"Proofs 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. 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. This 3-part work carefully balances Proofs,
Fundamentals, and Extras. 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. 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. This section includes a very informal
discussion of the Zermelo- Fraenkel Axioms for set theory. 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. 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. 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. 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. Next comes the section on the cardinality of
sets (which was originally the first section of the chapter); this
section gained proofs of the Schroeder-Bernstein theorem and the
Trichotomy Law for Sets, and lost most of the material about finite
and countable sets, which has now been moved to a new section
devoted to those two types of sets. The chapter concludes with the
section on the cardinality of the number systems. 3) The chapter on
the construction of the natural numbers, integers and rational
numbers from the Peano Postulates was removed entirely. That
material was originally included to provide the needed background
about the number systems, particularly for the discussion of the
cardinality of sets, but it was always somewhat out of place given
the level and scope of this text. The background material about the
natural numbers needed for the cardinality of sets has now been
summarized in a new section at the start of that chapter, making
the chapter both self-contained and more accessible than it
previously was. 4) The section on families of sets has been
thoroughly revised, with the focus being on families of sets in
general, not necessarily thought of as indexed. 5) A new section
about the convergence of sequences has been added to the chapter on
selected topics. This new section, which treats a topic from real
analysis, adds some diversity to the chapter, which had hitherto
contained selected topics of only an algebraic or combinatorial
nature. 6) A new section called ``You Are the Professor'' has been
added to the end of the last chapter. This new section, which
includes a number of attempted proofs taken from actual homework
exercises submitted by students, offers the reader the opportunity
to solidify her facility for writing proofs by critiquing these
submissions as if she were the instructor for the course. 7) All
known errors have been corrected. 8) Many minor adjustments of
wording have been made throughout the text, with the hope of
improving the exposition.
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