One of the main purposes of this book is to help you learn how to read and write proofs. To further this aim, much of the early material is familiar (even at the beginning, however, there are new and interesting things) to allow you to focus on giving complete and clear proofs without distractions.

A proof is an explanation why something is true. There is a notion of formal proof, which is essentially an explanation to a machine, but we are con- cerned here with giving proofs to humans. Just as one does not give the same explanation to a ten-year old that one gives to an adult, one's proof, one's ex- planation, depends on whom one is speaking to. The audience for all of your proofs is not your instructor (who already knows the reasons !); your expla- nations are to be directed towards students in class, one of whom is yourself. Adequate reasons must be given to defend assertions against any possible ob- jection; on the other hand, there is no need to explain why 3 == 3. Try your best to say enough to persuade, and try your best not to put others to sleep by belaboring the obvious. One role of the proofs in the text is to serve as models for your own proofs. Because one becomes more sophisticated as one learns, the proofs in the text also change; certain points made explicit in the beginning are later left unsaid.

Some people think that a proof must be full of symbols, looking like ancient Egyptian hieroglyphics. Not so. Look in any mathematics book, and you will find words. Your proofs should be written in complete sentences. Of course, you may use symbols and pictures if necessary, but remember that a symbol is like a pronoun; it means nothing unless it is specified. Just as you wouldn't begin a story by saying, "He gave some of it to him there," you must not begin a proof by saying that x == y^2 without telling what x and y are (are they numbers? real? rational? integers? positive?).

Even though the context of this course is largely elementary, do not be lulled into thinking that it is an easy course with an inevitable grade of A at its conclusion. There are challenges within. If one wants the reward Bacon mentions, then there is no alternative but to do some mathematics. The jour- ney may have some difficulties, but its goals are valuable. As Bacon says, the reward is understanding, subtlety and, we may add, pleasure.

Joseph Rotman has been on the faculty of the mathematics department of the University of Illinois, Urbana-Champaign since 1959, and has been Professor Emeritus since 2004. ...