The sections on induction and recursion have been slightly expanded, and have been relocated to an earlier place in the chapter (following the new section), both because they are more concrete than the material found in the other sections of the chapter, and because ideas from the sections on induction and recursion are used in the other sections. There is a new section at the start of the chapter that summarizes various properties of the set of natural numbers these properties play important roles subsequently in the chapter. Also included in this new section is a slightly expanded discussion of the Axiom of Choice, and new discussion of Zorn's Lemma, which is used later in the text.2) The chapter about the cardinality of sets has been rearranged and expanded. We do not make use of these axioms subsequently in the text, but it is valuable for any mathematician to be aware that an axiomatic basis for set theory exists. This section includes a very informal discussion of the Zermelo–Fraenkel Axioms for set theory.
A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised.New to the second edition:1) A new section about the foundations of set theory has been added at the end of the chapter about sets. Part 1 presents logic and basic proof techniques Part 2 thoroughly covers fundamental material such as sets, functions and relations and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences. This 3-part work carefully balances Proofs, Fundamentals, and Extras. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra and real analysis. If the reader finds any such errors-which will hopefully be few in number-it would be very helpful if you would send them ///bloch/proofs2errata.pdf.€œProofs and Fundamentals: A First Course in Abstract Mathematics†2nd edition is designed as a "transition" course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. Preface to the Second Edition xiii Errors Although all known errors from the first edition have been corrected, there are likely to be some remaining undetected errors, and, in spite of the author’s best effort, there are likely to be some errors in the new sections and revisions of older material that were written for the second edition. (9) Many minor adjustments of wording have been made throughout the text, with the hope of improving the exposition. (8) All known errors have been corrected. Whereas the author still finds the notation used in the first edi- tion superior in terms of avoiding confusion with inverse functions, he has deferred to requests from colleagues and reviewers to switch to the standard notation, with the hope that any confusion due to the standard notation will be outweighed by the benefit for students in preparing to read mathematical texts that use the standard notation. Axler Annandale-on-Hudson, NY 12504 San Francisco, CA 94132 USA ISBN 978-1-4419-7126-5 with any form of information storage and retrieval, electronic adaptation, computer software, or by similar e-ISBN 978-1-4419-7127-2 Springer is part of Springer Science+Business Media (DOI 10.1007/978-1-4419-7127-2 Ethan D. Ribet Mathematics Department San Francisco State University University of California at Berkeley Berkeley, CA USA USA This work may not be translated or copied in whole or in part without the written ISSN 0172-6056 permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY Bard College Mathematics Department S. Printed on acid-free paper not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are Springer New York Dordrecht Heidelberg London Editorial Board K.A.