Guide to discrete mathematics : an accessible introduction to the history, theory, logic and applications / / Gerard O'Regan

• Autore: O'Regan Gerard
• Pubblicazione: Cham, Switzerland : , : Springer, , [2021]; ©2021
• Edizione: 2nd ed.
• Descrizione fisica: 1 online resource (459 pages)
• Disciplina: 004.0151
• Soggetto topico: Computer science - Mathematics; Discrete mathematics
• Nota di contenuto: Intro -- Preface -- Overview -- Organization and Features -- Audience -- Contents -- 1 Mathematics in Civilization -- 1.1 Introduction -- 1.2 The Babylonians -- 1.3 The Egyptians -- 1.4 The Greeks -- 1.5 The Romans -- 1.6 Islamic Influence -- 1.7 Chinese and Indian Mathematics -- 1.8 Review Questions -- 1.9 Summary -- References -- 2 Sets, Relations and Functions -- 2.1 Introduction -- 2.2 Set Theory -- 2.2.1 Set Theoretical Operations -- 2.2.2 Properties of Set Theoretical Operations -- 2.2.3 Russell's Paradox -- 2.2.4 Computer Representation of Sets -- 2.3 Relations -- 2.3.1 Reflexive, Symmetric and Transitive Relations -- 2.3.2 Composition of Relations -- 2.3.3 Binary Relations -- 2.3.4 Applications of Relations -- 2.4 Functions -- 2.5 Application of Functions -- 2.6 Review Questions -- 2.7 Summary -- References -- 3 Number Theory -- 3.1 Introduction -- 3.2 Elementary Number Theory -- 3.3 Prime Number Theory -- 3.3.1 Algorithms -- 3.3.2 Greatest Common Divisors (GCD) -- 3.3.3 Least Common Multiple (LCM) -- 3.3.4 Euclid's Algorithm -- 3.3.5 Distribution of Primes -- 3.4 Theory of Congruences -- 3.5 Binary System and Computer Representation of Numbers -- 3.6 Review Questions -- 3.7 Summary -- References -- 4 Mathematical Induction and Recursion -- 4.1 Introduction -- 4.2 Strong Induction -- 4.3 Recursion -- 4.4 Structural Induction -- 4.5 Review Questions -- 4.6 Summary -- Reference -- 5 Sequences, Series, and Permutations and Combinations -- 5.1 Introduction -- 5.2 Sequences and Series -- 5.3 Arithmetic and Geometric Sequences -- 5.4 Arithmetic and Geometric Series -- 5.5 Simple and Compound Interest -- 5.6 Time Value of Money and Annuities -- 5.7 Permutations and Combinations -- 5.8 Review Questions -- 5.9 Summary -- 6 Algebra -- 6.1 Introduction -- 6.2 Simple and Simultaneous Equations -- 6.3 Quadratic Equations -- 6.4 Indices and Logarithms. ; 6.5 Horner's Method for Polynomials -- 6.6 Abstract Algebra -- 6.6.1 Monoids and Groups -- 6.6.2 Rings -- 6.6.3 Fields -- 6.6.4 Vector Spaces -- 6.7 Review Questions -- 6.8 Summary -- Reference -- 7 Automata Theory -- 7.1 Introduction -- 7.2 Finite-State Machines -- 7.3 Pushdown Automata -- 7.4 Turing Machines -- 7.5 Hybrid Automata -- 7.6 Review Questions -- 7.7 Summary -- Reference -- 8 Matrix Theory -- 8.1 Introduction -- 8.2 Two × Two Matrices -- 8.3 Matrix Operations -- 8.4 Determinants -- 8.5 Eigen Vectors and Values -- 8.6 Gaussian Elimination -- 8.7 Business Applications of Matrices -- 8.8 Review Questions -- 8.9 Summary -- References -- 9 Graph Theory -- 9.1 Introduction -- 9.2 Undirected Graphs -- 9.2.1 Hamiltonian Paths -- 9.3 Trees -- 9.3.1 Binary Trees -- 9.4 Graph Algorithms -- 9.5 Graph Colouring and Four-Colour Problem -- 9.6 Review Questions -- 9.7 Summary -- References -- 10 Cryptography -- 10.1 Introduction -- 10.2 Breaking the Enigma Codes -- 10.3 Cryptographic Systems -- 10.4 Symmetric