General Advice on the Writing Projects

This Web site contains some helpful advice and provides information resources -- including books, articles, and Internet resources -- to get you started. It also contains specific suggestions of where you might look when working on the various Writing Projects given in each chapter. We do not guarantee that you will find exactly what you are looking for if you consult the references or follow the links we suggest, but at least our pointers will start you in the right direction. (Some of our hot links on these pages are actually Web searches -- see below. Others are links to Web sites on which you will find information, applets, and more links. It can never hurt to follow a link -- your browser always lets you "go back" to the previous page.) Tracking down the information is half the fun!

This material is adapted from the Student Solutions Guide for Discrete Mathematics and Its Applications, fourth edition by Kenneth H. Rosen. This Guide contains the full solutions to all of the odd-numbered exercises in the textbook. These solutions explain why a particular method is used, and why it works. For some exercises, one or two other possible approaches are described to illustrate that a problem may be solved in several different ways. The Guide also contains a guide to writing proofs and a list of common errors that students make in discrete mathematics. In order to assist students in preparing for examinations, sample test questions and answers and sample crib sheets, are provided for each chapter. Students find the Guide very useful, and you may wish to order a copy [note: this link is to the third edition's Guide for sale at amazon.com; the fourth edition should be available there soon].

Gathering the information

• Explore all facets of the Internet, including e-mail, Usenet discussion groups (try the sci.math newsgroup in particular -- see Deja News for general information), gophers, the World Wide Web, and so on. You will find lots of sources of information, and you will get to communicate with other people who have the kinds of information you want, or know where to get it. Ask around -- people tend to be very friendly and helpful in this community. It's a fascinating social dynamic!

• The first place to search for material on any of the Writing Projects is probably the World Wide Web. One of the best ways to find things on the Web is to use a search engine. You type in one or more key words or phrases (such as "graph theory"), and the search engine looks over the millions of Web sites around the world to find those that mention these words or phrases; then you can visit those sites. The whole process takes only a few seconds, and the sites are listed in order, with the "best matches" -- the sites that are likely to contain the most relevant information -- at the top. One particularly good search engine is Metacrawler. It sends your request to half a dozen other search engines (such as Infoseek) and shows you the ten to twenty "best" hits (those sites that contain your words or phrases most prominently). In fact, the existence of the World Wide Web and search engines makes it embarrassingly easy to find information on any topic one desires. You will find that some of the hot links we provide on these pages are actually searches using metacrawler. For example, here is a search for the phrase "discrete mathematics."

• One trick to remember when using the Web is that if you have found a detailed site, you will often want to go to that site's "home" page. There is almost always a hot link to do so.

• There are some great mathematics sites on the Web, which you will want to bookmark. We like one originally called "Eric's Treasure Trove." (It is now part of the CRC Concise Encyclopedia of Mathematics, and as such has 10 letters per day randomly blocked, but it is still very useful.)

• Obviously you have already found the Web site for this textbook. Notice the Web icon throughout the textbook, which indicates relevant material on this Web site. The material is organized by chapter, and is also searchable. You should always check this site when embarking on the Writing Projects.

• Most libraries have on-line search facilities that allow you to look for key words in titles of books in their collection. For example, to find resources on fuzzy sets or fuzzy logic, you could search on the word "fuzzy." You can also search for authors or titles, of course. Ask a librarian for assistance if necessary. Also, catalogs to many university libraries (Oakland University, as a random example) and even the Library of Congress are available on the Internet. See the webCATS Web site for a comprehensive list.

• On-line bookstores, such as amazon.com, have good search engines. You can search by author, title, topic, or keyword, and once you find a book relevant to your subject, there will be links to similar books and related subjects. The list of topics is huge, including esoteric things relevant to these Projects, such as "automatic theorem proving." Often the description of the book will include reviews or other pertinent information, and if you actually want to buy the book, you can usually get it within a few days, at a reduced price. A few of the hot links we provide on these pages are subject or key-word searches to amazon.com, such as this one to the subject "discrete mathematics."

• We have provided an extensive list of references, and there are links to them from the Writing Projects where they might be helpful. In addition, you should consult the "Suggested Readings" section of the textbook (pages B-1 through B-7, near the back of the book). There are chapter notes annotating the alphabetical list of over 100 books and articles.

• The following library research technique should come in handy. If the source you are looking at does not deal in enough detail with the topic you are investigating, then consult the references given in that source. Continue this process backwards as deeply and broadly as necessary.

• There is a comprehensive set of brief summaries or reviews of essentially every mathematical research paper and book written since 1940 (and being kept up-to-date at the rate of about 50,000 items per year), in a journal called Mathematical Reviews (MR), published out of Ann Arbor, Michigan, by the American Mathematical Society. It comes in various forms -- in hardcover volumes, on CD-ROM, and on the Internet. Ask your librarian for the forms your school has access to. In the best formats, you can search for key words, as well as authors or titles. The hypertext version on the World Wide Web (called MathSciNet) lets you quickly follow leads from one review to another. The reviews in MR are organized by topic; it would be useful for students to browse the list of topics (known as the Mathematics Subject Classification and organized as a tree, of course) to get an overview of the broad range of mathematical research being conducted and how it all fits together. Most of discrete mathematics falls into category 05 Combinatorics, although you will also find material relevant to this course under 01 History and biography , 03 Mathematical logic and foundations , 04 Set theory, 06 Order, lattices, ordered algebraic structures, 11 Number theory , 15 Linear and multilinear algebra; matrix theory, 60 Probability theory and stochastic processes, 68 Computer Science, 90 Economics, operations research, programming, games, 94 Information and communication, circuits, and other headings.

