Mathematics for Computer Science by Eric Lehman, F Thomson Leighton, Albert R Meyer

By Eric Lehman, F Thomson Leighton, Albert R Meyer

This publication covers simple discrete arithmetic for desktop technological know-how and engineering. It emphasizes mathematical definitions and proofs in addition to appropriate tools. themes contain formal good judgment notation, evidence tools; induction, well-ordering; units, family members; uncomplicated graph idea; integer congruences; asymptotic notation and development of services; diversifications and mixtures, counting rules; discrete likelihood. extra chosen subject matters can also be lined, comparable to recursive definition and structural induction; kingdom machines and invariants; recurrences; producing functions.

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We list below a bunch of equivalence axioms with the symbol “ ! ” between equivalent formulas. These axioms are important because they are all that’s needed to prove every possible equivalence. We’ll start with some equivalences for AND’s “mcs” — 2011/6/2 — 10:18 — page 46 — #54 46 Chapter 3 Logical Formulas that look like the familiar ones for multiplication of numbers: ! B A AND B AND A ! 8) T AND A ! 9) F AND A ! 10) Three axioms that don’t directly correspond to number properties are A AND A !

21) At this point NOT only applies to variables, and we won’t need Demorgan’s Laws any further. 21) into a disjunctive form. 7) here to justify distributing over an AND from the right. 22) The last step is to turn each of these AND-terms into a disjunctive normal form with all three variables A, B, and C . B AND A/. This term needs to mention C to be in normal form. B AND A/ ! 2. Any propositional formula can be transformed into disjunctive normal form or a conjunctive normal form using the equivalences listed above.

There is a “there exists” notation, 9, to indicate that a predicate is true for at least one, but not necessarily all objects. So 9x 2 R: 5x 2 7 D 0 is true, while 8x 2 R: 5x 2 7D0 is not true. There are several ways to express the notions of “always true” and “sometimes true” in English. The table below gives some general formats on the left and specific examples using those formats on the right. You can expect to see such phrases hundreds of times in mathematical writing! 1P stands for problems whose instances can be solved in time that grows polynomially with the size of the instance.

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