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MI ~: www.EngineeringEBooksPdf.com List of Symbols Subject Symbol Logic Meaning Page not p pandq pvq porq p E q or p XOR q p or q but not both p and q pAq P Q P is logically equivalent to Q p q if pthenq 18 p q p if and only if q therefore 24 29 P(x) predicate in x 76 P(x) =t- Q(x) every element in the truth set for P(x) is in the truth set for Q(x) 84 P(x) Q (x P(x) and Q(x) have identical truth sets 84 V for all 78 A there exists 79 NOT-gate 46 AND-gate 46 ROR-gate 46 NAND-gate 54 NOR-gate 54 Sheffer stroke 54 Applications of Logic N NAND Peirce arrow 54 r2 number written in binary notation 58 nOo number written in decimal notation 58 n16 number written in hexadecimal notation 71 Number Theory and d In d%n d does not divide n 148 149 Applications n div d the integer quotient of n divided by d 158 n mod d the remainder of n divided by d 158 [xj the floor of x 165 Fxl the ceiling of x 165 W gcd(a, b) the absolute value of x the greatest common divisor of a and b 164 192 x := e x is assigned the value e 186 d divides n www.EngineeringEBooksPdf.com Page Subject Symbol Meaning Sequences and so forth 199 the summation from k equals m to n of ak 202 the product from k equals m to n of a, 205 n& n factorial 206 aE A a is an element of A 76 a0A a is not an element of A 76 (a,, a2 , , a.) the set with elements a,, a2 , , an 76 Ix E D I Plx)} the set of all x in D for which P(x) is true the sets of all real numbers, negative real numbers, positive real numbers, and nonnegative real numbers n E ak k=m n fl -k k=m Set Theory R, R-, R+, Ronneg 77 76, 77 Z' Z-' Z+ znonneg the sets of all integers, negative integers, positive integers, and nonnegative integers 76, 77 Q, Q- Q+, Qnonneg the sets of all rational numbers, negative rational numbers, positive rational numbers, and nonnegative rational numbers 76, 77 N the set of natural numbers A C B A is a subset of B 256 A g B A=B A =B A is not a subset of B 257 A equals B A UB A union B 258 260 A n B A intersect B 260 the difference of B minus A 260 the complement of A ordered pair 260 (xby) (XI, ordered n-tuple 264 the Cartesian product of A and B 265 A' X29 * * * , Xn) A x B A, x A x X An the Cartesian product of Al, A 77 264 A, 265 the empty set 262 97(A) the power set of A 264 www.EngineeringEBooksPdf.com List of Symbols Subject Symbol Meaning Counting and Probability N (A) P(A) the number of elements in a set A the probability of a set A 299 299 P(n, r) the number of r-permutations of a set of n elements 315 (n) n choose r, the number of r-combinations of a set of n elements, the number of r-element subsets of a set of n elements 334 multiset of size r 349 the probability of A given B 376 f is a function from X to Y 390 the value of f at x 390 x-f|y f sends x to y 390 f (A) the image of A 402 f 1(C) ix the inverse image of C the identity function on X 402 394 bx b raised to the power x 411 expb (x) b raised to the power x 411 1ogb (x) logarithm with base b of x the inverse function of F 395 415 the composition of g and f 432 x is approximately equal to y big-0 of f of x 206 519 (f (x)) big-Omega of f of x 519 J(f(x)) big-Theta of f of x 519 xRy x is related to y by R 572 r [x,1, Xi2 X iI P(A IB) Functions f: X f (x) -+ Y F-1 f og Algorithm Efficiency Relations x - y O(f(x)) Page R the inverse relation of R 578 m = n (mod d) m is congruent to n modulo d 597 [a] x -< y the equivalence class of a x is related to y by a partial order relation < 599 635 Continuedonfirst page of back endpapers www.EngineeringEBooksPdf.com DISCRETE MATHEMATICS WITH APPLICATIONS THIRD EDITION SUSANNI k S EPP DePaul Uuniversity TF-IcHIVIScON BROOKS/COLE Australia * Canada * Mexico * Singapore * Spain United Kingdom * United States www.EngineeringEBooksPdf.com THCOlVISCON BROOKS/COLE Cover Photo: The stones are discrete objects placed one on top of anotherlike a chain of careful reasoning A person who decides to build such a tower aspires to the heights and enjoys playing with a challengingproblem Choosing the stones takes both a scientific and an aesthetic sense Getting them to balance requires patient effort and careful thought And the tower that results is beautiful A perfect metaphorfor discrete mathematics! 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Thomson Learning Seneca, 53 Colonia Polanco 11560 Mexico D.F Mexico Spain/Portugal Paraninfo Calle Magallanes, 25 28015 Madrid, Spain www.EngineeringEBooksPdf.com To Jayne and Ernest www.EngineeringEBooksPdf.com CONTENTS Chapter The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence Statements; Compound Statements; Truth Values; Evaluating the Truth of More General Compound Statements; Logical Equivalence; Tautologies and Contradictions; Summary of Logical Equivalences 1.2 Conditional Statements Logical Equivalences Involving 17 ->; Representation of If-Then As Or; The Negation of a Conditional Statement; The Contrapositive of a Conditional Statement; The Converse and Inverse of a Conditional Statement; Only If and the Biconditional; Necessary and Sufficient Conditions; Remarks 1.3 Valid and Invalid Arguments 29 Modus Ponens and Modus Tollens; Additional Valid Argument Forms: Rules of Inference; Fallacies; Contradictions and Valid Arguments; Summary of Rules of Inference 1.4 Application: Digital Logic Circuits 43 Black Boxes and Gates; The Input/Output for a Circuit; The Boolean Expression Corresponding to a Circuit; The Circuit Corresponding to a Boolean Expression; Finding a Circuit That Corresponds to a Given Input/Output Table; Simplifying Combinational Circuits; NAND and NOR Gates 1.5 Application: Number Systems and Circuits for Addition 57 Binary Representation of Numbers; Binary Addition and Subtraction; Circuits for Computer Addition; Two's Complements and the Computer Representation of Negative Integers; 8-Bit Representation of a Number; Computer Addition with Negative Integers; Hexadecimal Notation Chapter The Logic of Quantified Statements 75 2.1 Introduction to Predicates and Quantified Statements I 75 The Universal Quantifier: V; The Existential Quantifier: B; Formal Versus Informal Language; Universal Conditional Statements; Equivalent Forms of the Universal and Existential Statements; Implicit Quantification; Tarski's World 2.2 Introduction to Predicates and Quantified Statements II 88 Negations of Quantified Statements; Negations of Universal Conditional Statements; The Relation among V, 3, A, and v; Vacuous Truth of Universal Statements; Variants of Universal Conditional Statements; Necessary and Sufficient Conditions, Only If iv www.EngineeringEBooksPdf.com Contents 2.3 Statements Containing Multiple Quantifiers V 97 Translating from Informal to Formal Language; Ambiguous Language; Negations of Multiply-Quantified Statements; Order of Quantifiers; Formal Logical Notation; Prolog 2.4 Arguments with Quantified Statements 111 Universal Modus Ponens; Use of Universal Modus Ponens in a Proof; Universal Modus Tollens; Proving Validity of Arguments with Quantified Statements; Using Diagrams to Test for Validity; Creating Additional Forms of Argument; Remark on the Converse and Inverse Errors Chapter Elementary Number Theory and Methods of Proof 125 3.1 Direct Proof and Counterexample I: Introduction 126 Definitions; Proving Existential Statements; Disproving Universal Statements by Counterexample; Proving Universal Statements; Directions for Writing Proofs of Universal Statements; Common Mistakes; Getting Proofs Started; Showing That an Existential Statement Is False; Conjecture, Proof, and Disproof 3.2 Direct Proof and Counterexample II: Rational Numbers 141 More on Generalizing from the Generic Particular; Proving Properties of Rational Numbers; Deriving New Mathematics from Old 3.3 Direct Proof and Counterexample Ill: Divisibility 148 Proving Properties of Divisibility; Counterexamples and Divisibility; The Unique Factorization Theorem 3.4 Direct Proof and Counterexample IV,Division into Cases and the Quotient-Remainder Theorem 156 Discussion of the Quotient-Remainder Theorem and Examples; div and mod; Alternative Representations of Integers and Applications to Number Theory 3.5 Direct Proof and Counterexample V Floor and Ceiling 164 Definition and Basic Properties; The Floor of n/2 3.6 Indirect Argument: Contradiction and Contraposition 171 Proof by Contradiction; Argument by Contraposition; Relation between Proof by Contradiction and Proof by Contraposition; Proof