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E-raamat: Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications

  • Formaat: PDF+DRM
  • Sari: Texts in Computer Science
  • Ilmumisaeg: 28-Oct-2021
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030815882
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Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and ApplicationsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Texts in Computer Science
  • Ilmumisaeg: 28-Oct-2021
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030815882
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This stimulating textbook presents a broad and accessible guide to the fundamentals of discrete mathematics, highlighting how the techniques may be applied to various exciting areas in computing. The text is designed to motivate and inspire the reader, encouraging further study in this important skill.





 





Features: This book provides an introduction to the building blocks of discrete mathematics, including sets, relations and functions; describes the basics of number theory, the techniques of induction and recursion, and the applications of mathematical sequences, series, permutations, and combinations; presents the essentials of algebra; explains the fundamentals of automata theory, matrices, graph theory, cryptography, coding theory, language theory, and the concepts of computability and decidability; reviews the history of logic, discussing propositional and predicate logic, as well as advanced topics such as the nature of theorem proving; examines the field of software engineering, including software reliability and dependability and describes formal methods; investigates probability and statistics and presents an overview of operations research and financial mathematics.
1 Mathematics in Civilization 1(26)
1.1 Introduction
1(3)
1.2 The Babylonians
4(2)
1.3 The Egyptians
6(3)
1.4 The Greeks
9(9)
1.5 The Romans
18(3)
1.6 Islamic Influence
21(2)
1.7 Chinese and Indian Mathematics
23(1)
1.8 Review Questions
24(1)
1.9 Summary
25(1)
References
25(2)
2 Sets, Relations and Functions 27(28)
2.1 Introduction
27(1)
2.2 Set Theory
28(8)
2.2.1 Set Theoretical Operations
30(3)
2.2.2 Properties of Set Theoretical Operations
33(1)
2.2.3 Russell's Paradox
34(1)
2.2.4 Computer Representation of Sets
35(1)
2.3 Relations
36(8)
2.3.1 Reflexive, Symmetric and Transitive Relations
37(3)
2.3.2 Composition of Relations
40(1)
2.3.3 Binary Relations
41(1)
2.3.4 Applications of Relations
42(2)
2.4 Functions
44(5)
2.5 Application of Functions
49(3)
2.6 Review Questions
52(1)
2.7 Summary
52(1)
References
53(2)
3 Number Theory 55(24)
3.1 Introduction
55(2)
3.2 Elementary Number Theory
57(4)
3.3 Prime Number Theory
61(9)
3.3.1 Algorithms
62(2)
3.3.2 Greatest Common Divisors (GCD)
64(1)
3.3.3 Least Common Multiple (LCM)
65(1)
3.3.4 Euclid's Algorithm
66(2)
3.3.5 Distribution of Primes
68(2)
3.4 Theory of Congruences
70(4)
3.5 Binary System and Computer Representation of Numbers
74(3)
3.6 Review Questions
77(1)
3.7 Summary
77(1)
References
