## ECS 20 — Possible Readings — Fall 2013

Several students have expressed surprise about my not following any particular book. (Get used to it; many professors prefer to find their own path.) I’ve never found a book for this class that I actually like, so it would not be appropriate for me to insist you buy something that I myself would not have wanted to read. The material presented is mostly standard, described in many books, but I do try to give interesting examples, which you will find in few. Here is a brief guide of where material I have spoken on can be found. I will add to it as we go along.
• Introductory lecture. Not aligned with any particular book. I did a proof sqrt(2) is irrational; matching upper and lower bounds for the Towers of Hanoi problem; and a clever proof that five shuffles won’t suffice to randomizae a deck of 52 cards.
• Propositional and first-order logic. The basic material is covered in Schaum’s, Chapter 4, and Biggs, Chapter 3. Vellerman Chapters 1-3 does a very nice job with logic and proofs (really very well written), but amounts to 160 pages to read. The Wikipedia entry for Propositional calculus explains proof systems, completeness, and soundness. There is a separate Wikipedia page for compactness. The application to the tiling problem is my own, as far as I know; the only writeup is in my notes. Rosen, Chapter 1, seems to be a pretty good treatment of propositional and first-order logic, but it will take you a hundred pages of tiresome reading and still won’t get to completness, soundness, or compactness. There is a Wikipedia page on First-order logic. There you can read many interesting things that we will not cover. Taking a nice class is logic is recommended.
• Set theory, formal languages, and a touch of group theory. Most books on discrete math have a chapter on naive set theory: in Schaum’s it’s Chapter 1 and in Biggs it’s Chapter 2. For an axiomatic treatment of set theory see the Wikipedia page on Set Theory and the one on Zermelo-Fraenkel set theory. Group axioms and examples groups can be found in the Wikipedia page on Group (mathematics). Computer-important finite sets like BYTES, WORDS32, and FLOAT64 are not normally featured in discrete-math books, but modular addition is instead described for general numbers (not necessarilly powers of two). There’s also a Wikipedia page on Excusive or that includes its bitwise operation and (under relation to modern algebra) that it’s a group. Operations on floating-point types are described in tt inhe Wikipedia article IEEE floating point. Alphabets, words, languages, and regular expressions are covered in Schaum’12.1-12.4.
• Natural numbers and induction. A list of axioms for number theory can be found in the Wikipedia article on the Peano axioms. Induction is discussed in covered in Schaum’s Chapter 1.8 and Biggs Chapter 4.3, which has further, basic information about the natural numbrs. The cake-cutting example is classical, and is nicely descried in notes by David Wagner. The envelope-exchange example and trominos examples might have first been found in an old edition of Goodaire and Parmenter; I don’t know, but have seen those examples in other books. Summation formulas involving induction are found in every discrete math, a dime a dozen.
• Equivalence relations. These are discussed in Schaum’s 2.8 (with background material we have mostly covered in 2.1-2.7) and Biggs 12.1, 12.2, and 12.4. Congruences are in Biggs 13.1 and 13.2 and Schaum’s 11.8 and 11.9. I have tried harder than most books to point out the relationship between equivalence relations and partitions, and to get across the meaning of “modding out” by an equivalence relation, something books seem to do a particularly poor job of.
• Functions. Basic material, like what are injective, surjective, and bijective functions, and frequently occurring functions, is found in Schaum’s 3.1-3.4 and Bigg’s Chapter 5. Wikipedia: Bijection, injection, and surjection.
• Infinite sets. See Schaum’s 3.7 and Biggs 6.5, 6.6, 9.7. Wikipedia entries: Cardinality, Aleph number, Georg Cantor, Hilbert's paradox of the Grand Hotel.
• Asymptotic notation. Big-O is on Scahum’s p.59. Some lecture notes by Daisy Tang look pretty good. The Wikipedia article Big O notation is fine. Notice that table “Famil of Bachman-Landau notations”.

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