Proofs and fundamentals : a first course in abstract mathematics
著者
書誌事項
Proofs and fundamentals : a first course in abstract mathematics
Birkhäuser, c2000
- : us
- : sz
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注記
Includes bibliographical references (p. [405]-412) and indexes
内容説明・目次
- 巻冊次
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: us ISBN 9780817641115
内容説明
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.
目次
- Part 1 Proofs: informal logic, introduction, statements, relations between statements, valid arguments quantifiers
- strategies for proofs
- mathematical proofs - what they are and why we need them
- direct proofs, proofs by contrapositive and contradiction
- cases, and if and only if, quantifiers in theorems, writing mathematics. Part 2 Fundamentals: sets, introduction, sets - basic definitions
- set operations, indexes families of sets
- functions, image and inverse image, composition and inverse functions, injectivity, surjectivity and bijectivity, set of functions
- relations, congruence, equivalence relations
- infinite and finite sets, cardinality of sets, cardinality of the number systems, mathematical induction, recursion. Part 3 Extras: selected topics, binary operations, groups, homomorphisms and isomorphism, partially ordered sets, lattices, counting products and sums, counting - permutations and combinations
- number systems, back to the beginning, the natural numbers, further properties of the natural numbers, the integers, the rational numbers, the real numbers and the complex numbers. Appendix: properties of numbers. Hints of selected exercises.
- 巻冊次
-
: sz ISBN 9783764341114
内容説明
This text is designed as a "transition" textbook to introduce undergraduates to the writing of vigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. It serves as a bridge between computational courses, e.g. calculus, and more theoretical, proofs-oriented courses such as linear devoted to the proper writing of proofs and over 400 problem sets, which are mostly proofs rather than example problems. Because of the exposition and choice of topics this book should be of interest for classroom use as well as for the general reader who wants to gain a deeper understanding of the language of mathematics. The material of this text was chosen because it is needed in the advanced mathematics curriculum, yet it is often not taught in any other course at the level of calculus or below.
目次
- Part 1 Proofs: informal logic, introduction, statements, relations between statements, valid arguments quantifiers
- strategies for proofs
- mathematical proofs - what they are and why we need them
- direct proofs, proofs by contrapositive and contradiction
- cases, and if and only if, quantifiers in theorems, writing mathematics. Part 2 Fundamentals: sets, introduction, sets - basic definitions
- set operations, indexes families of sets
- functions, image and inverse image, composition and inverse functions, injectivity, surjectivity and bijectivity, set of functions
- relations, congruence, equivalence relations
- infinite and finite sets, cardinality of sets, cardinality of the number systems, mathematical induction, recursion. Part 3 Extras: selected topics, binary operations, groups, homomorphisms and isomorphism, partially ordered sets, lattices, counting products and sums, counting - permutations and combinations
- number systems, back to the beginning, the natural numbers, further properties of the natural numbers, the integers, the rational numbers, the real numbers and the complex numbers. Appendix: properties of numbers. Hints of selected exercises.
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