as a Problem-Solving Tool 3.7 Two Classical Theorems 179 The Irrationality of vf2; The Infinitude of the Set of Prime Numbers; When to Use Indirect Proof; Open Questions in Number Theory www.EngineeringEBooksPdf.com vi Contents 186 3.8 Application: Algorithms An Algorithmic Language; A Notation for Algorithms; Trace Tables; The Division Algorithm; The Euclidean Algorithm Chapter Sequences and Mathematical Induction 4.1 Sequences 199 199 Explicit Formulas for Sequences; Summation Notation; Product Notation; Factorial Notation; Properties of Summations and Products; Change of Variable; Sequences in Computer Programming; Application: Algorithm to Convert from Base 10 to Base Using Repeated Division by 4.2 Mathematical Induction I 215 Principle of Mathematical Induction; Sum of the First n Integers; Sum of a Geometric Sequence 4.3 Mathematical Induction II 227 Comparison of Mathematical Induction and Inductive Reasoning; Proving Divisibility Properties; Proving Inequalities 4.4 Strong Mathematical Induction and the Well-Ordering Principle 235 The Principle of Strong Mathematical Induction; Binary Representation of Integers; The Well-Ordering Principle for the Integers 4.5 Application: Correctness of Algorithms 244 Assertions; Loop Invariants; Correctness of the Division Algorithm; Correctness of the Euclidean Algorithm Chapter Set Theory 255 5.1 Basic Definitions of Set Theory 255 Subsets; Set Equality; Operations on Sets; Venn Diagrams; The Empty Set; Partitions of Sets; Power Sets; Cartesian Products; An Algorithm to Check Whether One Set Is a Subset of Another (Optional) 5.2 Properties of Sets 269 Set Identities; Proving Set Identities; Proving That a Set Is the Empty Set 5.3 Disproofs, Algebraic Proofs, and Boolean Algebras 282 Disproving an Alleged Set Property; Problem-Solving Strategy; The Number of Subsets of a Set; "Algebraic" Proofs of Set Identities; Boolean Algebras www.EngineeringEBooksPdf.com 1-4 Index Counting (cont.), 297-388 subsets of a set, 334-349 walks of length n, 693-695 Courses required for degree, 644 CPM (Critical Path Method), 644-646 Critical path, 646 Critical Path Method (CPM), 644-646 Critical row, 31, 37 Cryptography defined, 611 number theory and, 629-630 public-key, 612, 630 RSA, 617, 623-630 Databases, n-ary relations and, 581 Data type, 186 Date, regular expression for, 744 Davis, Philip J., 164, 282 Day of the week, computing, 158-159 Decimal digits, 155 Decimal expansion of fractions, pigeonhole principle applied to, 423-425 Decimal notation, 57 conversions to and from, 58-59, 71-72, 211-213 Decimal representation, 155 Decision tree, 706 Decoding functions, 396 Decreasing function, 515-517 Decrypting, 611 with Caesar cipher, 611-612 using RSA cryptography, 624-625 Dedekind, Richard, 608 Deductive reasoning, 1, 227 universal instantiation and, 111 Degree of a vertex, 658-662 De Morgan, Augustus, 1, 10, 217 De Morgan's laws of logic, 14, 91 application of, 11 described, 10-11 inequalities and, 11-12 De Morgan's laws for sets, 272, 289 generalized, 505 proof of, 276-278 Denying the antecedent, fallacy of, 37-38 Derangement of set, 475 Descartes, Rend, 97, 510, 543 Descendant, 715 Diaconis, Persi, 300 Diagrams arrow, 390-392, 415, 574-575 Hasse, 636-639 to show invalidity, 117-118 to test for validity, 115-119 transition, 748-749 Dice, probability in rolling pair of, 300-301 Dictionary order, 635-636 Difference between sets, 260, 261 Difference rule, 322-326 Digital logic circuits, 43-57 background of, 43-45 black boxes and gates, 45-46 Boolean expressions and, 48-51 equivalence classes of, 606 equivalence of, 53, 598 input/output table for, 47-48, 51-52 simplifying combinational circuits, 52-53 Digraph, 653 Dijkstra, Edsger W., 244 Dirac, P.A.M., 584 Direct argument, pigeonhole principle and, 427 Directed edge, 653 Directed graphs, 234, 653 matrices and, 684-686 of partial order relation, 637, 638-639 of relation, 580 of relation, recovered from Hasse diagram, 638-639 Direct proof counterexample I and, 126-141 counterexample II and (rational numbers), 141-147 counterexample III and (divisibility), 148-156 counterexample IV and (division into cases and quotient-remainder theorem), 156-164 method of, 131 of a theorem, 132-134 Dirichlet, Lejeune, 390, 420 Dirichlet box principle See Pigeonhole principle Disconnected graphs, 669-670 Discourse on Method (Descartes), 510 Discovery, 126 Discrete mathematics, origin of name, 77 Disjoint events, mutually, 382 Disjoint sets, 262-263 counting elements of, 321-333 Disjunction, defined, 5-6 negation of, 10- 11 Disjunctive normal form, 52 Disproof of alleged property of floor, 166-167 of alleged set property, 282-284 by counterexample, 129-130 of existential statement, 137-138 Disquisitiones Arithmeticae (Gauss), 597 Distinct equivalence class, 601-604 Distinct-roots case, 488-494 Distinct-roots theorem, 491-494 Distributive law, 14, 272, 288, 289, 481 proof of, 273-276 div, 157-159, 170 as function, 389 Divide-and-conquer approach, 557 binary search, 557-564 merge sort, 564-568 Divides, defined, 148 "Divides" relation antisymmetry of, 633-634 Hasse diagram for, 636-637 on set of positive integers, 634-635 Divisibility of algebraic expressions, 149 checking nondivisibility, 149-150 counterexamples and, 152-153 mathematical induction to prove, 228-230 by prime numbers, 150, 151-152, 236 proving properties of, 148-156 transitivity of, 150-152 unique factorization theorem, 153-154 Division algorithm, 190-191 Division into cases, proof by, 35, 40 Division-into-cases method of analysis, 19 Division rule, 349 www.EngineeringEBooksPdf.com Divisor(s), 148 greatest common, 192-196 positive, 148-149 of zero and one, 148, 149 Dodecahedron, puzzle in shape of, 676-679 Domain, 390 of Boolean function, 397 co-, 390, 402 of predicate variable, 76 Dot product, 689 Double complement law, 272, 288 proof of, 289-290 Double counting, 346 Double negative law, 14 Double negative property, Double of rational number, 146 Doubly indexed sequence, 469-470 Drawing graphs, 652-653 Dual identity, 289 Duality principal for Boolean algebra, 289 Dummy variable, 208 in loop, 211 EBCDIC (Extended Binary-Coded Decimal Interchange Code), 453 Edge-endpoint function, 650 Edges adjacent, 650 defined, 482, 650 directed, 653 incident on its endpoints, 650 parallel, 650 Edinburgh Prolog, 107n Edison, Thomas Alva, 487 8-bit representation, 63-70 Einstein, Albert, 321 Element argument, 260, 269, 271 Elementary operations, 532-533 Elements class of, 599 comparable, 639 counting, 302-304 Elements of Geometry (Euclid), 180, 182 Elements of set comparable and noncomparable, 639 disjoint set, 321-333 greatest, 640, 641 least, 240-241, 641 maximal, 640, 641 methods of selection, 334-335 minimal, 641 permutations with repeated, 344-345 Elimination, 34, 40 Elkies, Noam, 139 Ellipsis, 76, 199 Empty graph, 650 Empty set, 262, 278-280 deriving set identity using properties of, 286-287 uniqueness of, 279 Empty tree, 705 Encoding and decoding functions, 396 Encrypting, 611 with Caesar cipher, 611-612 using RSA cryptography, 624 End of world, calculating, 460-464, 480-481 Endpoints, 650, 653 End while, 188, 189 Enumeration, complete, 335 Index Equality of functions, 393-394 of ordered n -tuple sets, 264 properties of, 589 of sets, 258-260 Equally likely probability formula, 299 Equivalence classes, 599-610 of congruence modulo 3, 604-606 defined on finite sets, 600 of digital logic circuits, 606 of identifiers, 601 of identity relation, 601-604 names of, 605 rational numbers as, 607-608 representative of, 606 Equivalence of states of finite-state automata, 764-767 Equivalence relations, 594-610 congruence modulo n as, 615-616 defined, 597-599 equivalence classes of, 599-610 graph isomorphism as, 699 modular, 613-614 on sets of states of automaton, 764-773 Equivalent automata, 771-773 Equivalent digital logic circuits, 53, 598 Eratosthenes, 179 Escape character, 739 Etchemendy, John, 85 Euclid, 153, 180, 182, 183, 192 Euclidean algorithm, 192-196 correctness of, 251-253 extended version of, 618-620, 631 Euclid's lemma, 625-626 Euler, Leonhard, 139, 665-666 Euler circuits, 670-676 See also Hamiltonian circuits Euler path, 675-676 Euler phi function, 401 Euler's conjecture, 139 Even integers countability of set of all, 447 defined, 127 deriving additional result about, 145-146 Goldbach's conjecture about, 139 square of, 175-176 sum of, 132-134 Even parity, strings with, 741 Event(s), 299 independent, 381-385 