Logic and discrete mathematics : a concise introduction. [Hauptbd.] ...

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Title:

Author:

Conradie, Willem (DE-601)82883458X (DE-588)1073308510

ISBN:

9781118751275

Personal Author:

Conradie, Willem

Publication Information:

Chichester Wiley 2015

Physical Description:

XIX, 426 S. Ill., graph. Darst

General Note:

4.1 Basic concepts of first-order logic.

Contents:

Cover; Title Page; Copyright; Contents; List of Boxes; Preface; Acknowledgements; About the Companion Website; Chapter 1 Preliminaries; 1.1 Sets; 1.1.1 Exercises; 1.2 Basics of logical connectives and expressions; 1.2.1 Propositions, logical connectives, truth tables, tautologies; 1.2.2 Individual variables and quantifiers; 1.2.3 Exercises; 1.3 Mathematical induction; 1.3.1 Exercises; Chapter 2 Sets, Relations, Orders; 2.1 Set inclusions and equalities; 2.1.1 Properties of the set theoretic operations; 2.1.2 Exercises; 2.2 Functions; 2.2.1 Functions and their inverses.

2.2.2 Composition of mappings2.2.3 Exercises; 2.3 Binary relations and operations on them; 2.3.1 Binary relations; 2.3.2 Matrix and graphical representations of relations on finite sets; 2.3.3 Boolean operations on binary relations; 2.3.4 Inverse and composition of relations; 2.3.5 Exercises; 2.4 Special binary relations; 2.4.1 Properties of binary relations; 2.4.2 Functions as relations; 2.4.3 Reflexive, symmetric and transitive closures of a relation; 2.4.4 Exercises; 2.5 Equivalence relations and partitions; 2.5.1 Equivalence relations; 2.5.2 Quotient sets and partitions.

2.5.3 The kernel equivalence of a mapping2.5.4 Exercises; 2.6 Ordered sets; 2.6.1 Pre-orders and partial orders; 2.6.2 Graphical representing posets: Hasse diagrams; 2.6.3 Lower and upper bounds. Minimal and maximal elements; 2.6.4 Well-ordered sets; 2.6.5 Exercises; 2.7 An introduction to cardinality; 2.7.1 Equinumerosity and cardinality; 2.7.2 Exercises; 2.8 Isomorphisms of ordered sets. Ordinal numbers; 2.8.1 Exercises; 2.9 Application: relational databases; 2.9.1 Exercises; Chapter 3 Propositional Logic; 3.1 Propositions, logical connectives, truth tables, tautologies.

3.1.1 Propositions and propositional connectives. Truth tables3.1.2 Some remarks on the meaning of the connectives; 3.1.3 Propositional formulae; 3.1.4 Construction and parsing tree of a propositional formula; 3.1.5 Truth tables of propositional formulae; 3.1.6 Tautologies; 3.1.7 A better idea: search for a falsifying truth assignment; 3.1.8 Exercises; 3.2 Propositional logical consequence. Valid and invalid propositional inferences; 3.2.1 Propositional logical consequence; 3.2.2 Logically sound rules of propositional inference. Logically correct propositional arguments.

3.2.3 Fallacies of the implication3.2.4 Exercises; 3.3 The concept and use of deductive systems; 3.4 Semantic tableaux; 3.4.1 Exercises; 3.5 Logical equivalences. Negating propositional formulae; 3.5.1 Logically equivalent propositional formulae; 3.5.2 Some important equivalences; 3.5.3 Exercises; 3.6 Normal forms. Propositional resolution; 3.6.1 Conjunctive and disjunctive normal forms of propositional formulae; 3.6.2 Clausal form. Clausal resolution; 3.6.3 Resolution-based derivations; 3.6.4 Optimizing the method of resolution; 3.6.5 Exercises; Chapter 4 First-Order Logic.

Abstract:

A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual. Key Features: Suitable for a variety of courses for students in both Mathematics and Computer Science. Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematicsConcise, clear and uncluttered presentation with numerous examples. Covers some applications including cryptographic systems, discrete probability and network algorithms. Logic and Discrete Mathematics: A Concise Introductionis aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study. Willem Conradie, University of Johannesburg, South AfricaValentin Goranko, University of Stockholm, Sweden

Subject Term:

Logic, Symbolic and mathematical -- Textbooks.

Computer science -- Mathematics -- Textbooks.

Added Author:

Goranko, Valentin,

Copies:

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Summary

A concise yet rigorous introduction to logic and discrete mathematics.

This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade.

The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual.

Key Features :

Suitable for a variety of courses for students in both Mathematics and Computer Science. Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematics Concise, clear and uncluttered presentation with numerous examples. Covers some applications including cryptographic systems, discrete probability and network algorithms.

Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study.