The Structure of Proof
With Logic and Set Theory
Published on 9. September 2002
Book
Hardback
418 pages
978-0-13-019077-2 (ISBN)
Description
For a one-semester freshman or sophomore level course on the fundamentals of proof writing or transition to advanced mathematics course.
Rather than teach mathematics and the structure of proofs simultaneously, this text first introduces logic as the foundation of proofs and then demonstrates how logic applies to mathematical topics. This method ensures that the students gain a firm understanding of how logic interacts with mathematics and empowers them to solve more complex problems in future math courses.
Rather than teach mathematics and the structure of proofs simultaneously, this text first introduces logic as the foundation of proofs and then demonstrates how logic applies to mathematical topics. This method ensures that the students gain a firm understanding of how logic interacts with mathematics and empowers them to solve more complex problems in future math courses.
More details
Language
English
Place of publication
United States
Publishing group
Pearson Education (US)
Target group
Professional and scholarly
Dimensions
Width: 243 mm
Thickness: 20 mm
Weight
804 gr
ISBN-13
978-0-13-019077-2 (9780130190772)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Content
Preface.
I. LOGICAL FOUNDATIONS.
1. Propositional Logic.
Propositions. Propositional Forms. Rules of Inference. Rules of Replacement.
2. Predicates and Proofs.
Predicates and Sets. Quantification. Negating Quantifiers. Proofs with Quantifiers. Direct and Indirect Proof. More Methods.
II. MAIN TOPICS.
3. Set Theory.
Set Basics. Subsets. Equality of Sets. Families of Sets. Generalized Union and Intersection. Chapter Exercises.
4. Mathematical Induction.
The First Principle. Combinatorics. The Second Principle. The Well-Ordering Principle. Chapter Exercises.
5. Number Theory.
Axioms. Divisibility. Primes. Congruences. Chapter Exercises.
6. Relations and Functions.
Relations. Equivalence Relations. Functions. Function Operations. One-to-one and Onto. Images and Inverse Images. Cardinality. Chapter Exercises.
III. COMING ATTRACTIONS.
7. Ring Theory.
Types of Rings. Subrings and Ideals. Factor Rings. Homomorphisms. Polynomials. Chapter Exercises
8. Topology.
Spaces. Open Sets. Closed Sets. Isometries. Limits. Chapter Exercises.
IV. APPENDICES.
Appendix A: Logic Summary.
Appendix B: Summation Notation.
Appendix C: Greek Alphabet.
Bibliography.
Selected Solutions.
Index.
I. LOGICAL FOUNDATIONS.
1. Propositional Logic.
Propositions. Propositional Forms. Rules of Inference. Rules of Replacement.
2. Predicates and Proofs.
Predicates and Sets. Quantification. Negating Quantifiers. Proofs with Quantifiers. Direct and Indirect Proof. More Methods.
II. MAIN TOPICS.
3. Set Theory.
Set Basics. Subsets. Equality of Sets. Families of Sets. Generalized Union and Intersection. Chapter Exercises.
4. Mathematical Induction.
The First Principle. Combinatorics. The Second Principle. The Well-Ordering Principle. Chapter Exercises.
5. Number Theory.
Axioms. Divisibility. Primes. Congruences. Chapter Exercises.
6. Relations and Functions.
Relations. Equivalence Relations. Functions. Function Operations. One-to-one and Onto. Images and Inverse Images. Cardinality. Chapter Exercises.
III. COMING ATTRACTIONS.
7. Ring Theory.
Types of Rings. Subrings and Ideals. Factor Rings. Homomorphisms. Polynomials. Chapter Exercises
8. Topology.
Spaces. Open Sets. Closed Sets. Isometries. Limits. Chapter Exercises.
IV. APPENDICES.
Appendix A: Logic Summary.
Appendix B: Summation Notation.
Appendix C: Greek Alphabet.
Bibliography.
Selected Solutions.
Index.