probability of complement of, 324, 370-371 probability of general union of two, 371-373 Eventual-state function, 751-752 Examples, arguing from, 135-136 Exclusive or, 5-6 Exhaustion, method of, 79, 130 Existence of graphs, 660-661 Existential quantifier, 79-80 implicit, 83 Existential statements defined, 80 disproving, 137-138 equivalent forms for, 83 negation of, 89 proving, 128-129 true or false, 80 Expanded form, sum given in, 202, 203-204 Expected value, 373-374 to find average-case order for insertion sort, 539-540 of tossing loaded coin twice, 383-384 Expert systems, 121 Explicit formula checking correctness of, 483-485 for Fibonacci sequence, 493-494 finding, 476-477 for geometric sequence, 222-225, 578-580 for sequence, 200 simplifying, 480-483 Exponential functions with base 6,411-412 graphs of, 544 one-to-oneness of, 412 Exponential orders, 549-554 Exponents laws of, 411 modular arithmetic computations using, 617-618 Expressions See Algebraic expressions; Boolean expressions; Regular expression(s) Extended Binary Coded Decimal Interchange Code (EBCDIC), 453 Extended Euclidean algorithm, 618-620, 631 Extension, axiom of, 255 Factor, 148 Factorial notation, 206-207 Factorization theorem for integers, unique, 625, 626 Fallacies, 36-38 False positives and false negatives, 379-380 False solution, 346 Fantasy rule for mathematical proof, 271 Fermat, Pierre de, 138, 148, 184, 217, 302 last theorem, 138, 138n, 184, 185, little theorem, 626-627 Fermat primes, 184 Fibonacci (Leonardo of Pisa), 464 Fibonacci numbers, 464-465 Fibonacci sequence, formula for, 493-494 Final term, 200 adding on, 204 separating off, 204 Finite relations antisymmetry of, 632-633 inverse of, 578-579 Finite sets composition of functions defined on, 433 defined, 428, 443 equivalence class of relation defined on, 600 onto function defined on, 408 functions and relations on, 576 one-to-one and onto for, 429 one-to-one function defined on, 403-404 properties of binary relations on, 585-587 Finite-state automata, 735, 745-775 definition of, 748-750 designing, 752-754 equivalence relations and, 764-773 eventual-state function, 751-752 language accepted by, 750-751 nondeterministic, 758-759 pigeonhole principle and, 759-760 www.EngineeringEBooksPdf.com 1-5 regular expressions and, 756-759 simplifying, 763-775 software to simulate, 754-756 First-order logic, language of, 107 Floor, 164-171 Floor function, 389, 512-513, 535 graph of, 512-513 Floyd, Robert W., 244 For all statement, 78-79 negation of, 90, 91 Forest, 705 Formal languages, 735-738 See also Computer languages over alphabet, 736 translating from, 80-81 translating to, 81, 100-102 Formal logical notation, 105-107 Forms of argument See Argument form Formulas See also Explicit formula composition of functions defined by, 432 functions defined by, 392-393 Pascal's, 358-361 for sequence, 200 substitutions in, 358 summation of first n integers, 218-222, 482-483 For-next loop, 187, 189, 211 Forster, E M., 43 Forward chaining, 276 Fractions, pigeonhole principle applied to decimal expansion of, 423-425 Frege, F L Gottlob, 78, 608 Friedl, Jeffrey E F., 756n Frye, Roger, 139 Full-adder, 61-63 Full binary tree, 716 existence of, determining, 718-720 Fuller, R Buckminster, 697 Function machines, 392-394 Functions, 389-456 arrow diagrams for, 390-392, 415 Boolean, 397-398 cardinality with applications to computability, 443-456 ceiling, 389, 512 composed of rational power functions, orders of, 528 composition of, 431-443 constant, 392, 393 decreasing, 515-517 defined, 390 defined on a power set, 395 defined on a set of strings, 395 defined on general sets, 389-402 defined recursively, 505 efficiency of algorithms, 531-543, 557-570 encoding and decoding, 396 equality of, 393-394 eventual-state, 751-752 examples of, 394-396 exponential, 411-412 finite-state automata, 735, 745-775 floor, 389, 512-513, 535 f(x), 390 f(x) is o(g(x)), 531 graphing, 513-515 Hamming distance, 396 1-6 Index Functions (cont.), hash, 406-407 identity, on a set, 394 increasing, 515-517 inverse, 415-417 logarithmic, 395, 411-412, 413 multiple of, 514-515, 516-517 noncomputable, 454 one-to-one, 402-407 O-notation, 518-531 onto, 407-411 pigeonhole principle and, 420-431 power, 511-512, 522, 527-528 propositional 76 real-valued, of real variable, 510-518 recursive, 505-507 relations and, 575-577 sequences as, 389, 394 squaring, 392, 393 successor, 392, 393 well-defined, 398 Fundamental property of ordered pairs, 575 Fundamental theorem of arithmetic, 153-154 Galileo Galilei, 443 Gambler's ruin (example), 497 Gardner, Martin, 39n Gates Boolean expression and, 50-51 digital logic circuits and, 45-46 Gauss, Carl Friedrich, 153, 221, 597 General formula for sequence, 200 Generalization, 33, 40 Generalized associative law, 287 Generalized pigeonhole principle, 425-427 Generalizing from the generic particular, method of, 130-131, 138, 143 General partial order relation, notation for, 635 General polynomial functions, order of, 526 General recursive definitions, 499-509 Geometric sequence See also Arithmetic sequence defined, 479 explicit formula for, 578-580 formula for sum of, 222-225 "Geometry" (Descartes), 510 Gilbert, William S., 362 Glaser, 57 Gleick, James, 139 Godel, Escher Bach (Hofstadter), 271, 501 Godel, Kurt, 294 Goldbach, Chnstian, 139 Goldbach conjecture, 138-139 Golden ratio, 494, 499 Grammars, 735 Graph(s), 649-704 See also Directed graphs bipartite, 664 circuit-free, 705 complement of, 664 complete, 656-657 complete bipartite, 657 connected and disconnected, 669-670 defined, 511, 650 degree of a vertex, 658-662 drawing, 652-653 empty, 650 empty tree, 705 of equation, 650 examples of, 654-655 existence of, determining, 660-662 of exponential functions, 544 of f, 511 of floor function, 512-513 forest, 705 of function, 511, 650 isomorphic, 697-704 of logarithmic functions, 544-547 matrix representations of, 683-697 of multiple of function, 514-515 nonempty, 650 nonexistent, 660-662 paths and circuits in, 665-683 pictorial representation of, 652-653 of power function, 511-512 real-valued functions of real variable and, 510-518 simple, 656 subgraph, 657-658 terminology, 650-651 total degree of, 658-660 total weight of, 725 tree, 705 weighted, 725 Graph theory, origin of, 665 Greatest common divisor (gcd), 192-196 computing, by subtraction, 197 as linear combination, 619-620 Greatest element, 640, 641 Gries, David, 97, 244 Griggs, Jerrold, 258 Growth factor, 466 Guard, 187, 246 eventual falsity of, 247 Hairs on heads, example involving, 420-421 Half-adder, 61, 62 Halting problem, 294-296 Hamilton, Sir William Rowan, 676 Hamiltonian circuits, 676-679 See also Euler circuits Hamming, Richard W., 396 Hamming distance function, 396 Handshake theorem, 659 Hanoi, Tower of, 460-464, 480-481 Hardy, G H., 171, 199, 611, 629-630 Harmonic sums, 553-555 Hash functions, 406-407 Hasse, Helmut, 637 Hasse diagrams, 636-639 sideways, 645 Hausdorff, Felix, 575 Height of rooted tree, 714, 715 Hersh, Reuben, 164, 282 Hexadecimal notation, 70-73 Hilbert, David, 747 Hoare, C.A.R., 247 Hofstadter, Douglas, 179, 269, 271, 499, 501 Horizontal axes, 510 Honmer's rule, 543 Hydrocarbon molecules, structure of, 707-708 Hypothesis, 17, 18, 29 See also Inductive hypothesis inductive, 218, 235 Idempotent law, 14, 272, 288 proof of, 290 www.EngineeringEBooksPdf.com Identifiers binary relation on set of, 599 counting the number of Python, 324-325 equivalence and, 599 equivalence classes of, 601 Identities, set See Set identities Identity, defined, 272 Identity function composition with, 433435 on a set (example), 394 Identity laws, 14, 272, 288, 289 Identity matrices, 691, 692-693 Identity relation, equivalence classes of, 601-604 If, misuse of, 136-137 If-and-only-if statement, 23-25 necessary and sufficient conditions for, 25 If-then-else statements, 160, 186-187 execution in computer algorithms, 187 If-then statements, 186-187 chains of, 34 converse and inverse of, 27 converting only-if to, 23-25 execution in computer algorithms, 187 logical equivalences and, 19-20 necessary conditions in, 25, 26 negations of, 21, 91 or statements and, 19-20 sufficient conditions in, 25 true or false, 17-18, 82 truth tables for, 18 ijth entry of a matnx, 684 of a power of an adjacency matrix, 694-695 Illegal expressions (Boolean), 500 Image, 402 inverse, 390, 402 Images of X under f, 390 Implication arrow, 523 Implicit universal quantification, 83-85 Inaccessible states