Book of Proof
The text is very suitable for an "introduction to proofs/transitions" course. I have used this book as the primary text for such a course twice, a course with two main goals: prepare the student for proof-centric classes like abstract algebra and real analysis, and introduce the student to what the major ought to look like and what mathematics hopes to achieve beyond the calculus.
On the first role, the book really shines in its treatment of logic -- sentences with quantifiers and their negations -- methods of proof, induction(basic and general), equivalence relations, functions, and cardinality. Numerous examples are intertwined with introduction of concepts and thoughtful exercises echo the themes of each section. A high point is that the text ends with a rigorous treatment of the serious and magical results of Cantor on cardinality in addition to the Schroeder-Bernstein theorem. Some instructors might see a lack of an introduction to delta-epsilon arguments as a weak point. Others might see the lack of delineation between logic and axiomatics as a weakness.
On the second role, the book lacks a sense of what the major might expect out of a mathematics degree and so when I use this book in a course I normally assign a cheap Dover secondary text for this purpose, along the lines of Ian Stewart's "Concepts of Modern Mathematics," the chapters of which naturally complement those of this text.
Errors are rare, content is accurate.
There is no shortage of such texts on the book market yet I don't see myself changing this choice of text for my course anytime soon. The text also allows for a variety of pedagogical styles -- with a nice mixture of good direct writing, examples, and a lot of relevant problems.
The writing style of the text is best described as direct. Students, who were expected to read considerable sections of the text before coming to class also reported that the text was very good (and they liked that the price was right!).
The book is consistent in terms of terminology.
While almost every chapter depends on chapters preceding it there are pockets that I think are optional. I value the Euclidean algorithm and Bezout's Theorem ("the gcd of two integers can always be written as the integer linear combination of those two integers" and its corollaries) but I don't like the proof presented here and I think the topics can be held back until a course in number theory or in the opening weeks of abstract algebra. Likewise, the perfect number theorem's proof felt like a jump too high for many students so if time is pressing one could opt to postpone these topics in Chapters 7 and 8 respectively. A reversing of the order of Chapters 2 and 3 is also something I would recommend.
The topics are presented in a clear fashion with themes in each section clearly stated and how one sections theme builds upon previously developed themes.
The text is available online (for free) or for hardcopy purchase (~$15) and the two versions line up. The online interface is a plain pdf that appears just as you would expect from the hardcopy. In the long run, the text might benefit from a mathjax-designed interface like that of, say, Judson's "Abstract Algebra."
No grammatical errors.
For better or for worse, cultural relevance does not typically play into a mathematics text and this text is no different. I think this is a real shame -- a price we have paid collectively for emphasizing mathematics chiefly as a technocratic and scientific problem solving discipline as opposed to a humanistic and democratic problem framing one -- but this is not a stick I wish to beat this text with, or at least this text alone. As another reviewer pointed out, some of the problems in the logic section -- negate "you can fool some people all the time, all...." -- are perhaps a bit unusual but my students and I appreciated the ambiguity present in the real world and these problems presented an opportunity to contrast mathematical definitions with the ambiguity of language more generally.