Mathematical Foundations of Software Engineering: A Practical Guide to Essentials

Preface
	Overview
	Organization and Features
	Audience
Acknowledgments
Contents
Abbreviations
List of Figures
List of Tables
978-3-031-26212-8_1.pdf
	1 Fundamentals of Software Engineering
		1.1 Introduction
		1.2 What Is Software Engineering?
		1.3 Challenges in Software Engineering
		1.4 Software Processes and Lifecycles
			1.4.1 Waterfall Lifecycle
			1.4.2 Spiral Lifecycles
			1.4.3 Rational Unified Process
			1.4.4 Agile Development
			1.4.5 Continuous Software Development
		1.5 Activities in Software Development
			1.5.1 Requirements Definition
			1.5.2 Design
			1.5.3 Implementation
			1.5.4 Software Testing
			1.5.5 Support and Maintenance
		1.6 Software Inspections
		1.7 Software Project Management
		1.8 CMMI Maturity Model
		1.9 Formal Methods
		1.10 Review Questions
		1.11 Summary
		References
978-3-031-26212-8_2.pdf
	2 Software Engineering Mathematics
		2.1 Introduction
		2.2 Early Software Engineering Mathematics
		2.3 Debate on Mathematics in Software Engineering
		2.4 The Emergence of Formal Methods
		2.5 What Mathematics Do Software Engineers Need?
		2.6 Review Questions
		2.7 Summary
		References
978-3-031-26212-8_3.pdf
	3 Mathematical Prerequisites for Software Engineers
		3.1 Introduction
		3.2 Set Theory
			3.2.1 Set Theoretical Operations
			3.2.2 Computer Representation of Sets
		3.3 Relations
			3.3.1 Reflexive, Symmetric and Transitive Relations
			3.3.2 Composition of Relations
			3.3.3 Binary Relations
		3.4 Functions
		3.5 Arithmetic
			3.5.1 Fractions and Decimals
			3.5.2 Prime Number Theory
			3.5.3 Greatest Common Divisors (GCD)
			3.5.4 Least Common Multiple (LCM)
			3.5.5 Ratios and Proportions
			3.5.6 Percentages
		3.6 Trigonometry
			3.6.1 Definition of Sine, Cosine, and Tangent
			3.6.2 Sine and Cosine Rules
			3.6.3 Trigonometric Identities
			3.6.4 Degrees and Radians
			3.6.5 Periodic Functions and Sketch of Sine and Cosine Functions
			3.6.6 Power Series for Sine and Cosine
		3.7 Cartesian Coordinates
		3.8 Review Questions
		3.9 Summary
		References
978-3-031-26212-8_4.pdf
	4 Introduction to Algorithms
		4.1 Introduction
		4.2 Early Algorithms
			4.2.1 Greatest Common Divisors (GCD)
			4.2.2 Euclid’s Greatest Common Divisor Algorithm
			4.2.3 Sieve of Eratosthenes Algorithm
			4.2.4 Early Cipher Algorithms
		4.3 Sorting Algorithms
		4.4 Binary Trees and Graph Theory
		4.5 Modern Cryptographic Algorithms
		4.6 Algorithms in Numerical Analysis
		4.7 Computational Complexity
		4.8 Review Questions
		4.9 Summary
		References
978-3-031-26212-8_5.pdf
	5 Algebra
		5.1 Introduction
		5.2 Simplification of Algebraic Expressions
		5.3 Simple and Simultaneous Equations
			5.3.1 Simultaneous Equations
		5.4 Quadratic Equations
		5.5 Indices and Logarithms
		5.6 Exponentials and Natural Logarithms
		5.7 Horner’s Method for Polynomials
		5.8 Abstract Algebra
			5.8.1 Monoids and Groups
			5.8.2 Rings
			5.8.3 Fields
			5.8.4 Vector Spaces
		5.9 Review Questions
		5.10 Summary
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	6 Mathematical Induction and Recursion
		6.1 Introduction
		6.2 Strong Induction
		6.3 Recursion
		6.4 Structural Induction
		6.5 Review Questions
		6.6 Summary
		Reference
978-3-031-26212-8_7.pdf
	7 Graph Theory
		7.1 Introduction
		7.2 Undirected Graphs
			7.2.1 Hamiltonian Paths
		7.3 Trees
			7.3.1 Binary Trees
		7.4 Graph Algorithms
		7.5 Graph Colouring and Four-Colour Problem
		7.6 Review Questions
		7.7 Summary
		Reference
978-3-031-26212-8_8.pdf
	8 Sequences, Series, and Permutations and Combinations
		8.1 Introduction
		8.2 Sequences and Series
		8.3 Arithmetic and Geometric Sequences
		8.4 Arithmetic and Geometric Series
		8.5 Permutations and Combinations
		8.6 Review Questions
		8.7 Summary
978-3-031-26212-8_9.pdf
	9 A Short History of Logic
		9.1 Introduction