of finite-state automata, 773 Incident on (edge), 650 Inclusion/exclusion rule, 326-330 Inclusion in union, 269 proof of, 270 Inclusion of intersection, 269 proof of, 270-271 Increasing function, 515-517 Independent events, 381-385 Index,200 Index of summation, 202, 208 Index variable, 558 Indirect argument contradiction and contraposition and, 171-179 examples of, 179- 183 when to use, 183 Induction, 227-228 See also Mathematical induction Inductive hypothesis, 218, 235 Inductive property, 247 Inductive step, 218 Inequality(ies) De Morgan's law and, 11-12 logarithmic, 551-552 mathematical induction to prove, 230-232 notation for, Index polynomial, 523 triangle, 164 Inference, rules of, 33-36, 40 See also Modus ponens; Modus tollens application of, 35-36 Inferred truth, 30 Infinite relations, inverse of, 579 Infinite sequence, 200 Infinite set(s) cardinality of, 445, 446 countability of, 445-447 defined, 428, 443 onto functions defined on, 409-411 one-to-one functions defined on, 404-406 relations on, 589-592 Infinities, search for larger, 447-452 Infinitude of the set of prime numbers, 182-183 Infix notation, 737 Informal language translating from, 81, 100-102 translating to, 80-81 Informal negations of universal statements, 90 In general, mathematical definition of, 92-93 Initial conditions, 458, 459 Initial state, 748 Initial term, 200 explicit formula to fit given, 201-202 Injective function See One-to-one function Input alphabet, 748 Input/output devices finite-state automata as, 771 functions as, 392, 393 Inputloutput tables for circuit with two input signals, 310-311 designing circuit for, 51-52 for digital logic circuits, 45,47-48, 51-52 as function, 389, 397 Inputs, 390 Insertion sort algorithm, 536-540 Integer(s) See also Rational numbers; Real numbers binary notation for, 58 binary representation of, 57-59, 238-240 consecutive, 141, 155 consecutive, with opposite parity, 159-161 countability of sets of, 446-447 counting the number of, divisible by five, 322 defined as equivalence classes of ordered pairs of positive integers, 608 divisibility by prime numbers, 236 even, 127 expressing I as linear combination of relatively prime, 621-622 formula for sum of first n, 218-222, 482-483, 527-528 graphing functions defined on sets of, 513-514 greatest, 172 linear combination of, 619-620 negative, computer addition with, 66-70 negative, two's complements and computer representation of, 63-66 number of bits to represent in binary notation, 58, 547-548 odd, 127 order for sum of first n, 527-528 parity of, 159-161 pigeonhole principle and, 422-423 prime and composite, 127-128 quotients of, 141-147 representations of, 159-163 set of all (Z), 76 square of an odd, 161-163 study of properties of, 148-156 sum of rational and irrational, 173-175 unique factorization theorem for, 153-154, 625, 626 well-ordering principle for, 240-242 Integer powers of real numbers, nonnegative, 364 Integer variables, order for functions of, 527-528 Integral solutions of equation, counting number of, 353-354 Internal vertex, 710-711 Internet, searching on (example), Internet addresses, 325-326 Intersection(s) counting number of elements in, 329-330 empty set and, 262 of equivalence classes, 603-604 inclusion of, 269, 270 of independent events, probability of, 382-383 recursive definition of, 504-505 of sets, 260, 261 and union with subset, 278 Intervals and set operations, 261 Intractable algorithms, 568 Invalid argument form, 30-31 Inverse of conditional statements, 22-23 of relation, 578-579 of universal conditional statement, 93-94 Inverse error, 37-38 quantified form, 118, 121 Inverse functions, 415-417 composing a function with, 435-436 Inverse image, 390, 402 Inverter, 45-46 Irrational numbers defined, 141 determining rational number versus, 142-143 irrationality of square root of two, 180-182 summed with rational, 173-175 Isolated vertex, 650 Isomers, 707, 708 Isomorphic graphs, 697-704 Isomorphic invariants, 700-701 Isomorphic structures, 773 Iterations counting the number of, in a loop, 353 counting the number of, in a nested loop, 311 solving recurrence relations by, 475-487 Iterative statements, 187-189 ith row of matrix, 683 Job scheduling problem, 644-646 jth row of matrix, 683 Jumping to a conclusion, 37, 136 www.EngineeringEBooksPdf.com 1-7 Kant, Immanuel, 1, 723 k-equivalence class, finding, 766-767 k-equivalent states of finite-state automata, 765-767 Killian, Charles, 258 Kirchoff, Gustav, 707 Kleene, Stephen C., 734, 736, 738, 756 Kleene closure of I, 736 Kleene closure of L, 738 Kleene closure of r, 738 Kleene's Theorem, 756-759 Knights and knaves example, 39-40 Knowledge base of information, example of, 654-655 Knuth, Donald E., 133, 364n, 518, 531 Kolmogorov, Andrei Nikolaevich, 299, 370, 371 Kbnigsberg, Bridges of (puzzle), 665-666 Kripke, Saul, 296 Kronecker, Leopold, 692 Kronecker delta, 692 Kruskal, Joseph B., 725, 726 Kruskal's algorithm, 726-728 Kuratowski, Kazimierez, 575 Lagrange, Joseph Louis, 202 Lame, Gabriel, 194 Language of First-OrderLogic, The (Barwise and Etchemendy), 85 Languages See also Computer languages; Formal languages accepted by automaton, 750-751 accepted by quotient automata, 769, 771 defined by regular expression, 738-742 of first-order logic, 107 natural, 80-81 nonregular, 759-760 not accepted by any finite-state automaton, 759-760 regular, 735, 759-760 Laplace, Pierre-Simon, 302, 370, 375 Last theorem, Fermat's, 184, 185 Laws of exponents, 411 Leaf, 710-711 Least common multiple (Icm), 198 Least element, 641 finding, 240-241 Least nonnegative residues modulo n, 614 Left child, 716 Left subtree, 716 Legal expressions (Boolean), 500 Leibniz, Gottfried Wilhelm, 1, 116 Lemma, 193 Euclid's, 625-626 handshake, 659 Length of chain, 640 of string, 310, 736 of walk, 693-695 Leonardo of Pisa, 464 Less than, properties of, 590 "Less than or equal to" relation, 635 Level of vertex, 715 Lexical scanner, 735 Lexicographic order, 635-636 Limit, definition of, 101-102 Linear, 487 1-8 Index Linear combination of integers, 619 greatest common divisor as, 619-620 Linear combinations to satisfy initial conditions, 490-493 Linguistics, 707 List, counting elements of, 302-304 Little theorem, Fermat's, 626-627 Lobachevsky, Nicolai Ivanovitch, 632 Lob's paradox, 296 Logarithmic functions with base b, 395, 411-412, 413, 545 with bases greater than one, 552-553 graphs of, 544-547 Logarithmic inequalities, deriving order from, 551 -552 Logarithmic orders, 549-554 Logarithms common, 413 natural, 413 properties, 412, 419 to solve recurrence relations, 548-549 Logic defined, I limits of, 8, 12 Logical connectives gates and, 46 notation for, 17 switching devices and, 44 Logical equivalence conditional statements and, 19 defined, 8-9 double negative property, expressed as tautologies, 12-13 involving if-then statements, 19-20 involving tautologies and contradictions, 13 for quantified statements, 88 showing nonequivalence, 9-10 summary of, 13- 15 types of, 14 Logical expression, Logical form of argument, 1-2 Logical inference, 17 Logical operators, order of operations for, 24 Loop See also Nested loop counting the number of iterations in a, 353 defined, 650 Loop invariants, 246-251 defined, 247 procedure, 245 theorem, 247 Lottery, expected value of, 373-374 Lovelace, Countess of, 186 Lower limit of summation, 202 Lucas, tdouard, 460, 461 Lukasiewicz, Jan, 737 Lynch, John, 138n McCarthy, John, 506 McCarthy's 91 function, 506 McCulloch, Warren S., 734 Mach, Ernst, 571 Main diagonal of matrix, 684 Manin, I., 227 Mathematical induction, 199, 215-244 See also Strong mathematical induction; Well-ordering principle to check explicit formulas, 483-485 defined, 216-217 defining sequences and, 458 geometric sequence, formula for, 222-225 method of proof by, 218 principle of, 217 proving divisibility property, 228-230 proving inequality, 230-232 proving property of sequence, 232-233, 236-237 for recursively defined sets, 502 strong, 235-240 summation of first n integers, formula for, 218-222 Mathematical proofs See Proof(s) Mathematical structure, 773 Matrix (matrices) adjacency, 684-685, 686-687 connected components, 687-688 defined, 683 directed graphs and, 684-686 identity, 691, 692-693 multiplication, 689-693 powers of, 692-693 products of, 689-690 representations of graphs, 683-697 square, 684 symmetric, 687 undirected graphs and, 686-687 Maurolico, Francesco, 217 Maximal element, 640, 641 Memory circuit, computer, 746 