Mathematical Proofs: A Transition to Advanced Mathematics / Edition 2

by Gary Chartrand, Albert D. Polimeni, Ping Zhang
ISBN-10:
0321390539
ISBN-13:
9780321390530
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Overview

Mathematical Proofs: A Transition to Advanced Mathematics / Edition 2

Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses.

Product Details

ISBN-13: 9780321390530
Publisher: Pearson
Publication date: 10/17/2007
Edition description: Older Edition
Pages: 384
Product dimensions: 7.70(w) x 9.40(h) x 0.80(d)

About the Author


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.

1. Sets.
Describing a Set.
Special Sets.
Subsets.
Set Operations.
Indexed Collections of Sets.
Partitions of Sets.
Cartesian Products of Sets.

2. 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.

3. Direct Proof and Proof by Contrapositive.
Trivial and Vacuous Proofs.
Direct Proofs.
Proof by Contrapositive.
Proof by Cases.
Proof Evaluations.

4. 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.

5. 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.

6. Prove or Disprove.
Conjectures in Mathematics.
A Review of Quantifiers.
Existence Proofs.
A Review of Negations of QuantifiedStatements.
Counterexamples.
Disproving Statements.
Testing Statements.
A Quiz of "Prove or Disprove" Problems.

7. Equivalence Relations.
Relations.
Reflexive, Symmetric, and Transitive Relations.
Equivalence Relations.
Properties of Equivalence Classes.
Congruence Modulo n.
The Integers Modulo n.

8. 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.

9. 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.

10. Cardinalities of Sets.
Numerically Equivalent Sets.
Denumerable Sets.
Uncountable Sets.
Comparing Cardinalities of Sets.
The Schroder-Bernstein Theorem.

11. 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.

12. Proofs in Calculus.
Limits of Sequences.
Infinite Series.
Limits of Functions.
Fundamental Properties of Limits of Functions.
Continuity.
Differentiability.

13. 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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