Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography
Springer Science & Business Media, 6. pro 2012. - Broj stranica: 415
This modem introduction to the foundations of logic, mathematics, and computer science answers frequent questions that mysteriously remain mostly unanswered in other texts: • Why is the truth table for the logical implication so unintuitive? • Why are there no recipes to design proofs? • Where do these numerous mathematical rules come from? • What are the applications of formal logic and abstract mathematics? • What issues in logic, mathematics, and computer science still remain unresolved? Answers to such questions must necessarily present both theory and significant applica tions, which explains the length of the book. The text first shows how real life provides some guidance for the selection of axioms for the basis of a logical system, for instance, Boolean, classical, intuitionistic, or minimalistic logic. From such axioms, the text then derives de tailed explanations of the elements of modem logic and mathematics: set theory, arithmetic, number theory, combinatorics, probability, and graph theory, with applications to computer science. The motivation for such detail, and for the organization of the material, lies in a continuous thread from logic and mathematics to their uses in everyday life.
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Foundations of Logic and Mathematics: Applications to Computer Science and ...
Ograničeni pregled - 2002
A X B algorithm alphabet arithmetic atoms axiom of extensionality axiom of pairing axiom P1 bijection check digit common commutes Consequently consists contains contraposition converse law Deduction Theorem defined definition denoted directed graph disjoint divisor double negation edge empty set ENIGMA machine Example False finite following theorem shows function F graph G Hence implicational calculus induction hypothesis injective instance integer intuitionistic logic inverse Karnaugh table law of contraposition letters logical connective logical formula logical implication logically equivalent mathematical modulo Modus Ponens multiplication natural numbers non-empty non-negative notation ordinal P)A Q pairs path-connected Peirce's law permutation positive integer proceeds by induction Proof Apply proof proceeds propositional calculus propositional form rational numbers rotor sequence set theory ſº subset surjective tautology theorem 175 theorem holds transitive transposition True Truth tables Truth values variables verify vertex vertices VX(P VX(Q well-formed sets well-ordered whence