MATHOGRAPHICAL REMARK xv

to also attempt exercises rated PG (parental guidance), but may sometimes want

to consult their instructor for a hint. It is also a good idea to double-check your

solution with the instructor, especially if it looks trivial to you. Exercises rated R

(restricted) are intended for mature audiences. The X-rated problems must not be

attempted by anyone easily offended or discouraged.

There is another important reason why we do not recommend skipping chapters.

Mathematical formalism is a good thing, but it is secondary to the development

of the ideas that are being formalised. In the spirit of A. W. Hurwitz, we shall

introduce many of the more difficult concepts in stages: first intuitively, perhaps by

a suggestive analogy, and later in the text with full mathematical rigor. By skipping

ahead, you may miss the more rigorous treatments of a concept and become stuck

with some vague intuitive notions that you should have long outgrown. The latter

problem may be alleviated by making good use of the index. Also, a footnote often

alerts the reader when telling the whole truth is postponed.

If Theorem 4 of Chapter 17 is referred to in Chapter 17 itself, it will be called

just Theorem 4. Outside of Chapter 17, it will be called Theorem 17.4.

The end of a proof is usually marked by a • .

Mathographical Remark

The quotes of Rota and Kac are taken from Discrete thoughts, by Mark Kac,

Gian-Carlo Rota and Jacob T. Schwartz, Birkhauser, Boston 1992, pages ix and

15.