		9.2 Syllogistic Logic
		9.3 Paradoxes and Fallacies
		9.4 Stoic Logic
		9.5 Boole’s Symbolic Logic
			9.5.1 Switching Circuits and Boolean Algebra
		9.6 Frege
		9.7 Review Questions
		9.8 Summary
		References
978-3-031-26212-8_10.pdf
	10 Propositional and Predicate Logic
		10.1 Introduction
		10.2 Propositional Logic
			10.2.1 Truth Tables
			10.2.2 Properties of Propositional Calculus
			10.2.3 Proof in Propositional Calculus
			10.2.4 Semantic Tableaux in Propositional Logic
			10.2.5 Natural Deduction
			10.2.6 Sketch of Formalization of Propositional Calculus
			10.2.7 Applications of Propositional Calculus
			10.2.8 Limitations of Propositional Calculus
		10.3 Predicate Calculus
			10.3.1 Sketch of Formalization of Predicate Calculus
			10.3.2 Interpretation and Valuation Functions
			10.3.3 Properties of Predicate Calculus
			10.3.4 Applications of Predicate Calculus
			10.3.5 Semantic Tableaux in Predicate Calculus
		10.4 Review Questions
		10.5 Summary
		References
978-3-031-26212-8_11.pdf
	11 Advanced Topics in Logic
		11.1 Introduction
		11.2 Fuzzy Logic
		11.3 Temporal Logic
		11.4 Intuitionist Logic
		11.5 Undefined Values
			11.5.1 Logic of Partial Functions
			11.5.2 Parnas Logic
			11.5.3 Dijkstra and Undefinedness
		11.6 Logic and AI
		11.7 Review Questions
		11.8 Summary
		References
978-3-031-26212-8_12.pdf
	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
978-3-031-26212-8_13.pdf
	13 Automata Theory
		13.1 Introduction
		13.2 Finite-State Machines
		13.3 Pushdown Automata
		13.4 Turing Machines
		13.5 Review Questions
		13.6 Summary
		Reference
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	14 Computability and Decidability
		14.1 Introduction
		14.2 Logicism and Formalism
		14.3 Decidability
		14.4 Computability
		14.5 Computational Complexity
		14.6 Review Questions
		14.7 Summary
		Reference
978-3-031-26212-8_15.pdf
	15 Software Reliability and Dependability
		15.1 Introduction
		15.2 Software Reliability
			15.2.1 Software Reliability and Defects
			15.2.2 Cleanroom Methodology
			15.2.3 Software Reliability Models
		15.3 Dependability
		15.4 Computer Security
		15.5 System Availability
		15.6 Safety Critical Systems
		15.7 Review Questions
		15.8 Summary
		References
978-3-031-26212-8_16.pdf
	16 Overview of Formal Methods
		16.1 Introduction
		16.2 Why Should We Use Formal Methods?
		16.3 Industrial Applications of Formal Methods
		16.4 Industrial Tools for Formal Methods
		16.5 Approaches to Formal Methods
			16.5.1 Model-Oriented Approach
			16.5.2 Axiomatic Approach
		16.6 Proof and Formal Methods
		16.7 Debate on Formal Methods in Software Engineering
		16.8 The Vienna Development Method
		16.9 VDM♣, the Irish School of VDM
		16.10 The Z Specification Language
		16.11 The B-Method
		16.12 Predicate Transformers and Weakest Preconditions
		16.13 The Process Calculi
		16.14 Finite-State Machines
		16.15 The Parnas Way
		16.16 Model Checking
		16.17 Usability of Formal Methods
		16.18 Review Questions
		16.19 Summary
		References
978-3-031-26212-8_17.pdf
	17 Z Formal Specification Language
		17.1 Introduction
		17.2 Sets
		17.3 Relations
		17.4 Functions
		17.5 Sequences
		17.6 Bags
		17.7 Schemas and Schema Composition
		17.8 Reification and Decomposition
		17.9 Proof in Z
		17.10 Industrial Applications of Z
		17.11 Review Questions
		17.12 Summary
		Reference
978-3-031-26212-8_18.pdf
	18 Model Checking
		18.1 Introduction
		18.2 Modelling Concurrent Systems
		18.3 Linear Temporal Logic
		18.4 Computational Tree Logic
		18.5 Tools for Model Checking
		18.6 Industrial Applications of Model Checking
		18.7 Review Questions
		18.8 Summary
		References
978-3-031-26212-8_19.pdf
	19 The Nature of Theorem Proving
		19.1 Introduction
		19.2 Early Automation of Proof
		19.3 Interactive Theorem Provers
		19.4 A Selection of Theorem Provers
		19.5 Review Questions