78(1)
4 Mathematical Induction and Recursion 79(10)
4.1 Introduction
79(3)
4.2 Strong Induction
82(2)
4.3 Recursion
84(2)
4.4 Structural Induction
86(1)
4.5 Review Questions
87(1)
4.6 Summary
87(1)
Reference
88(1)
5 Sequences, Series, and Permutations and Combinations 89(14)
5.1 Introduction
89(1)
5.2 Sequences and Series
89(1)
5.3 Arithmetic and Geometric Sequences
90(1)
5.4 Arithmetic and Geometric Series
91(2)
5.5 Simple and Compound Interest
93(1)
5.6 Time Value of Money and Annuities
94(2)
5.7 Permutations and Combinations
96(4)
5.8 Review Questions
100(1)
5.9 Summary
101(2)
6 Algebra 103(18)
6.1 Introduction
103(1)
6.2 Simple and Simultaneous Equations
103(4)
6.3 Quadratic Equations
107(2)
6.4 Indices and Logarithms
109(2)
6.5 Homer's Method for Polynomials
111(1)
6.6 Abstract Algebra
112(6)
6.6.1 Monoids and Groups
113(1)
6.6.2 Rings
114(1)
6.6.3 Fields
115(1)
6.6.4 Vector Spaces
116(2)
6.7 Review Questions
118(1)
6.8 Summary
119(1)
Reference
119(2)
7 Automata Theory 121(12)
7.1 Introduction
121(1)
7.2 Finite-State Machines
122(3)
7.3 Pushdown Automata
125(2)
7.4 Turing Machines
127(2)
7.5 Hybrid Automata
129(1)
7.6 Review Questions
130(1)
7.7 Summary
131(1)
Reference
131(2)
8 Matrix Theory 133(14)
8.1 Introduction
133(1)
8.2 Two x Two Matrices
134(3)
8.3 Matrix Operations
137(2)
8.4 Determinants
139(2)
8.5 Eigen Vectors and Values
141(1)
8.6 Gaussian Elimination
142(2)
8.7 Business Applications of Matrices
144(1)
8.8 Review Questions
145(1)
8.9 Summary
145(1)
References
146(1)
9 Graph Theory 147(14)
9.1 Introduction
147(1)
9.2 Undirected Graphs
148(6)
9.2.1 Hamiltonian Paths
153(1)
9.3 Trees
154(1)
9.3.1 Binary Trees
155(1)
9.4 Graph Algorithms
155(1)
9.5 Graph Colouring and Four-Colour Problem
156(1)
9.6 Review Questions
157(1)
9.7 Summary
158(1)
References
159(2)
10 Cryptography 161(16)
10.1 Introduction
161(2)
10.2 Breaking the Enigma Codes
163(3)
10.3 Cryptographic Systems
166(1)
10.4 Symmetric Key Systems
166(5)
10.5 Public Key Systems
171(4)
10.5.1 RSA Public Key Cryptosystem
173(1)
10.5.2 Digital Signatures
174(1)
10.6 Review Questions
175(1)
10.7 Summary
175(1)
References
176(1)
11 Coding Theory 177(14)
11.1 Introduction
177(1)
11.2 Mathematical Foundations
178(1)
11.3 Simple Channel Code
179(1)
11.4 Block Codes
180(3)
11.4.1 Error Detection and Correction
182(1)
11.5 Linear Block Codes
183(5)
11.5.1 Parity Check Matrix
186(1)
11.5.2 Binary Hamming Code
186(2)
11.5.3 Binary Parity-Check Code
188(1)
11.6 Miscellaneous Codes in Use
188(1)
11.7 Review Questions
188(1)
11.8 Summary
189(1)
References
189(2)
12 Language Theory and Semantics 191(22)
12.1 Introduction
191(1)
12.2 Alphabets and Words
192(1)
12.3 Grammars
193(5)
12.3.1 Backus Naur Form
195(2)
12.3.2 Parse Trees and Derivations
197(1)
12.4 Programming Language Semantics
198(7)
12.4.1 Axiomatic Semantics
200(1)
12.4.2 Operational Semantics
201(1)
12.4.3 Denotational Semantics
202(1)
12.5 Lambda Calculus
203(2)
12.6 Lattices and Order
205(6)
12.6.1 Partially Ordered Sets
205(2)
12.6.2 Lattices
207(2)
12.6.3 Complete Partial Orders
209(1)
12.6.4 Recursion
