USA.: Department of Mathematics. Grand Valley State University (GVSU), 2014.- 591 p. -
ISBN 9781492103851, (Series: OPEN EDUCATION MATERIALS). eBook. English.
(This work may be copied, distributed, and/or modified under the conditions of the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License).Description.
Mathematical Reasoning: Writing and Proof is designed to be a text for the first course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics.
The primary goals of the text are to help students:
· Develop logical thinking skills and to develop the ability to think more abstractly in a proof oriented setting.
· Develop the ability to construct and write mathematical proofs using standard methods of mathematical proof including direct proofs, proof by contradiction, mathematical induction, case analysis, and counterexamples.
· Develop the ability to read and understand written mathematical proofs.
· Develop talents for creative thinking and problem solving.
· Improve their quality of communication in mathematics. This includes improving writing techniques, reading comprehension, and oral communication in mathematics.
· Better understand the nature of mathematics and its language.
This text also provides students with material that will be needed for their further study of mathematics.Contents.
Note to Students.
Preface.
Introduction to Writing Proofs in Mathematics.
Statements and Conditional Statements.
Constructing Direct Proofs.
Summary.
Logical Reasoning.
Statements and Logical Operators.
Logically Equivalent Statements.
Open Sentences and Sets.
Quantifiers and Negations.
Summary.
Constructing and Writing Proofs in Mathematics.
Direct Proofs.
More Methods of Proof.
Proof by Contradiction.
Using Cases in Proofs.
The Division Algorithm and Congruence.
Review of Proof Methods.
Summary.
Mathematical Induction.
The Principle of Mathematical Induction.
Other Forms of Mathematical Induction.
Induction and Recursion .
Summary.
Set Theory.
Sets and Operations on Sets.
Proving Set Relationships.
Properties of Set Operations.
Cartesian Products.
Indexed Families of Sets.
Summary.
Functions.
Introduction to Functions.
More about Functions.
Injections, Surjections, and Bijections.
Composition of Functions.
Inverse Functions.
Functions Acting on Sets.
Summary.
Equivalence Relations.
Relations.
Equivalence Relations.
Equivalence Classes.
Modular Arithmetic.
Summary.
Topics in Number Theory.
The Greatest Common Divisor.
Prime Numbers and Prime Factorizations.
Linear Diophantine Equations.
Summary.
Finite and Infinite Sets.
Finite Sets.
Countable Sets.
Uncountable Sets.
Summary.
Appendix A. Guidelines for Writing Mathematical Proofs.
Appendix B. Answers for the Progress Checks.
Appendix C. Answers and Hints for Selected Exercises.
Appendix D. List of Symbols.
Index.
