A Short Course In Discrete Mathematics

, by Bender, Edward A.; Williamson, S. Gill

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ISBN: 9780486439464 | 0486439461
Cover: Paperback
Copyright: 12/13/2004

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What sort of mathematics do I need for computer science? In response to this frequently asked question, a pair of professors at the University of California at San Diego created this text. Its sources are two of the university's most basic courses: Discrete Mathematics, and Mathematics for Algorithm and System Analysis. Intended for use by sophomores in the first of a two-quarter sequence, the text assumes some familiarity with calculus. Topics include Boolean functions and computer arithmetic; logic; number theory and cryptography; sets and functions; equivalence and order; and induction, sequences, and series. Multiple choice questions for review appear throughout the text. Original 2005 ed. Notation Index. Subject Index.

Unit BF: Boolean Functions and Computer Arithmetic

Section 1: Boolean Functions

(26)

Boolean function, binary operator, unary operator, not (~), and (^), or (V), exclusive or ( ), truth table, disjunctive normal form, conjunctive normal form

Section 2: Number Systems and Computer Arithmetic

(14)

digit symbols, digit symbol of index or rank i, base-b number, binary arithmetic, two's complement, logic gate, half adder, full adder

Multiple Choice Questions for Review

(4)

Unit Lo: Logic

Section 1: Propositional Logic

(11)

truth table, statement forms, tautology, contradiction, implication, conditional, contrapositive, double implication, biconditional, converse, inverse, if, only if, sufficient, necessary, unless

Section 2: Predicate Logic

(11)

predicate, truth set, prime, composite, Fermat number, Mersenne number, perfect numbers, Goldbach conjecture, Fermat's Last Theorem, Marin Mersenne (1588-1648), Pierre de Fermat (1601-4665), Christian Goldbach (1690-1764), Leonhard Euler (1707-1783), Karl Friedrich Gauss (1777-1855)

Multiple Choice Questions for Review

(4)

Unit NT: Number Theory and Cryptography

Section 1: Basic Facts About Numbers

(12)

rational numbers, irrational numbers, prime, composite, odd, even, n divides m, prime factorization, infinitely many primes, perfect squares, irrationality of integral square roots, residue classes mod d, mod as binary operator, mod as equivalence relation, modular arithmetic, modular addition, modular multiplication, floor function, ceiling function, diagonalization proofs

Section 2: Cryptography and Secrecy

(16)

plaintext, ciphertext, key, espionage, greatest common divisor, least common multiple, gcd(m, n) as linear combination of m and n, Euclidean algorithm, Euler ø function, public key, symmetric encryption, discrete log problem, Diffie-Hellman algorithm, *RSA algorithm Multiple Choice Questions for Review

(3)

Unit SF: Sets and Functions

Section 1: Sets

(14)

intersection, union, difference, complement, symmetric difference, product, Cartesian product, binomial coefficients, C(n, k) = (nk), algebraic rules, associative rule, distributive rule, idempotent rule, DeMorgan's rule, absorption rule, commutative rule, lexicographic order, power set, characteristic function, set partitions, Bell numbers, refinement

Section 2: Functions

(14)

function, domain, range, codomain, image, relation, functional relation, one-line notation, surjection, onto, injection, one-to-one, bijection, permutation, two-line notation, composition of functions, cycle form of permutation, image of function, inverse image, coimage, set partitions, Stirling numbers S(n, k), Bell numbers

Multiple Choice Questions for Review

109

(4)

Unit EO: Equivalence and Order

Section 1: Equivalence

113

(11)

equivalence relation, equivalence class, blocks of a partition, coimage, reflexive, symmetric, transitive, relational description, coimage description, pigeonhole principle, subset sums, monotone subsequences, extended pigeonhole principle, transpositions

Section 2: Order

124

(22)

antisymmetric relation, order relation, partially ordered set, poset, incidence matrices, total ordering, linear ordering, set inclusion, lattice of subsets, incomparable subsets, refinement relation, subposet, direct product, Cartesian product, coordinate order, characteristic function, directed graph diagrams, transitive closure, Boolean product, Boolean sum, covering relation, Hasse diagram, chain, least element, greatest element, maximal element, minimal element, lexicographic (lex) order, length-first lex order, lexicographic bucket sort, comparison sort, tiling problems, domino coverings, rotation-reflection relation, linear extensions, topological sorting

Multiple Choice Questions for Review

146

(3)

Unit IS: Induction, Sequences and Series

Section 1: Induction

149

(11)

induction, strong induction, simple induction, induction hypothesis, induction step, base case, product of primes, sum of first n integers, *sums of kth powers, *differences

Section 2: Infinite Sequences

160

(8)

infinite sequence, limit of sequence, convergent sequence, bounded sequence, mono-tone sequence, convergent to infinity

Section 3: Infinite Series

168

(11)

infinite series, telescoping series, geometric series, harmonic series, alternating harmonic series, alternating series, generalized alternating series, absolute convergence, conditional convergent, tails of series, integral test, general harmonic series, convergence and intuition, the size of primes

Multiple Choice Questions for Review

179

(4)

Solutions to Exercises

183

(48)

Notation Index

231

(2)

Subject Index

233

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A Short Course In Discrete Mathematics

Summary

Table of Contents

Supplemental Materials