Mathematical Proofs : A Transition to Advanced Mathematics,9780321390530

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Mathematical Proofs : A Transition to Advanced Mathematics

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

Edition:

2nd

ISBN13:

9780321390530

ISBN10:

0321390539

Format:

Hardcover

Pub. Date:

10/3/2007

Publisher(s):

Pearson

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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.

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
Table of Contents provided by Publisher. All Rights Reserved.

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