Comprehensive Mathematics for Computer Scientists 1

This two-volume textbook, of which this is the first volume, is a self-contained comprehensive presentation of mathematics for computer scientists. It includes coverage of sets, numbers, graphs, algebra, logic, grammars and machines. It also deals with linear geometry, calculus, ODEs, and special themes such as neural networks, Fourier theory, wavelets, numerical issues, statistics, categories, and manifolds. This text is complemented by an online university course which covers the same theoretical content in a totally different presentation. The student or working scientist who gets involved with this text may at any time consult the online interface which contains applets and other interactive tools. For the second edition the entire text has been carefully re-written, and many examples have been added, as well as illustrations and explications to statements and proofs which were exposed in a too short a style. This makes the book easier for both instructors and students.

Includes supplementary material: sn.pub/extras

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This two-volume textbook Comprehensive Mathematics for Computer Scientists is a self-contained comprehensive presentation of mathematics including sets, numbers, graphs, algebra, logic, grammars, machines, linear geometry, calculus, ODEs, and special themes such as neural networks, Fourier theory, wavelets, numerical issues, statistics, categories, and manifolds. The concept framework is streamlined but defining and proving virtually everything. The style implicitly follows the spirit of recent topos-oriented theoretical computer science. Despite the theoretical soundness, the material stresses a large number of core computer science subjects, such as, for example, a discussion of floating point arithmetic, Backus-Naur normal forms, L-systems, Chomsky hierarchies, algorithms for data encoding, e.g., the Reed-Solomon code. The numerous course examples are motivated by computer science and bear a generic scientific meaning.

For the second edition the entire text has been carefully reread, and many examples have been added, as well as illustrations and explications to statements and proofs which were exposed in a too shorthand style. This makes the book more comfortable for instructors as well as for students to handle.

Contenu
Sets, Numbers, and Graphs.- FundamentalsConcepts and Logic.- Axiomatic Set Theory.- Boolean Set Algebra.- Functions and Relations.- Ordinal and Natural Numbers.- Recursion Theorem and Universal Properties.- Natural Arithmetic.- Infinities.- The Classical Number Domains Z, Q, R, and C.- Categories of Graphs.- Construction of Graphs.- Some Special Graphs.- Planarity.- First Advanced Topic.- Algebra, Formal Logic, and Linear Geometry.- Monoids, Groups, Rings, and Fields.- Primes.- Formal Propositional Logic.- Formal Predicate Logic.- Languages, Grammars, and Automata.- Categories of Matrixes.- Modules and Vector Spaces.- Linear Dependence, Bases, and Dimension.- Algorithms in Linear Algebra.- Linear Geometry.- Eigenvalues, the Vector Product, and Quaternions.- Second Advanced Topic.