209(2)
12.7 Review Questions
211(1)
12.8 Summary
211(1)
References
212(1)
13 Computability and Decidability 213(12)
13.1 Introduction
213(1)
13.2 Logicism and Formalism
214(2)
13.3 Decidability
216(2)
13.4 Computability
218(4)
13.5 Computational Complexity
222(1)
13.6 Review Questions
223(1)
13.7 Summary
223(1)
References
224(1)
14 A Short History of Logic 225(16)
14.1 Introduction
225(1)
14.2 Syllogistic Logic
226(1)
14.3 Paradoxes and Fallacies
227(2)
14.4 Stoic Logic
229(2)
14.5 Boole's Symbolic Logic
231(4)
14.5.1 Switching Circuits and Boolean Algebra
233(2)
14.6 Application of Symbolic Logic to Digital Computing
235(1)
14.7 Frege
236(1)
14.8 Review Questions
237(1)
14.9 Summary
238(1)
References
239(2)
15 Propositional and Predicate Logic 241(28)
15.1 Introduction
241(1)
15.2 Propositional Logic
242(14)
15.2.1 Truth Tables
243(2)
15.2.2 Properties of Propositional Calculus
245(2)
15.2.3 Proof in Propositional Calculus
247(3)
15.2.4 Semantic Tableaux in Propositional Logic
250(2)
15.2.5 Natural Deduction
252(1)
15.2.6 Sketch of Formalization of Propositional Calculus
253(1)
15.2.7 Applications of Propositional Calculus
254(2)
15.2.8 Limitations of Propositional Calculus
256(1)
15.3 Predicate Calculus
256(9)
15.3.1 Sketch of Formalization of Predicate Calculus
259(2)
15.3.2 Interpretation and Valuation Functions
261(1)
15.3.3 Properties of Predicate Calculus
262(1)
15.3.4 Applications of Predicate Calculus
262(1)
15.3.5 Semantic Tableaux in Predicate Calculus
263(2)
15.4 Review Questions
265(1)
15.5 Summary
266(1)
References
267(2)
16 Advanced Topics in Logic 269(18)
16.1 Introduction
269(1)
16.2 Fuzzy Logic
269(2)
16.3 Temporal Logic
271(2)
16.4 Intuitionist Logic
273(1)
16.5 Undefined Values
274(6)
16.5.1 Logic of Partial Functions
275(2)
16.5.2 Parnas Logic
277(1)
16.5.3 Dijkstra and Undefinedness
278(2)
16.6 Logic and AI
280(4)
16.7 Review Questions
284(1)
16.8 Summary
284(1)
References
285(2)
17 The Nature of Theorem Proving 287(10)
17.1 Introduction
287(2)
17.2 Early Automation of Proof
289(2)
17.3 Interactive Theorem Provers
291(3)
17.4 A Selection of Theorem Provers
294(1)
17.5 Review Questions
294(1)
17.6 Summary
295(1)
References
296(1)
18 Software Engineering Mathematics 297(16)
18.1 Introduction
297(2)
18.2 What is Software Engineering?
299(5)
18.3 Early Software Engineering Mathematics
304(3)
18.4 Mathematics in Software Engineering
307(1)
18.5 Software Inspections and Testing
308(1)
18.6 Process Maturity Models
309(1)
18.7 Review Questions
310(1)
18.8 Summary
311(1)
References
311(2)
19 Software Reliability and Dependability 313(14)
19.1 Introduction
313(1)
19.2 Software Reliability
314(6)
19.2.1 Software Reliability and Defects
315(2)
19.2.2 Cleanroom Methodology
317(1)
19.2.3 Software Reliability Models
318(2)
19.3 Dependability
320(2)
19.4 Computer Security
322(1)
19.5 System Availability
323(1)
19.6 Safety-Critical Systems
324(1)
19.7 Review Questions
325(1)
19.8 Summary
325(1)
References
326(1)
20 Formal Methods 327(20)
20.1 Introduction
327(2)
20.1.1 Definition 20.1 (Formal Specification)
327(2)
