Comprehensive Mathematics for Computer Scientists 1: Sets and Numbers, Graphs and Algebra, Logic and Machines, Linear Geometry (Universitext)
Gérard Milmeister, Guerino Mazzola, Jody Weissmann
Language: English
Pages: 389
ISBN: 2:00090697
Format: PDF / Kindle (mobi) / ePub
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. This text is complemented by an online university course which covers the same theoretical content, however, in a totally different presentation. The student or working scientist who once gets involved in this text may at any time consult the online interface which comprises applets and other interactive tools.
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, T ) where F is the name of a field of class C, and where T is the type class of F , or the pairs (C, S), where S is the direct superclass of C; we set head Λ (C, F , T ) = T and tail Λ (C, F , T ) = C, or, for superclass arrows, head Λ (C, S) = S and tail Λ (C, S) = C. If one forgets about the direction in a digraph, the remaining structure is that of an “undirected” graph, or simply “graph”. We shall henceforth always write digraph for a directed graph, and graph for an undirected graph, if
, 11) (s2 , 00) (s1 , 11) s1 s2 (s2 , 01) (s1 , 01) (s2 , 10) (s1 , 10) Fig. 12.3. The Moore graph Moore(M). Here is the description of paths in the Moore graph of a sequential machine: 12.2 Moore Graphs 141 Proposition 106 For a sequential machine M : S × Qn → S, a canonical bijection P W : Path(Moore(M)) → S × Word(Qn ) is given as follows: If p = s1 (s1 ,q1 ) s2 (s2 ,q2 ) s3 · · · (sm−1 ,qm−1 ) sm , then P W (p) = (s1 , q1 q2 . . . qm−1 ). Under this bijection, for a given
here several concepts which will only be explained rigorously in the chapter on topology in the second volume of this book. However, the elementary character of the results and the important problem of drawing graphs suggests a preliminary treatment of the subject in this first part of the course. 13.1 Euler’s Formula for Polyhedra To begin with, this chapter only deals with undirected graphs which have no loops and no multiple edges. In fact, drawing such graphs immediately implies drawing of
subset M ⊂ M such that for all m, n ∈ M , m ∗ n ∈ M and e ∈ M , while the multiplication ∗ is the restriction of ∗ to M . A submonoid therefore gives rise to the evident embedding homomorphism M. iM : M Exercise 55 Given a monoid (M, ∗) and a (possibly empty) subset S ⊂ M, there is a unique minimal submonoid M of M such that S ⊂ M . It is denoted by S and is called the submonoid generated by S. Show that S consists of all finite products s1 ∗ s2 ∗ . . . sn , si ∈ S, and of the neutral element e.
a) and (x ∈ c and x ∈ b)} = {x | x ∈ c − a and x ∈ c − b} (1) (2) (3) (4) 2.2 Basic Concepts and Results = {x | x ∈ c − a} ∩ {x | x ∈ c − b} 23 (5) = (c − a) ∩ (c − b) Equalities (1) and (3) hold because AND is associative. Equality (2) holds because P AND Q = P AND P AND Q for any truth values P and Q. Equality (4) is the definition of the set difference. Equality (5) is the definition of the intersection of two sets. C HAPTER 3 Boolean Set Algebra In this chapter, we shall give a
