The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same.
Rezensionen / Stimmen
"... Proofs and Fundamentals has many strengths. One notable, strength, is its excellent organization. The book begins with a three-part preface, which makes its aims very clear. There are large exercise sets throughout the book ... Exercises are well integrated with the text and vary appropriately from easy to hard ... Topics in Part III are quite varied, mostly independent from each other, and truly dependent on Parts I and II. At the end of the book there are useful hints to selected exercises. Perhaps the book's greatest strength is the author's zeal and skill for helping students write mathematics better. Careful guidance is given throughout the book. Basic issues like not abusing equal signs are treated explicitly. Attention is given to even relatively small issues, like not placing a mathematical symbol directly after a punctuation mark. Throughout the book, theorems are often followed first by informative 'scratch work' and only then by proofs. Thus students can see many examples of what they should think, what they should write, and how these are usually not the same." -MAA Online "This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a 'transition' course." ---Zentralblatt Math
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Research
Illustrationen
Maße
Höhe: 23.5 cm
Breite: 15.5 cm
Dicke: 25 mm
Gewicht
ISBN-13
978-0-8176-4111-5 (9780817641115)
DOI
10.1007/978-1-4612-2130-2
Schweitzer Klassifikation
I PROOFS.- 1 Informal Logic.- 1.1 Introduction.- 1.2 Statements.- 1.3 Relations Between Statements.- 1.4 Valid Arguments.- 1.5 Quantifiers.- 2 Strategies for Proofs.- 2.1 Mathematical Proofs - What They Are and Why We Need Them.- 2.2 Direct Proofs.- 2.3 Proofs by Contrapositive and Contradiction.- 2.4 Cases, and If and Only If.- 2.5 Quantifiers in Theorems.- 2.6 Writing Mathematics.- II Fundamentals.- 3 Sets.- 3.1 Introduction.- 3.2 Sets - Basic Definitions.- 3.3 Set Operations.- 3.4 Indexed Families of Sets.- 4 Functions.- 4.1 Functions.- 4.2 Image and Inverse Image.- 4.3 Composition and Inverse Functions.- 4.4 Injectivity, Surjectivity and Bijectivity.- 4.5 Sets of Functions.- 5 Relations.- 5.1 Relations.- 5.2 Congruence.- 5.3 Equivalence Relations.- 6 Infinite and Finite Sets.- 6.1 Cardinality of Sets.- 6.2 Cardinality of the Number Systems.- 6.3 Mathematical Induction.- 6.4 Recursion.- III Extras.- 7 Selected Topics.- 7.1 Binary Operations.- 7.2 Groups.- 7.3 Homomorphisms and Isomorphisms.- 7.4 Partially Ordered Sets.- 7.5 Lattices.- 7.6 Counting: Products and Sums.- 7.7 Counting: Permutations and Combinations.- 8 Number Systems.- 8.1 Back to the Beginning.- 8.2 The Natural Numbers.- 8.3 Further Properties of the Natural Numbers.- 8.4 The Integers.- 8.5 The Rational Numbers.- 8.6 The Real Numbers and the Complex Numbers.- 8.7 Appendix: Proof of Theorem 8.2.1.- 9 Explorations.- 9.1 Introduction.- 9.2 Greatest Common Divisors.- 9.3 Divisibility Tests.- 9.4 Real-Valued Functions.- 9.5 Iterations of Functions.- 9.6 Fibonacci Numbers and Lucas Numbers.- 9.7 Fuzzy Sets.- Appendix: Properties of Numbers.- Hints for Selected Exercises.- References.