Memory dump, reading, 72-73 Menge, 255 Merge sort, 564-568 Mersenne, Marin, 183 Mersenne primes, 183-184 Messages, coding, 396 Method of complete enumeration, 335 of direct proof, 131 of division into cases, 35, 40 of exhaustion, 79, 130 of generalizing from generic particular, 130-131, 138, 143 of iteration, 475-480 of proof by contradiction, 171-175 of proof by contraposition, 175-178 Middle elements of array, 557, 558-559 Mill, John Stuart, 111 Minimal element, 641 Minimal spanning trees, 725-731 See also Kruskal's algorithm; Prim's algorithm MIU-system, 501 mod/modulo, 157-159, 161, 170 See also Modular arithmetic congruence modulo relation, 573 congruence modulo relation, 590-592, 597, 604-606 congruence modulo n, 613-615, 626 as function, 389 inverse modulo n, 621-623 notation for congruence relations, 597 Modular arithmetic, 615-618 applications to cryptography, 623-630 inverse modulo n in, 621-623 practical use of, 616 www.EngineeringEBooksPdf.com Modular equivalence relations, 613-614 Modus ponens, 31-32, 33,40 universal, 112-114, 115 validity of, 112, 113, 115 Modus tollens, 32-33, 40 universal, 114, 119 Monty Hall problem, 301 Multiple of a function, 514-517 of integer, 148 least common (1cm), 198 Multiple-input AND-gate, 51 Multiple-input OR-gate, 51 Multiplication rule, 307-320 as difficult or impossible to apply, 311-313 Multiplications matrix, 689-693 needed to multiply n distinct numbers, number of, 238 Multiplicative identity, 185 of matrix, 691-692 Multiply-quantified statements interpreting, 99-100 negation of, 102-104 translating from informal to formal language, 100-102 truth value of, 100, 104 writing, 97 Multiset of size r, 349-356 Mutually disjoint subsets, union of, 263 Mutually disjoint events, independence and, 382 Mutually disjoint sets, 262-263 Mutually independent events, 384 NAND-gate, 54 Napier, John, 544, 545 n-ary relations, 572, 580-581 National Security Agency, 611 Natural language translating from, 81 translating to, 80-81 Natural logarithms, 413 Natural numbers, set of (N), 77 Naur, Peter, 707 Necessary conditions, 25-26 for universal conditional statements, 94-95 Negation(s) in Boolean algebra, 288 of multiply-quantified statements, 102-104 of quantified statements, 88-90 Negation law, 10-11,14 Negation of statement, truth value for, 4-5 Negative integers computer addition with, 66-70 two's complements and computer representation of, 63-66 Neither-nor, 3-4 Nested loop number of iterations of, 311 order of an algorithm with, 534-535 Newton, Isaac, 116 Index Next-state function, 748 Next-states of star(*)-equivalent states, 768 Next-state table, 747, 748 annotated, 748, 749-750 Nicomachus, 627 n! (n factorial), 206 Nonaccepting states of automaton, 750 Noncomparable elements, 639 Nonconstructive proof of existence, 129 Nondeterministic finite-state automata, 758-759 Nondeterministic polynomial-time algorithm (NP), 568n Nondivisibility, checking, 149-150 Nonempty graph, 650 Nonequivalence, showing, 9-10 Nonexistent graph, 660-662 Nonfunction, 392 Nonisomorphic graphs, 701-702 Nonisomorphic trees, 712-714 Nonnegative integer powers of real numbers, 364 Nonnegative integers, set of, 256 Nonoverlapping sets, 262-263 Nonregular language, 759-760 Non-trees, 705 Nontrivial circuit, 669 Nontrivial Hamiltonian circuit, 678 NOR-gate, 54 Not, symbol for, Notation for algorithms, 190 Backus-Naur, 707, 735 binary, 57-59 for congruence relations, 597 decimal, 57, 58-59 for equivalence class, 600 to facilitate working with regular expressions, 742-743 factorial, 206-207 for formal language, 736 formal logical, 105-107 for functions, 390, 393 for general partial order relations, 635 hexadecimal, 70-73 for implicit quantification, 84-85 infix, 737 for language, 707 octal, 74 postfix, 737 prefix, 737 product, 205 for relations, 572 for sets, 255-256 for sets, to describelanguage defined by regular expression, 739-740 for star(*)-equivalence classes, 768 summation, 202-205 for walks, 668 NOT-gate, 45-46 Not well defined function, 398 recursive "function" as, 507 NP-complete, 568 n-tuples, 397 ordered, 264-265, 310 Null set, 262, 278-280 deriving set identity using properties of, 286-287 Null string, 310, 736, 742 Number of elements in set, 428 Numbers See Integer(s); Rational numbers; Real numbers Number theory See also Modular arithmetic algorithms and, 186-198 cryptography and, 629-630 defined, 148 divisibility, 148-156 Euclid's lemma and, 625-626 floor and ceiling, 164-171 open questions in, 183-184 properties of integers, 148-156 properties of rational numbers, 144-145 quotient-remainder theorem, 156- 157 Octal notation, 74 Odd integers defined, 127 deriving additional result about, 145-146 squares of, 161-163 Of order at least g, 519 Of order at most g, 518, 519 Of order g, 518-519 Of order g(n), 533 Omega-notation (Q-notation), 518-531, 551 approximation for polynomial with some negative coefficients, 525-526 definitions to show order of polynomial function with positive coefficients, 523-524 properties of, 521 translating to, 520 One-dimensional arrays, 210-211 counting elements of, 303-304 1-equivalence classes, finding, 767, 772 One's complement, 64 One-to-one correspondences, 402,413-414 inverse function for, 415-417 One-to-one function, 402-407 See also Pigeonhole principle composition of, 436-438 exponential and logarithmic functions as, 412 for finite sets, 429 not possible, 420 One-to-one property, 402 Only if biconditional and, 23-25 universal conditional statements and, 94, 95 O-notation, 518-531 description of, 518-521 exponential and logarithmic orders, 550-552 for polynomial with negative coefficients, 524 polynomial orders, 523-527 translating to, 520 Onto functions, 407-411 composition of, 438-441 for finite sets, 429 Onto property, 402 Open sentences, 76 Operations, order of, 3, 18, 24 Operations on sets, 260-261 Optimistic approach to problem solving, 284 www.EngineeringEBooksPdf.com 1-9 or exclusive, 5-6 symbol for, when to use, 12 Order, algonthm, 532-535 Ordered 4-tuples, 308 Ordered n-tuple, 264-265, 310 Ordered pairs, 264 fundamental property of, 575 vertices of, 653 Ordered selection of elements, 334-335 Ordered triple, 264 Ordinal number, 443 OR-gate, 45-46 multiple-input, 51 Ongin, 77, 510 or statement See also and statement; If-then statements negation of, 10-11, 91 switching circuit for, 44 Outputs, 390 Pairwise disjoint sets, 262-263 Pairwise independent events, 383-384 Palindrome, 736 Parallel, switches in, 43-44 Parallel adder, 63 Parallel edges, 650 Parallel processing of data, 569 Parent, 715 Parenthesis structure, 501-502 property of set of, 502-503 Parity of integers, 159-161 Parity property, 159-161 Parse trees, 706-707 Partially ordered set, 640 Partial order relations, 632-648 compatible, 640-641 defined, 634 fundamental, 634-635 Hasse diagrams, 636-639 lexicographic order, 635-636 partially and totally ordered sets, 639-641 PERT and CPM, 644-646 topological sorting, 641-644 Partitions of sets, 262-264 relations induced by, 595-597 Pascal, Blaise, 141, 217, 302, 358 Pascal's formula, 358-361 Pascal's triangle, 358-359 Passwords, counting, 321 Paths in graphs, 665-683 Peano, Giuseppe, 260, 608 Peirce, Charles Sanders, 54, 78, 572 Peirce arrow, 54 Perfect square, 87, 139 Periodical publications, knowledge base about, 654-655 Permutations, 313-317 defined, 313 of letters in word, 314 of objects around circle, 314 relation between combinations and, 335-337 r-permutation, 315-317 of selected letters of word, 317 of a set with repeated elements, 344-345 1-10 Index Personal identification numbers (PINs), counting the number of, 308, 309, 323-324 PERT (Program Evaluation and Review Technique), 644-646 Pessimistic approach to problem solving, 284 Pi, 205 Piaget, Jean, 17 Pictorial representations of graphs, 652-653 Pigeonhole principle, 420-431 application of, 420-423 application to decimal expansions of fractions, 423-425 application to finite-state automata, 759-760 generalized, 425-427 proof of, 428-429 Pitts, Walter, 734 Plaintext, 611 Poker hand problems, example, 343-344 Polish notation, 737 Polynomial, root of, 147 Polynomial evaluation, term-by-term, 543 Polynomial functions See also Power functions limitations on orders of, 526-527 with negative coefficients, approximations for, 524-525 orders of, 523-527 Polynomial inequality, 523 Polynomial-time algorithms, 568 Poset, 640 Positive closure of S, 736 Possibility trees, 306-320 double counting on, avoiding, 