• In addition to the American Mathematical Society (AMS), there are good sites for the other two main American mathematics organizations, the Mathematical Association of America (MAA) and the Society for Industrial and Applied Mathematics (SIAM), as well as the Association for Computing Machinery (ACM). These sites have fantastic amounts of information about mathematics and related fields, with excellent links. The MAA also has a subsite especially geared toward students.

• We shouldn't need to mention obvious things, like using the index and table of contents of any book you consult. When looking up items in an index, don't forget to try possible variations of what you are looking for (e.g., you may find one of the entries "induction, mathematical" or "mathematical induction" but not the other).

• An excellent source for many of these Writing Projects is Applications of Discrete Mathematics, edited by John G. Michaels and Kenneth H. Rosen (McGraw-Hill, 1991). This is listed as item [MiRo] in our bibliography. It has articles covering various parts of pure and applied discrete mathematics, at levels varying from elementary to intermediate. It is worth browsing through this book, even if you do not find anything in it relevant to a Project you are working on. It will give you a feeling for the breadth of the subject you are studying.

• Popular accounts of mathematical topics often make their way into The New York Times. This premier of American newspapers has a detailed index, which is available in most libraries that carry the newspaper. One prolific writer of mathematical articles is Gina Kolata. The Times also produces on a regular basis a special edition of mathematics-related articles; ask your mathematics department whether they have a recent issue.

• Many of the essays assigned in this textbook deal with the history of mathematical topics. Most books on the general history of mathematics are filed under the call letters QA 21. See [Bo4] and [Ev3] for two good sources. There are also wonderful extensive collections of essays about mathematics, both historical and expository. A classic is the four-volume treatise [Ne]. A more recent one of high quality is [DaHe]. Perhaps the best resource for the history of mathematics is the MacTutor History of Mathematics archive on the Web. It has biographies of hundreds of mathematicians, as well as references, articles, links, and an unbelievable amount of information.

• Some of these Projects go into depth on various topics in discrete mathematics. There are several good, more advanced textbooks on combinatorics and graph theory, such as [Bo1], [BoMu], [Br2], [ChLe], [GrYe], [Ro1], [Tu1], [We]. The library classifications here are QA 164 and QA 166, where you will also find specialized books, research monographs, and conference proceedings. In addition, there are dozens of other discrete mathematics textbooks at a level comparable with or slightly more or less advanced than your textbook. An excellent one is [Gr2]. It has comprehensive discussions of most discrete mathematical topics and a wide variety of interesting problems, including some challenging and open-ended ones. It also has a bibliography of 335 books and articles, and a detailed index that will lead you to the right source for further reading. Another, slightly different, more advanced book to take a look at (if nothing else, for its style!) is [GrKn].

• There is an intimate relationship between discrete mathematics and computer science. Computer science books of all sorts, whether dealing with hardware and circuit issues, programming, data structures, algorithms, complexity, theoretical foundations, operating systems, compilers, artificial intelligence, or other topics, may well be relevant to many of these Projects. QA 76 is where many such books are housed in the library, although specialized topics will have their own call numbers (e.g., Q 335 for artificial intelligence or the high TK 7800's for circuit design). Our list that follows includes several textbooks on data structures and algorithms. Another lively source is [De2], a collection of essays on various aspects of computer science and related mathematics, each with references for further reading. You will find those essays relevant to a large number of the Writing Projects, and you should definitely try to have a look at this collection.

Writing your report

• All the rules and advice you have learned over the years about good writing apply to technical writing as well as to other forms of prose. It is often more difficult to express mathematical ideas clearly and precisely, so do not expect these Writing Projects to be easy.

• Know your reader! Keep in mind for whom you are writing, and pitch the level to that audience (your instructor, fellow students, intelligent laypersons?). Think about how much you will assume your reader knows and how much you will need to fill in. (When in doubt, do not assume the reader knows much.)

• Organize, organize, organize! Essays need to have an introduction, a body, and a conclusion. If the work is going to be long, it probably makes sense to have labeled sections covering the different points. Make an outline of what you plan to say, and think a lot about how to order it, both before you start writing and throughout the process.

• Use a word processor if you have access to one. This makes it much easier to revise and edit your work numerous times, until it is just the way you want it, and of course the final product looks much more professional. Make sure to take advantage of special features like spelling, grammar, and usage checkers. Pay some attention to the format (fonts, spacing, layout, etc.); most word processors let you design a very pleasing document. Print your essay on a laser-quality printer if you can. If your essay will contain much mathematical symbolism, try to use a mathematical word processor or typesetter. The best of these is TeX, which it would definitely be worth your while to learn to use (although it is not easy). To give you an idea of how nice TeX can look, realize that the textbook itself was typeset using it. You can find out more about TeX from the TeX Users Group (TUG).

• Give proper credit. Taking someone else's words, pictures, or ideas without acknowledging the source is a serious breach of academic honesty known as plagiarism. Doing this can get you a failing grade on the Project or worse (e.g., suspension). Make sure to properly list the books, articles, and Web sites you have used in preparing your report, following whatever style your instructor requires. Make sure to footnote or otherwise identify passages that come from outside sources, and put the material in quotation marks if it is used verbatim.