		19.6 Summary
		References
978-3-031-26212-8_20.pdf
	20 Cryptography
		20.1 Introduction
		20.2 Breaking the Enigma Codes
		20.3 Cryptographic Systems
		20.4 Symmetric Key Systems
		20.5 Public Key Systems
			20.5.1 RSA Public Key Cryptosystem
			20.5.2 Digital Signatures
		20.6 Review Questions
		20.7 Summary
		References
978-3-031-26212-8_21.pdf
	21 Coding Theory
		21.1 Introduction
		21.2 Mathematical Foundations of Coding Theory
		21.3 Simple Channel Code
		21.4 Block Codes
			21.4.1 Error Detection and Correction
		21.5 Linear Block Codes
			21.5.1 Parity Check Matrix
			21.5.2 Binary Hamming Code
			21.5.3 Binary Parity Check Code
		21.6 Miscellaneous Codes in Use
		21.7 Review Questions
		21.8 Summary
		References
978-3-031-26212-8_22.pdf
	22 Introduction to 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
978-3-031-26212-8_23.pdf
	23 Introduction to 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 Bayesian Statistics
		23.7 Review Questions
		23.8 Summary
		References
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	24 Introduction to Data Science
		24.1 Introduction
		24.2 Ethics of Data Science
			24.2.1 Data Science and Data Scientists
			24.2.2 Data Science and Society
		24.3 What Is Data Analytics?
			24.3.1 Business Analytics and Business Intelligence
			24.3.2 Big Data and Data Mining
			24.3.3 Data Analytics for Social Media
			24.3.4 Sources of Data
		24.4 Mathematics Used in Data Science
		24.5 Review Questions
		24.6 Summary
		Reference
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	25 Calculus I
		25.1 Introduction
		25.2 Differentiation
			25.2.1 Rules of Differentiation
		25.3 Integration
			25.3.1 Definite Integrals
			25.3.2 Fundamental Theorems of Integral Calculus
		25.4 Review Questions
		25.5 Summary
		Reference
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	26 Calculus II
		26.1 Introduction
		26.2 Applications of Calculus
		26.3 Fourier Series
		26.4 The Laplace Transform
		26.5 Differential Equations
		26.6 Review Questions
		26.7 Summary
		Reference
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	27 Matrix Theory
		27.1 Introduction
		27.2 Two × Two Matrices
		27.3 Matrix Operations
		27.4 Determinants
		27.5 Eigen Vectors and Values
		27.6 Gaussian Elimination
		27.7 Review Questions
		27.8 Summary
		Reference
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	28 Complex Numbers and Quaternions
		28.1 Introduction
		28.2 Complex Numbers
		28.3 Quaternions
		28.4 Quaternion Algebra
			28.4.1 Quaternions and Rotations
		28.5 Review Questions
		28.6 Summary
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	29 Vectors
		29.1 Introduction
		29.2 Vectors in Euclidean Space
			29.2.1 Dot Product
			29.2.2 Cross Product
			29.2.3 Linear Dependent and Independent Vectors
		29.3 Review Questions
		29.4 Summary
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	30 Basic Financial Mathematics
		30.1 Introduction
		30.2 Simple Interest
			30.2.1 Computing Future and Present Values
			30.2.2 Computing Future Value
			30.2.3 Computing Present Values
		30.3 Compound Interest
			30.3.1 Present Value Under Compound Interest
			30.3.2 Equivalent Values
		30.4 Basic Mathematics of Annuities
		30.5 Loans and Mortgages
		30.6 Review Questions
		30.7 Summary
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	31 Introduction to Operations Research
		31.1 Introduction
		31.2 Linear Programming
			31.2.1 Linear Programming Example
			31.2.2 General Formulation of LP Problem
		31.3 Cost Volume Profit Analysis
		31.4 Game Theory
		31.5 Review Questions
		31.6 Summary
		References
978-3-031-26212-8_32.pdf
	32 Mathematical Software for Software Engineers
		32.1 Introduction
		32.2 Microsoft Excel
		32.3 Python
		32.4 Maple
		32.5 Minitab Statistical Software
		32.6 R Statistical Software Environment
		32.7 Mathematica
		32.8 MATLAB
		32.9 Summary
33.pdf
	Index