346 multiplication rule and, 307-320 Post, Emil, 734 Post-conditions algorithm, 245-246 for loop, 246, 247 Postfix notation, 737 Power functions, 511 -512 defined, 511 graphs of, 511-512 orders of, 522, 527-528 rational, orders of functions composed of, 528 Powers of matrix, 692-693 of ten, 480 Power sets, 264 function defined on, 395 Pre-conditions, algorithm 245-246 for the loop, 246, 247 Predicate calculus, 75 Predicates defined, 76 and quantified statements 1, 75-88 and quantified statements II, 88-97 truth values of, finding, 76-77 Predicate symbols, 76 Predicate variables, 76 Prefix notation, 737 Preimage, 390 Premises, 29 ambiguous, using, 36 defined, I majorandminor, 112, 114, 115, 117 Prim, Robert C., 725, 729 Prime integers, defined, 127-128 Primeness, relative, 621-622 Prime numbers, 83 divisibility by, 150, 151-152,236 Fermat primes, 184 infinitude of set of, 182-183 Mersenne primes, 183-184 twin primes conjecture, 184 Prim's algorithm, 729-731 Principle of mathematical induction, 217 Printing problem, 296 Probability(ies) binomial, 385 of complement of event, 324, 370-371 conditional, 375-378 counting and, 298-306 for deck of cards, 299-300 equally likely probability formula, 299 of general union of two events, 371-373 of intersections of independent events, 382-383 Probability axioms, 370-373 Problem-solving strategies, 284 Problem-solving tool, proof as a, 177-178 Procedural versions of set definitions, 270 Product(s) Cartesian, 264-265, 309, 397, 572-573, 580-581 correctness of loop to compute, 249-251 dot, 689 of matrices, 689-699 notation, 205 properties of, 207-208 recursive definition of, 503-504 scalar, 689 Productions, 707 Product modulo n, computing, 617 Product rule See Multiplication rule Program Evaluation and Review Technique (PERT), 644-646 Projection onto number line, 452 Prolog (programming language), 107-108 Proof(s) See also Algorithms; Mathematical induction algebraic, for set properties, 360 beginnings of, 137 combinatorial, 357, 360-361, 367 common mistakes in, 135-137 by contradiction, method of, 171-175 by contraposition, method of, 175-178 defined, 125 direct, and counterexample I, 126-141 direct, and counterexample II (rational numbers), 141-147 direct, and counterexample III (divisibility), 148-156 direct, and counterexample IV (division into cases and quotient-remainder theorem), 156-164 direct, and counterexample V (floor and ceiling), 164-171 direct, of a theorem, 132-134 and discovery, 126 disproof by counterexample, 129-130 disproof of existential statement, 137-138 by division into cases, 35, 40 of existential statements, 128-129 of famous theorems, 179-183 www.EngineeringEBooksPdf.com floor and ceiling, 164-171 indirect, 171-179 indirect, when to use, 183 method of generalizing from the generic particular, 143 as a problem-solving tool, 177-178 of properties of divisibility, 150-152 of properties of rational numbers, 144-145 universal modus ponens in, 13-114 of universal statements, 130-135 variations among, 135 writing, for universal statements, 134-135 Proper subset, 257 Properties of functions, 402-419 of sets, 269-282 Proposition, 2, 176 Propositional calculus, 75 Propositional forms See Statement forms Propositional functions, 76 Pseudocode, 186 Public key, 623 Public-key cryptography, 612, 630 Push-down automaton, 735 P vs NP problem, 568 Pythagoras, 180 Pythagorean theorem, 180 Python identifiers, counting, 324-325 Q.E.D (quod erat demonstrandum), 134 Quantified statements, 75-124 arguments with, 111- 121 implicit quantification, 83-85 logical equivalence for, 88 multiply-, 97-100 negation of, 88-90 predicates and statements I, 75-88 predicates and statements II, 88-97 validity of arguments with, 115-119 Quantifiers defined, 78 existential, 79-80 order of, 97, 104-105 universal, 78-79 Quaternary relations, 581 Quine, Willard VanOrman, 18 Quod erat demonstrandum(Q.E.D.), 134 Quotient(s), 156 of integers, 141-147 Quotient automaton, 764, 768-771 Quotient-remainder theorem, 156-157 existence part, 241-242 Rabbits, calculating reproductive rates of, 464-465 Ralston, Anthony, 215 Random process, 298-299 Range, 390, 391,402 Rational numbers Archimedean property for, 243 defined, 141 direct proof and, 141-147 double of, 146 as equivalence classes, 607-608 set of all positive, countability of, 448-452 set of all (Q), 76 Index summed with irrational, 173-175 sum of rationals is rational, 144-145 Rational power functions, orders of functions composed of, 528 r-combinations, 334 with repetition allowed, 349-356 Real numbers See also Integer(s) between and 1, 450-451, 452 cardinality of set of all, 452 functions and relations on sets of, 576-577 nonnegative integer powers of, 364 reciprocal of, 100 relation to decimals, 449-450 relation to number line, 77 set of all (R), 76 trichotomy property of, 35 Real-valued functions of real variable, 510-518 Reciprocal, 100, 179 Recognizer, 49 Recurrence relations, 457-458, 459-460, 470-472, 561-562 logarithms used to solve, 548-549 second-order linear homogeneous, 487-499 Recursion, 457-509 in merge sort algorithm, 565 recursive definition of sums and products, 503-504 recursive functions, 505-507 Recursion, recursively defined sequences, 457-475 recursively defined sets, 500-503 second-order linear homogeneous recurrence relation, 487-499 solving by iteration, 475-487 Recursive definition for factorial, 206 for product notation, 205 of summation, 204 Recursive leap of faith, 460 Recursive paradigm, 460 Reduce a number modulo n, 614 Reductio ad absurdum, 171 Reduction ad impossible, 171 Reflexive property of cardinality, 444 Reflexivity, 584-594 See also Equivalence relations; Partial order relations Regular expressionss, 735, 738-744 over alphabet, 738-739 finite-state automaton and, 756-759 language defined by, 738-742 order of precedence for operations in, 739 practical uses of, 742-744 Regular languages, 735, 759-760 Relational database theory, 580-581 Relation of equality, 589-590 Relations, 571-648 antisymmetry property, 632-634 arrow diagrams of, 574-575 binary See Binary relations defined,572 directed graph of, 580 equivalence, 594-610 equivalence, finite-state automata and, 764-773 functions and, 575-577 identity, equivalence classes of, 601-604 inverse, 578-579 n-ary, 572, 580-581 notation for, 572 partial order, 632-648 properties of general binary, 584-594 quarternary, 581 recurrence, 459-460, 470-472, 548-549, 561-562 reflexivity, symmetry, and transitivity and,584-594 second-order linear homogeneous recurrence, 487-499 on sets, 571-583 ternary, 581 total order, 639-640 Relative complement of set, 260 Relative primeness, 621 expressing as linear combination of relatively prime integers, 621-622 Remainder, 156-157 Repeating decimal, 423 Representative of equivalence class, 606 Residue of a, 614 Residues modulo n, 614 Restriction, 500 of partial order relation, 648 Reverse Polish notation, 737 Ribet, Kenneth, 138 Right child, 716 Right subtree, 716 Rivest, Ronald, 612 Rooted trees, 714-716 Root of polynomial, 147 Roussel, P., 107 Rows, multiplying, 689 r-permutation, 315-317 RSA cryptography, 617, 623-630 Chinese remainder theorem and, 627-628 cipher, why it works, 628-629 decrypting message using, 624-625 encrypting message using, 624 Fermat's little theorem and, 626-627 Rules addition, 321-322 difference, 322-326 division, 349 inclusion/exclusion, 326-330 multiplication, 307-320 of universal instantiation, 111 -112 Russell, Bertrand, 235, 293, 294, 296, 475, 518 Russell's paradox, 293-294, 296 Sample space, 299 Savage, Carla, 258 Sawyer, W W., 665 Scalar product, 689 Schroeder-Bernstein theorem, 456 Schultz, James E., 431 n Search algorithm binary, 557-564 sequential, 536 Second order, defined, 487 Second-order linear homogeneous recurrence relation with constant coefficients, 487-499 Selection sort algorithm, 542 Semantics, 707 www.EngineeringEBooksPdf.com 1-11 Sequences, 199-215 See also Recursion alternating, 200-201 arithmetic, 577-578 change of variable, 208-2 10 in computer programming, 210-211 defined, 200 doubly indexed, 469-470 factorial notation and, 206-207 finding terms given by explicit formulas, 200-202 as functions, 389, 394 geometric, 222-225, 479, 578-580 linear combinations of, to satisfy initial conditions, 490-493 product notation and, 205 properties of summations and products, 207-208 proving property of, 232-233, 236-237 singly indexed, 469 summation notation and, 202-205 Sequential circuits, 