Key Systems -- 10.5 Public Key Systems -- 10.5.1 RSA Public Key Cryptosystem -- 10.5.2 Digital Signatures -- 10.6 Review Questions -- 10.7 Summary -- References -- 11 Coding Theory -- 11.1 Introduction -- 11.2 Mathematical Foundations -- 11.3 Simple Channel Code -- 11.4 Block Codes -- 11.4.1 Error Detection and Correction -- 11.5 Linear Block Codes -- 11.5.1 Parity Check Matrix -- 11.5.2 Binary Hamming Code -- 11.5.3 Binary Parity-Check Code -- 11.6 Miscellaneous Codes in Use -- 11.7 Review Questions -- 11.8 Summary -- References -- 12 Language Theory and Semantics -- 12.1 Introduction -- 12.2 Alphabets and Words -- 12.3 Grammars -- 12.3.1 Backus Naur Form -- 12.3.2 Parse Trees and Derivations -- 12.4 Programming Language Semantics -- 12.4.1 Axiomatic Semantics -- 12.4.2 Operational Semantics -- 12.4.3 Denotational Semantics -- 12.5 Lambda Calculus. ; 12.6 Lattices and Order -- 12.6.1 Partially Ordered Sets -- 12.6.2 Lattices -- 12.6.3 Complete Partial Orders -- 12.6.4 Recursion -- 12.7 Review Questions -- 12.8 Summary -- References -- 13 Computability and Decidability -- 13.1 Introduction -- 13.2 Logicism and Formalism -- 13.3 Decidability -- 13.4 Computability -- 13.5 Computational Complexity -- 13.6 Review Questions -- 13.7 Summary -- References -- 14 A Short History of Logic -- 14.1 Introduction -- 14.2 Syllogistic Logic -- 14.3 Paradoxes and Fallacies -- 14.4 Stoic Logic -- 14.5 Boole's Symbolic Logic -- 14.5.1 Switching Circuits and Boolean Algebra -- 14.6 Application of Symbolic Logic to Digital Computing -- 14.7 Frege -- 14.8 Review Questions -- 14.9 Summary -- References -- 15 Propositional and Predicate Logic -- 15.1 Introduction -- 15.2 Propositional Logic -- 15.2.1 Truth Tables -- 15.2.2 Properties of Propositional Calculus -- 15.2.3 Proof in Propositional Calculus -- 15.2.4 Semantic Tableaux in Propositional Logic -- 15.2.5 Natural Deduction -- 15.2.6 Sketch of Formalization of Propositional Calculus -- 15.2.7 Applications of Propositional Calculus -- 15.2.8 Limitations of Propositional Calculus -- 15.3 Predicate Calculus -- 15.3.1 Sketch of Formalization of Predicate Calculus -- 15.3.2 Interpretation and Valuation Functions -- 15.3.3 Properties of Predicate Calculus -- 15.3.4 Applications of Predicate Calculus -- 15.3.5 Semantic Tableaux in Predicate Calculus -- 15.4 Review Questions -- 15.5 Summary -- References -- 16 Advanced Topics in Logic -- 16.1 Introduction -- 16.2 Fuzzy Logic -- 16.3 Temporal Logic -- 16.4 Intuitionist Logic -- 16.5 Undefined Values -- 16.5.1 Logic of Partial Functions -- 16.5.2 Parnas Logic -- 16.5.3 Dijkstra and Undefinedness -- 16.6 Logic and AI -- 16.7 Review Questions -- 16.8 Summary -- References -- 17 The Nature of Theorem Proving -- 17.1 Introduction. ; 17.2 Early Automation of Proof -- 17.3 Interactive Theorem Provers -- 17.4 A Selection of Theorem Provers -- 17.5 Review Questions -- 17.6 Summary -- References -- 18 Software Engineering Mathematics -- 18.1 Introduction -- 18.2 What is Software Engineering? -- 18.3 Early Software Engineering Mathematics -- 18.4 Mathematics in Software Engineering -- 18.5 Software Inspections and Testing -- 18.6 Process Maturity Models -- 18.7 Review Questions -- 18.8 Summary -- References -- 19 Software Reliability and Dependability -- 19.1 Introduction -- 19.2 Software