor for a series of books on special topics in mathematics. She has supervised 7 doctoral students and has taught a wide variety of undergraduate and graduate mathematics courses including courses on introduction to research. She has given over 60 lectures at regional, national and international conferences. She is a council member of the Institute of Combinatorics and Its Applications and a member of the American Mathematical Society and the Association of Women in Mathematics.

Table of Contents

Communicating Mathematics

Learning Mathematics

What Others Have Said About Writing

Mathematical Writing

Using Symbols

Writing Mathematical Expressions

Common Words and Phrases in Mathematics

Some Closing Comments About Writing

Sets

Describing a Set

Special Sets

Subsets

Set Operations

Indexed Collections of Sets

Partitions of Sets

Cartesian Products of Sets

Logic

Statements

The Negation of a Statement

The Disjunction and Conjunction of Statements

The Implication

More On Implications

The Biconditional

Tautologies and Contradictions

Logical Equivalence

Some Fundamental Properties of Logical Equivalence

Characterizations of Statements

Quantified Statements and Their Negations

Direct Proof and Proof by Contrapositive

Trivial and Vacuous Proofs

Direct Proofs

Proof by Contrapositive

Proof by Cases

Proof Evaluations

More on Direct Proof and Proof by Contrapositive

Proofs Involving Divisibility of Integers

Proofs Involving Congruence of Integers

Proofs Involving Real Numbers

Proofs Involving Sets

Fundamental Properties of Set Operations

Proofs Involving Cartesian Products of Sets

Proof by Contradiction

Proof by Contradiction

Examples of Proof by Contradiction

The Three Prisoners Problem

Other Examples of Proof by Contradiction

The Irrationality of …À2

A Review of the Three Proof Techniques

Prove or Disprove

Conjectures in Mathematics

A Review of Quantifiers

Existence Proofs

A Review of Negations of Quantified Statements

Counterexamples

Disproving Statements

Testing Statements

A Quiz of “Prove or Disprove” Problems

Equivalence Relations

Relations

Reflexive, Symmetric, and Transitive Relations

Equivalence Relations

Properties of Equivalence Classes

Congruence Modulo n

The Integers Modulo n

Functions

The Definition of function

The Set of All Functions From A to B

One-to-one and Onto Functions

Bijective Functions

Composition of Functions

Inverse Functions

Permutations

Mathematical Induction

The Well-Ordering Principle

The Principle of Mathematical Induction

Mathematical Induction and Sums of Numbers

Mathematical Induction and Inequalities

Mathematical Induction and Divisibility

Other Examples of Induction Proofs

Proof By Minimum Counterexample

The Strong Form of Induction

Cardinalities of Sets

Numerically Equivalent Sets

Denumerable Sets

Uncountable Sets

Comparing Cardinalities of Sets

The Schroder-Bernstein Theorem

Proofs in Number Theory

Divisibility Properties of Integers

The Division Algorithm

Greatest Common Divisors

The Euclidean Algorithm

Relatively Prime Integers

The Fundamental Theorem of Arithmetic

Concepts Involving Sums of Divisors

Proofs in Calculus

Limits of Sequences

Infinite Series

Limits of Functions

Fundamental Properties of Limits of Functions

Continuity

Differentiability

Proofs in Group Theory

Binary Operations

Groups

Permutation Groups

Fundamental Properties of Groups

Subgroups

Isomorphic Groups

Answers and Hints to Selected Odd-Numbered Exercises

References Index of Symbols

Index of Mathematical Terms

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