46, 746 Sequential search algorithm, 536 Series, switches in, 43-44 Set(s) See also Subsets of accepting states, 748 Boolean algebra and, 287-290 Cartesian products, 264-265 countable, 445-447, 451 countably infinite, 445-446 counting subsets of a, 334-349, 368 defining, 76 derangement of, 475 disjoint, 262-263 empty, 262, 278-280 formal languages, 735-738 given by defining property, 256 with no elements, 262, 278-280 number of elements in, 428 number of partitions of, into subsets, 469-472 number of subsets of, 284-285 operations on, 260-261 partially and totally ordered, 639-641 partitions of, 262-264 power, 264 properties of, 269-282 properties of, disproofs of, 282-284 recursively defined, 500-503 relations and, 258, 571-583 with same cardinality, 443-445 of strings, function defined on, 395 of strings over alphabet, 500-501 symmetric difference of, 292 uncountable, 446, 450-451, 452 with uncountable subsets, 451 universal, 260 Set difference law, 272 Set difference property, deriving, 286 Set equality, 258-260 Set identities, 272-278 "algebraic" proofs of, 286-287 counterexample for, 283-284 proving, 273-278 Set notation, 255-256 to describe language defined by regular expression, 739-740 Set theory, 255-296 I 1-12 Index Shamir, Adi, 612 Shannon, Claude, 43, 734 Sheffer, H M., 54 Sheffer stroke, 54-55 Siblings, 715 Sieve of Eratosthenes, 179 Sigma, 202 Signal bit, 45 Simple circuit, 667 Simple conditional, 26 Simple graphs, 656 isomorphic, 702-703 nonexistent, 660-662 Simple path, 667 Singh, Simon, 138n Single-root case, 494-497 Single-root theorem, 495-496 Singly indexed sequence, 469 Size of matrix, 684 Smullyan, Raymond, 39 Socks, example of matching, 422 Software to simulate finite-state automaton, 754-756 Solution to recurrence relation, 475 Sorting, methods for insertion sort, 536-540 merge sort, 564-568 selection sort, 542 topological sorting, 641-644 Space efficiency of algorithms, 569 Spanning trees, 723-733 minimal, 725-731 Specialization, 33-34, 40 Square matrix, 684 Square of an odd integer, 161-163 Square root of two, irrationality of, 180-182 Squaring function, 392, 393 Standard factored form of integer, 154 *-equivalence (star equivalence) classes, finding, 767-768 *-equivalent (star equivalent) states of finite-state automata, 765 Statement calculus, 75 Statement forms defined, logically equivalent, 8-9 simplifying, 14 Statements compound, 3-8 conditional, 17-29 containing multiple quantifiers, 97-100 contradictory, 12 defined, 2-3 logically equivalent, predicates and, 75-97 quantified, 75-124 true or false, States of automaton, 748 (State)-transition diagram See Transition diagram Stevin, Simon, 449 Stirling, James, 469 Stirling numbers of the second kind, 469-470 String of characters of alphabet, 736 String-reversing function, 414 Strings, 310-311 See also Bit string accepted by automaton, 752-754 with even parity, 741 function defined on set of, 395 individual, in language defined by regular expression, 740-741 one-to-one correspondences involving, 413-414 recursively defined sets of, 500-501 relation on set of, 574 Strong mathematical induction, 235-240 See also Well-ordering principle Structural induction, 502 Structures, mathematical, 773 Subgraphs, 657-658 See also Spanning trees connected, 670 Sublist, counting elements of, 302-303 Subscript, 200 Subset relation, 256, 634 Hasse diagram for, 637-638 Subsets algorithm for checking for, 266-267 chain of, 640 characteristic function of, 401 of countable sets, 451 counting, of a set, 334-349, 368 defined, 256-257 intersection and union with, 269 number of partitions of set into, 469-472 of partially ordered sets, 640 proof of subset relation, 270-271 proper, 257 of set, number of, 284-285 totally ordered, 640 Substitutions into binomial theorem, 367 in formulas, 358 Subtraction in binary notation, 60 computing gcd's by, 197 Subtrees, 716 Successor function, 392, 393 Sufficient conditions, 25-26 for universal conditional statements and, 94-95 Sum, 61 Summands, 140 Summations binomial theorem to simplify sum, 368-369 computing, 202-203 defined, 202 expanded form of, 202, 203-204 of first n integers, 218-222, 482-483, 527-528 of geometric sequence, 222-225 harmonic, 553-555 notation, 202-205 properties of, 207-208 recursive definition of, 204, 503-504 of sums (example), 503-504 Sum-of-products form, 52 Sun-Tsu, 627, 631 Surjective function See Onto functions Swift, Jonathan, 457 Switches in parallel, 43-44 in series, 43-44 Syllogism, 31 Symbolic Logic (Carroll), 123 Symbols, order of in expressions, www.EngineeringEBooksPdf.com Symmetric difference of sets, 292 Symmetric matrices, 687 Symmetric property of cardinality, 444 Symmetry, 584-594 See also Equivalence relations Syntactic analyzer, 735 Syntactic derivation tree, 706-707 Syntax, 706, 707 Taniyama-Shimura conjecture, 138 Tarski, Alfred, 85 Tarski's World (computer program), 85 evaluating argument for, 120-121 formalizing statements in, 105-107 negating statements in, 103-104 quantifier order in, 104-105 truth of multiply-quantified statements in, 98-99 Tautologies, 12-13 defined, 13 logical equivalence and, 13 negation of, 14 Teams, calculating number of, 337-342 Telescoping sum, 205 Term-by-term polynomial evaluation, 543 Terminal vertices, 710-711 maximum number of, 718-720 Terminating decimal, 423 Term of sequence, 200 Ternary relation, 581 Theorem definition of, 133 direct proof of a, 132-134 There exists, symbol for, 79 "There exists a unique" notation, 110 There exists statement, negation of, 91 Therefore, symbol for, 29 Theta-notation (8-notation), 518-531 definitions to show order of polynomial function with positive coefficients, 523-524 for functions of integer variables, 527-528 harmonic sums, order of, 553-555 logarithmic inequalities, deriving order from, 551-552 logarithmic orders, 552-553 properties of, 521 Thinking Machines Corp., 139 Thoreau, Henry David, 763 3x + I problem, 507 Time efficiency of algorithm, 532-535 Topological sorting, 641-644 Total degree of graph, 658-660 Totally ordered set, 640 Total order relation, 639-640 Tournament play possibilities, 306-307 Tower of Hanoi (example), 460-464 explicit formula for, 480-481 Trace table, 188-189, 191 for insertion sort, 538 Tractable algorithms, 568 Tranpose of matrix, 697 Transforming summations, 209 Transition diagram, 748-749 Transitive closure of a relation, 587-588 Transitive property of cardinality, 444 Transitivity, 34, 40, 119 " Index See also Equivalence relations; Partial order relations of divisibility, 150-152 relations and, 584-594 of subsets, 269 universal, 120-121 Traveling salesman problem, 568, 678-679 Trees, 705-733 binary, 716-721 characterizing, 708-714 examples of, 705-708 full binary, 716, 718-719 multiplication rule and possibility, 306-320 nonisomorphic, 712-714 non-trees and, 705 rooted, 714-716 spanning, 723-733 syntactic derivation or parse, 706-707 theorems about, 710-711, 714 trivial, 705 Trefethen, Lloyd, 300 Trefethen, Nick, 300 Triangle inequality, 164 Trichotomy property of real numbers, 35 Trivial circuit, 669 Trivial tree, 705 Trivial walk, 667 True by default, 18, 92-93 Truth set of predicates, 77-78 Truth tables for compound statement, 6-8 for conjunction, for disjunction, for exclusive or, 6-7 as functions, 389 for if-then statements, 18 Truth values, 4-6 for compound statement, 6-8 for negation, 4-5 order of, of predicate, finding, 76-77 for statement, 4-5 for statement form, Tucker, Alan, 349 Tukey, John, 45 Turing, Alan M., 294, 295, 734, 747 Turing machine, 734 Twin primes conjecture, 184 2-equivalence class, finding, 767, 772 Two-dimensional arrays See also Matrix (matrices) to program automaton, 755-756 Two-dimensional Cartesian coordinate system, 510 Two's complements computer addition with negative integers and, 66-70 computer representation of negative integers and, 63-66 Uncountable set, 446, 450-451, 452 Undirected graphs, matrices and, 686-687 Union counting elements of general, 327-329 of equivalence classes, 603-604 inclusion of, 269, 270 and intersection with subset, 278 of languages, 738 of mutually disjointed subsets, 263 recursive definitions of, 504-505 of sets, 260, 261 of two events, probability of general, 371-373 Unique factorization theorem, 153-154, 625, 626 Universal bound laws, 14, 272, 288 Universal conditional statements contrapositives, converses, and inverses of, 93-94 necessary and sufficient conditions for, 94-95 negation of, 90-91 variants