A Transition to Advanced Mathematics by Douglas Smith is a comprehensive and accessible textbook designed for undergraduate students making the transition from calculus to advanced mathematics. The book provides a solid foundation in the concepts and techniques of advanced mathematics and prepares students for further study in mathematics, physics, engineering, and other related disciplines.
The author, Douglas Smith, has a clear and accessible writing style, which makes the material easy to understand for students with varying levels of mathematical background. The book covers a wide range of topics, including set theory, number theory, relations, functions, and linear algebra. Additionally, the book includes numerous examples and exercises, which provide opportunities for students to practice what they have learned and apply the concepts covered in the book to new problems.
TABLE OF CONTENTS
Chapter 1 Logic and Proofs
- Propositions and Connectives
- Conditionals and Biconditionals
- Quantifiers
- Mathematical Proofs
- Proofs Involving Quantifiers
Chapter 2 Set Theory
- Basic Notions of Set Theory
- Set Operations
- Extended Set Operations and Indexed Families of Sets
- Induction
- Equivalent Forms of Induction
- Principles of Counting
Chapter3 Relations
- Cartesian Products and Relations
- Equivalence Relations
- Partitions
- Ordering Relations Graphs of Relations
Chapter4 Functions
- Functions as Relations
- Constructions of Functions
- Functions that Are Onto, One-to-One Functions
- Induced Set Functions
Chapter 5 Cardinality
- Equivalent Sets, Finite Sets
- Infinite Sets
- Countable Sets
- The Ordering of Cardinal Numbers
- Comparability of Cardinal Numbers and the Axiom of Choice
Chapter 6 Concepts of Algebra: Groups
- Algebraic Structures
- Groups
- Examples of Groups
- Subgroups
- Cosets and Lagrange’s Theorem
- Quotient Groups
- Isomorphism; The Fundamental Theorem of Group Homomorphism
- Application
Chapter7 Concepts of Analysis: Completeness of the Real Numbers
- Ordered Field Properties of the Real Numbers
- The Hiene-Borel Theorem
- The Bolzano Weierstrass Theorem
- The Bounded Monotone Sequence Theorem
- Equivalents of Completeness
Download Now Book in PDF
A Transition to Advanced Mathematics by Douglas Smith is notable for its emphasis on problem-solving and its focus on applications, which helps students to see the practical applications of advanced mathematics and understand how the concepts covered in the book can be used in real-world situations.
Whether you are a student looking to make the transition to advanced mathematics or a professional seeking to refresh your knowledge of the subject, A Transition to Advanced Mathematics by Douglas Smith is an excellent resource that is sure to meet your needs. With its comprehensive coverage of the subject and its focus on problem-solving and applications, this textbook is sure to provide a valuable resource for anyone interested in learning about advanced mathematics.