20.2 Why Should We Use Formal Methods?
329(2)
20.2.1 Comment 20.1 (Missile Safety)
330(1)
20.3 Applications of Formal Methods
331(1)
20.4 Tools for Formal Methods
331(1)
20.5 Approaches to Formal Methods
332(2)
20.5.1 Model-Oriented Approach
332(1)
20.5.2 Axiomatic Approach
333(1)
20.5.3 Comment 20.2 (Axiomatic Approach)
333(1)
20.6 Proof and Formal Methods
334(1)
20.7 The Future of Formal Methods
335(1)
20.8 The Vienna Development Method
335(1)
20.9 VDM, the Irish School of VDM
336(2)
20.10 The Z Specification Language
338(1)
20.11 The B Method
339(1)
20.12 Predicate Transformers and Weakest Preconditions
340(1)
20.13 The Process Calculi
340(1)
20.14 The Parnas Way
341(1)
20.15 Usability of Formal Methods
342(3)
20.15.1 Why are Formal Methods difficult?
343(1)
20.15.2 Characteristics of a Usable Formal Method
344(1)
20.16 Review Questions
345(1)
20.17 Summary
345(2)
21 Z Formal Specification Language 347(16)
21.1 Introduction
347(2)
21.2 Sets
349(2)
21.3 Relations
351(2)
21.4 Functions
353(1)
21.5 Sequences
354(1)
21.6 Bags
355(2)
21.7 Schemas and Schema Composition
357(3)
21.8 Reification and Decomposition
360(1)
21.9 Proof in Z
361(1)
21.10 Review Questions
361(1)
21.11 Summary
362(1)
Reference
362(1)
22 Statistics 363(20)
22.1 Introduction
363(1)
22.2 Basic Statistics
364(1)
22.2.1 Abuse of Statistics
364(1)
22.2.2 Statistical Sampling and Data Collection
365(1)
22.3 Frequency Distribution and Charts
365(4)
22.4 Statistical Measures
369(4)
22.4.1 Arithmetic Mean
370(1)
22.4.2 Mode
371(1)
22.4.3 Median
372(1)
22.5 Variance and Standard Deviation
373(1)
22.6 Correlation and Regression
374(3)
22.6.1 Regression
377(1)
22.7 Statistical Inference and Hypothesis Testing
377(3)
22.8 Review Questions
380(1)
22.9 Summary
380(1)
References
381(2)
23 Probability Theory 383(24)
23.1 Introduction
383(1)
23.2 Basic Probability Theory
384(4)
23.2.1 Laws of Probability
385(1)
23.2.2 Bayes' Formula
386(2)
23.3 Random Variables
388(2)
23.4 Binomial and Poisson Distributions
390(3)
23.5 The Normal Distribution
393(6)
23.5.1 Unit Normal Distribution
394(1)
23.5.2 Confidence Intervals and Tests of Significance
395(2)
23.5.3 The Central Limit Theorem
397(2)
23.6 Bayesianism
399(2)
23.7 Queueing Theory
401(2)
23.8 Review Questions
403(1)
23.9 Summary
404(1)
References
405(2)
24 Operations Research 407(16)
24.1 Introduction
407(2)
24.2 Linear Programming
409(6)
24.2.1 Linear Programming Example
410(4)
24.2.2 General Formulation of LP Problem
414(1)
24.3 Cost-Volume-Profit Analysis
415(3)
24.4 Game Theory
418(3)
24.5 Review Questions
421(1)
24.6 Summary
421(1)
References
422(1)
25 Basic Financial Mathematics 423(22)
25.1 Introduction
423(1)
25.2 Simple Interest
424(4)
25.2.1 Computing Future and Present Values
425(1)
25.2.2 Computing Future Value
425(1)
25.2.3 Computing Present Values
426(2)
25.3 Compound Interest
428(7)
25.3.1 Present Value Under Compound Interest
431(1)
25.3.2 Equivalent Values
432(3)
25.4 Basic Mathematics of Annuities
435(5)
25.5 Loans and Mortgages
440(3)
25.6 Review Questions
443(1)
25.7 Summary
444(1)
Glossary 445(4)
Index 449
Dr. Gerard O'Regan is a CMMI software process improvement consultant with research interests including software quality and software process improvement, mathematical approaches to software quality, and the history of computing. He is the author of such Springer titles as Introduction to the History of Computing, Pillars of Computing, Introduction to Software Quality, Giants of Computing, and Mathematics in Computing.
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