of, 93-94 writing, 81-82 Universal instantiation rule of, 111 112 validity of, 111, 115 Universal modus ponens, 112-114, 115 Universal modus tollens, 114, 119 Universal quantifiers, 78-79 implicit, 83 Universal set, 260 Universal statements defined, 78 disproving, by counterexample, 129-130 equivalent forms for, 83 negation of, 88, 90 proving, 130-135 true or false, 78-79 vacuous truth of, 92-93 writing proofs for, 134-135 Universal transitivity, 120-121 Universe of discourse, 260 UNIX utilities, 735, 742 Unless statement, 28 Unordered selection of elements, 335 Upper limit of summation, 202 Vacuously true statement, 92-93 conditional, 18 Vacuous truth of universal statements, 92-93 Valid argument form, 31, 40, 115 Valid arguments, defined, 29-30, 115 See also Arguments, valid and invalid Validity of arguments with quantified statements, 115-119 www.EngineeringEBooksPdf.com 1-13 Value expected, 373-374 of a function, 390 Vandermonde, Alexander, 362 Vandermonde convolution, 362 Variables Boolean, 48, 186 change of in sums, 208-210 in computer languages, 186 dummy, 208, 211 predicate, 76 Vegetarians and cannibals example, 655 Vending machine example, 746-747 Venn, John, 257 Venn diagrams for operations on sets, 257-258, 271 Vertex (vertices) adjacent, 650, 667 branch, 710-711 connected, 650 defined, 482, 650 degree of, 658-662 internal, 710-711 isolated, 650 level of, 715 with odd degree, 662 ordered pairs of, 653 terminal, 710-711 Vertical axes, 510 Volterra, Vito, 389 Walks, 667 closed, 667 counting, of length n, 693-695 notation for, 668 trivial, 667 Way, 57 Weighted graph, 725 Well-defined function, 398 Well-ordering principle, 180n, 240-242 Wheeler, Anna Pell, 156, 306, 402 While loop, 187-188, 189 maximum number of iterations in binary search, 560 Whitehead, Alfred North, 126, 356, 431, 649 Wiener, Norbert, 575, 745 Wiles, Andrew, 138 Worst-case order(s) of g(n), 533 for insertion sort, 538-539 for sequential search, 536 XML, 735 0-equivalence class, finding, 766, 772 0-equivalent states of finite-state automata, 765-766 Zero factorial (0!), 206 Zero product property, 143 CREDITS This page constitutes an extension of the copyright page We have made every effort to trace the ownership of all copyrighted material and to secure permission from copyright holders In the event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in future printings Thanks are due to the following authors, publishers, and agents for permission to use the material indicated Chapter 1 Bettmann/CORBIS; 10 Culver Pictures; 39 Indiana University Archives; 43 M.I.T Museum; 44 Courtesy of Intel Corporation; 45 Courtesy of IBM; 48 CORBIS; 54 Courtesy of Harvard University Archives Chapter 78 (top) Culver Pictures; 78 (bottom) Friedrich Schiller, Universitat Jena/Thuringer Universitats/und Landesbibliothek Unfversitatsarchiv; 85 Public Domain; 102 Public Domain; 116 Culver Pictures Chapter 138 (top) CORBIS; 138 (bottom) Andrew Wiles/Princeton University; 180 CORBIS; 186 Getty Images; 190 Suleymaniye Kutuphanesi, Istanbul,Turkey Chapter 202 CORBIS; 244 (top) Courtesy of Christiane Floyd; 244 (bottom) Center for American History/The University of Texas at Austin; 247 Courtesy of Tony Hoare Chapter 255 David Eugene Smith Collection, Rare Book and Manuscript Library, Columbia University; 257 The President and Council of the Royal Society of London; 260 Stock Montage; 293 Bettmann/CORBIS; 294 Public Domain Chapter 298 Reprinted by permission of United Feature Syndicate, Inc.; 302 Bettmann/CORBIS; 358 Hulton-Deutch Collection/CORBIS; 362 Academie Royale de Belgique; 370 Andrei Nikolavich Kolmogorov-Yevgney Khaldei/CORBIS; 379 Public Domain Chapter 390 Stock Montage; 396 Courtesy of The U.S Naval Academy; 443 Bettmann/Corbis; 449 (top) Public Domain; 449 (bottom) Getty Images Chapter 461 (top) Courtesy of Francis Lucas; 461 (middle) Courtesy of Paul Stockmeyer/www.cs.wm.edu/-pkstoc/toh.html; 464 CORBIS; 506 (top) Roger Russmeyer/CORBIS Chapter 510 CORBIS; 531 CORBIS; 532 Courtesy of Donald Knuth; 545 CORBIS Chapters 597 Bettmann/CORBIS; 612 RSA 2003/Courtesy of Leonard Adleman Chapter 11 666 Bettmann/CORBIS; 676 Bettmann/CORBIS; 707 (top) Courtesy of IBM Corporation © 2003; 707 (bottom) Courtesy of Peter Naur; 708 CORBIS; 726 Courtesy of Joseph Kruskal; 729 Courtesy of Lucent Technologies Inc C 2003, Lucent Technologies Inc All rights reserved Chapter 12 735 Christopher Felver/CORBIS; 736 University of Wisconsin; 747 (top) David Eugene Smith Collection, Rare Book and Manuscript Library, Columbia University; 747 (bottom) Godfrey Argent www.EngineeringEBooksPdf.com List of Symbols Subject Symbol Meaning Formal Languages and Finite-State Automata E an alphabet of a language the null string the set of all strings over E of length n En Graphs and Trees 736 310 736 the set of all strings over E 736 the set of all strings over E with length at least I 736 LL' the concatenation of languages L and L' 738 L* the Kleene closure of L 738 (rs), (r I s), (r*) regular expressions 738 [XI - Xn], [ ^X xn] x+, x?, x{n}, x{m n} character classes shorthand notations for regular expressions N(s, m) the value of the next-state function for a state s and input symbol m 748 742, 743 743 initial state 748 accepting state 748 L(A) language accepted by A 750 N*(s, w) the value of the eventual-state function for a state s and input string w 751 s R* t s and t are *-equivalent s Rk t s and t are k-equivalent 764 765 A the quotient automaton of A 768 A matrix 683 I identity matrix 692 A+ B AB sum of matrices A and B product of matrices A and B 697 689 An matrix A to the power n 693 V (G) E(G) the set of vertices of a graph G the set of edges of a graph G 650 650 -i Matrices Page {v, w} the edge joining v and w in a simple graph 656 K, complete graph on n vertices 656 K,,, complete bipartite graph on (m, n) vertices 657 degree of vertex v 658 awalk from vo to v, 667 deg(v) voeev e2 e, v w(e) the weight of edge e 725 w(G) the total weight of graph G 725 www.EngineeringEBooksPdf.com Reference Formulas Topic Name Formula Logic De Morgan's law -'(p A q) -p V -q De Morgan's law -(p V q) -p Negation of- Sums (p - Page A -q 10 q 20 q) =pA Equivalence of a conditional and its contrapositive p - q Nonequivalence of a conditional and its converse p - q Nonequivalence of a conditional and its inverse p - q Negation of a universal statement -(Vx in D, Q(x)) 3x in D such that -Q(x) 88 Negation of an existential statement -(3x in D such that Q(x)) - Vx in D, -Q(x) 89 Sum of the first n integers I + 2+ - -q 10 q 23 p - p 21 p -* -q - -+ 23 n= ( + r2 + + Sum of powers of r l Counting and Probability in the P(E) = N(E) Probability equally likely case ) n~+l r = 218 I 222 299 N(S) Number of r-permutations of a set with n elements P(n, r) -= Number of elements in a union N(A U B) Number of subsets of size r of a set with (nf 315 (n -r)! - N(A) + N(B) -N(A "'r1 n! r!(n -r)! (n ) n B) 327 337 n elements Pascal's formula (r) www.EngineeringEBooksPdf.com )+ (r) 358 Topic Name Formula Counting and Probability Binomial theorem (a + Page Z (, ) a bh 364 k- Probability of the complement of an event P(A') = I -P(A) Probability of a union P(A U B) Conditional probability P(A B) 324 P(A) F + P(B) P(A P (B) - P(A n B) B) 376 Bayes' formula IA) Laws of Exponents 379 P(A B")P(B1 ) P(A B1 )P(B,) + P(A I B2)P(B2 ) + - ± P(A I B,) P(B11 ) Li0 I 411 411 'L"L =// bl Vb Properties of Logarithms 371 Y1!= 411 b`+V bu= bU 411 (bu)'s - bUI!t 411 (bc)' = bI C O b" = b' V u = u 411 y b =x logX log(xy) logbx) log5(y) 412 419 log,(x) 419 log, () log(x)j 412 = a logb(x) = log,(x) -log 419 l og() 419 log 5(v)=-u = V 412 logb(c) Iog,,(u) (y) www.EngineeringEBooksPdf.com ISBN :e u S - 0~ * www.EngineeringEBooksPdf.com * 0-534-35945-D Ivii IYONZ S f 1g U45411 i ... 599 635 Continuedonfirst page of back endpapers www.EngineeringEBooksPdf.com DISCRETE MATHEMATICS WITH APPLICATIONS THIRD EDITION SUSANNI k S EPP DePaul Uuniversity TF-IcHIVIScON BROOKS/COLE... their interest Many of the changes in this edition have resulted from continuing interaction with students Themes of a Discrete Mathematics Course Discrete mathematics describes processes that consist... www.EngineeringEBooksPdf.com A-4 PREFACE My purpose in writing this book was to provide a clear, accessible treatment of discrete mathematics for students majoring or minoring in computer science, mathematics, mathematics
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