Reading, Writing, and Proving A Closer Look at Mathematics /

• Κύριοι συγγραφείς: Daepp, Ulrich (Συγγραφέας), Gorkin, Pamela (Συγγραφέας)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: New York, NY : Springer New York, 2003.
• Σειρά: Undergraduate Texts in Mathematics,
• Θέματα: Mathematics.; Mathematical analysis.; Analysis (Mathematics).; Mathematical logic.; Number theory.; Mathematical Logic and Foundations.; Analysis.; Number Theory.

  This book, which is based on Pólya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics. It ends by providing projects for independent study. Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis. 

 The How, When, and Why of Mathematics -- Logically Speaking -- Introducing the Contrapositive and Converse -- Set Notation and Quantifiers -- Proof Techniques -- Sets -- Operations on Sets -- More on Operations on Sets -- The Power Set and the Cartesian Product -- Relations -- Partitions -- Order in the Reals -- Functions, Domain, and Range -- Functions, One-to-One, and Onto -- Inverses -- Images and Inverse Images -- Mathematical Induction -- Sequences -- Convergence of Sequences of Real Numbers -- Equivalent Sets -- Finite Sets and an Infinite Set -- Countable and Uncountable Sets -- Metric Spaces -- Getting to Know Open and Closed Sets -- Modular Arithmetic -- Fermat’s Little Theorem -- Projects.