Comprehensive mathematics for computer scientists. 1: Sets and numbers, graphs and algebra, logic and machines, linear geometry
With ``Comprehensive Mathematics for Computer Scientists'' the authors try to present the entire extended landscape of mathematics needed in computer science within two small paperback textbooks. In the first volume they present mathematical core theory from axiomatic set theory to numbers, graphs, algebraic and logical structures, linear geometry. In the second volume they plan to present topology and calculus, differential equations, and more specialized and current subjects as neural networks, fractals, numerics, Fourier theory, wavelets, probability and statistics, manifolds, and categories. The price they have to pay is to omit any details, any discussions, any ``ramification in mathematical theorization''. While this provides no problem for well educated mathematicians there is the danger that mathematically uneducated people have almost no chance to learn how mathematics works at all. Mathematical thinking and understanding has to be trained by dealing with much more material and formal details, by proving theorems and propositions, and by solving exercises. However, the aim of the authors writing this book was not to publish another textbook in mathematics. Their book is ``merely'' part of a larger formal online training project which is hosted by the Department of Informatics at the University of Zürich. The online counterpart of the text includes various interactive tools for examples and exercises. In this way a multi media surface is provided to the ``reader''who can switch between the textbook and the online learning environment. To what extent such new combinations really help to deepen the student's mathematical knowledge remains to prove in future. Volume 1 of the textbook has 356 pages. It consists of two parts with altogether 26 chapters and includes an appendix with hints for further reading, bibliography and index. In Part I ``Sets, Numbers, and Graphs'' 14 chapters cover: Fundamentals -- Concepts and Logic, Axiomatic Set Theory, Boolean Set Algebra, Functions and Relations, Ordinal and Natural Numbers, Recursion Theorem and Universal Properties, Natural Arithmetics, Infinities, Classical Number Domains, Categories of Graphs, Construction of Graphs, Some Special Graphs, Planarity, First Advanced Topic. The 12 chapters of Part II cover 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.
scientific article; zbMATH DE number 2120296