Reliability -- 19.2.1 Software Reliability and Defects -- 19.2.2 Cleanroom Methodology -- 19.2.3 Software Reliability Models -- 19.3 Dependability -- 19.4 Computer Security -- 19.5 System Availability -- 19.6 Safety-Critical Systems -- 19.7 Review Questions -- 19.8 Summary -- References -- 20 Formal Methods -- 20.1 Introduction -- 20.1.1 Definition 20.1 (Formal Specification) -- 20.2 Why Should We Use Formal Methods? -- 20.2.1 Comment 20.1 (Missile Safety) -- 20.3 Applications of Formal Methods -- 20.4 Tools for Formal Methods -- 20.5 Approaches to Formal Methods -- 20.5.1 Model-Oriented Approach -- 20.5.2 Axiomatic Approach -- 20.5.3 Comment 20.2 (Axiomatic Approach) -- 20.6 Proof and Formal Methods -- 20.7 The Future of Formal Methods -- 20.8 The Vienna Development Method -- 20.9 VDM♣, the Irish School of VDM -- 20.10 The Z Specification Language -- 20.11 The B Method -- 20.12 Predicate Transformers and Weakest Preconditions -- 20.13 The Process Calculi -- 20.14 The Parnas Way -- 20.15 Usability of Formal Methods -- 20.15.1 Why are Formal Methods difficult? -- 20.15.2 Characteristics of a Usable Formal Method -- 20.16 Review Questions -- 20.17 Summary -- 21 Z Formal Specification Language -- 21.1 Introduction -- 21.2 Sets -- 21.3 Relations -- 21.4 Functions -- 21.5 Sequences -- 21.6 Bags. ; 21.7 Schemas and Schema Composition -- 21.8 Reification and Decomposition -- 21.9 Proof in Z -- 21.10 Review Questions -- 21.11 Summary -- Reference -- 22 Statistics -- 22.1 Introduction -- 22.2 Basic Statistics -- 22.2.1 Abuse of Statistics -- 22.2.2 Statistical Sampling and Data Collection -- 22.3 Frequency Distribution and Charts -- 22.4 Statistical Measures -- 22.4.1 Arithmetic Mean -- 22.4.2 Mode -- 22.4.3 Median -- 22.5 Variance and Standard Deviation -- 22.6 Correlation and Regression -- 22.6.1 Regression -- 22.7 Statistical Inference and Hypothesis Testing -- 22.8 Review Questions -- 22.9 Summary -- References -- 23 Probability Theory -- 23.1 Introduction -- 23.2 Basic Probability Theory -- 23.2.1 Laws of Probability -- 23.2.2 Bayes' Formula -- 23.3 Random Variables -- 23.4 Binomial and Poisson Distributions -- 23.5 The Normal Distribution -- 23.5.1 Unit Normal Distribution -- 23.5.2 Confidence Intervals and Tests of Significance -- 23.5.3 The Central Limit Theorem -- 23.6 Bayesianism -- 23.7 Queueing Theory -- 23.8 Review Questions -- 23.9 Summary -- References -- 24 Operations Research -- 24.1 Introduction -- 24.2 Linear Programming -- 24.2.1 Linear Programming Example -- 24.2.2 General Formulation of LP Problem -- 24.3 Cost-Volume-Profit Analysis -- 24.4 Game Theory -- 24.5 Review Questions -- 24.6 Summary -- References -- 25 Basic Financial Mathematics -- 25.1 Introduction -- 25.2 Simple Interest -- 25.2.1 Computing Future and Present Values -- 25.2.2 Computing Future Value -- 25.2.3 Computing Present Values -- 25.3 Compound Interest -- 25.3.1 Present Value Under Compound Interest -- 25.3.2 Equivalent Values -- 25.4 Basic Mathematics of Annuities -- 25.5 Loans and Mortgages -- 25.6 Review Questions -- 25.7 Summary -- Glossary -- Index.
• Titolo autorizzato: Guide to Discrete Mathematics
• ISBN: 3-030-81588-9
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996464400303316
• Lo trovi qui: Univ. di Salerno
• Serie: Texts in computer science.