## Comprehensive Mathematics for Computer Scientists 1: Sets and Numbers, Graphs and Algebra, Logic and Machines, Linear GeometryA second edition of a book is a success and an obligation at the same time. We are satis ed that a number of university courses have been orga› nized on the basis of the rst volume of Comprehensive Mathematics for Computer Scientists. The instructors recognized that the self›contained presentation of a broad specturm of mathematical core topics is a rm point of departure for a sustainable formal education in computer sci› ence. We feel obliged to meet the valuable feedback of the responsible in› structors of such courses, in particular of Joel Young (Computer Science Department, Brown University) who has provided us with numerous re› marks on misprints, errors, or obscurities. We would like to express our gratitude for these collaborative contributions. We have reread the entire text and not only eliminated identi ed errors, but also given some addi› tional examples and explications to statements and proofs which were exposed in a too shorthand style. A second edition of the second volume will be published as soon as the errata, the suggestions for improvements, and the publisher’s strategy are in harmony. |

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### Contents

FundamentalsConcepts and Logic | 3 |

Axiomatic Set Theory | 17 |

Ordinal and Natural Numbers | 45 |

Copyright | |

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### Common terms and phrases

acceptor addition algebra algorithm alphabet arithmetic arrow associated automata automaton axiom basis bijection binary Boolean called Cartesian product Cauchy sequence chapter coefficient column commutative ring complex numbers concept construction Corollary cycle defined Definition denoted det(M digraph directed graph empty equation equipollent equivalence relation Euclidean space example Exercise fact field formal formula function geometry given group homomorphism Heyting algebra implies induction injection integer inverse isomorphism Ker(f Lang(JA language lemma linear logic mathematical matrix module monoid Moore graph morphism natural numbers neutral element non-empty non-zero notation ordinal pair permutation phrase structure grammar polynomial Proof proposition quaternion R-module rational numbers real numbers recursion representation ring homomorphism rotation S(EX sentences Show solution sorite subgroup subset Suppose symbols theorem tion truth values Turing machine uniquely determined universal property variables vector space vertex whence Word(JA words write x e a x e c zero