"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.
