This modern introduction to the foundations of logic and mathematics not only takes theory into account, but also treats in some detail applications that have a substantial impact on everyday life (loans and mortgages, bar codes, public-key cryptography). A first college-level introduction to logic, proofs, sets, number theory, and graph theory, and an excellent self-study reference and resource for instructors. This modern introduction to the foundations of logic and mathematics answers many questions that usually remain unanswered: Why is the truth table for logical implication so unintuitive? Why are there no recipes to design proofs? Where do these numerous mathematical rules come from? What issues in logic, mathematics, and computer science still remain unresolved? And the perennial question: In what ways are we going to use this material?
'Foundations of Logic and Mathematics' takes into account not only theory, but also treats in some detail applications that have substantial impact on everyday life---for example, financial loans and mortgages, bar codes (Universal Product Codes), public-key cryptography (Rivest--Shamir--Adelman codes), and transportation networks.
Topics covered: truth tables, propositional and predicate calculi, set theory, theory and practice of basic arithmetic, cardinality, well-formed sets, completeness and incompleteness of various logic, number theory, combinatorics, and graph theory. One of the key strengths of the presentation is the continuous thread from logic to applications, without gaps. Thus, material that is necessary for logical coherence---omitted from most texts---is found here.
'Foundations of Logic and Mathematics' takes into account not only theory, but also treats in some detail applications that have substantial impact on everyday life---for example, financial loans and mortgages, bar codes (Universal Product Codes), public-key cryptography (Rivest--Shamir--Adelman codes), and transportation networks.
Topics covered: truth tables, propositional and predicate calculi, set theory, theory and practice of basic arithmetic, cardinality, well-formed sets, completeness and incompleteness of various logic, number theory, combinatorics, and graph theory. One of the key strengths of the presentation is the continuous thread from logic to applications, without gaps. Thus, material that is necessary for logical coherence---omitted from most texts---is found here.
From the reviews:
"The book under review covers the topics which can usually be found in textbooks of discrete mathematics for students in computer science or mathematics (Boolean logic, predicate calculus, sets and functions, induction, integers, rational numbers, cardinality, modular arithmetic, cryptography, combinatorics, probability, graphs) as well as more advanced topics in mathematical logic (intuitionistic logic, transfinite induction). While the range of topics is relatively standard, the way they are presented is highly original. The author has chosen a strictly formal and axiomatic approach. All the results are proved in full detail from first principles . . . remarkably, all the arithmetic laws on the rational numbers are proved, step after step, starting from the very definitions! . . . a valuable reference text and a useful companion for anybody wondering how the basic mathematical concepts can be rigorously developed within set theory. The author has managed to combine the foundational approach with a careful treatment of many applications. More than 1000 exercises complete the text. ...All the results are proved in full detail from first principles...remarkably, the arithmetic laws on the rational numbers are proved, step after step, starting from the very definitions!...This is a valuable reference text and a useful companion for anybody wondering how basic mathematical concepts can be rigorously developed within set theory."
-MATHEMATICAL REVIEWS
"In order to give an idea of the originality of this book in combining theoretical and applied issues, having a source in everyday life and a strong impact on civilization, let us mention that the first section of Part A includes a typology of proofs and of theorems. The section on induction includes arithmetic in finance. The section on decidability includes a chapter on automatic theorem proving. Section 6 in Part B describes the Enigma machines: how the German machine cipher was broken and how it was read by the Allies in World War Two. Rigorous and modern in its theoretical aspect, attractive as a detective novel in its applied aspects, this paper book deserves the attention of both beginners and advanced students in mathematics, logic and computer sciences as well as in social sciences."
---Zentralblatt MATH
"This book could serve as a text and as a reference; it has a different presentation style. ... The book is divided in two parts; Part A is titled Theory and Part B Applications. It has a five and a half page bibliography, and a 13 page extensive index. ... The book provides detailed coverage of topics on Logic and Mathematics. The two part presentation of theory and applications is well thought out. ... the exercises provided are exhaustive." (Saif Terai, The Book Review Column, 2011)
"The book under review covers the topics which can usually be found in textbooks of discrete mathematics for students in computer science or mathematics (Boolean logic, predicate calculus, sets and functions, induction, integers, rational numbers, cardinality, modular arithmetic, cryptography, combinatorics, probability, graphs) as well as more advanced topics in mathematical logic (intuitionistic logic, transfinite induction). While the range of topics is relatively standard, the way they are presented is highly original. The author has chosen a strictly formal and axiomatic approach. All the results are proved in full detail from first principles . . . remarkably, all the arithmetic laws on the rational numbers are proved, step after step, starting from the very definitions! . . . a valuable reference text and a useful companion for anybody wondering how the basic mathematical concepts can be rigorously developed within set theory. The author has managed to combine the foundational approach with a careful treatment of many applications. More than 1000 exercises complete the text. ...All the results are proved in full detail from first principles...remarkably, the arithmetic laws on the rational numbers are proved, step after step, starting from the very definitions!...This is a valuable reference text and a useful companion for anybody wondering how basic mathematical concepts can be rigorously developed within set theory."
-MATHEMATICAL REVIEWS
"In order to give an idea of the originality of this book in combining theoretical and applied issues, having a source in everyday life and a strong impact on civilization, let us mention that the first section of Part A includes a typology of proofs and of theorems. The section on induction includes arithmetic in finance. The section on decidability includes a chapter on automatic theorem proving. Section 6 in Part B describes the Enigma machines: how the German machine cipher was broken and how it was read by the Allies in World War Two. Rigorous and modern in its theoretical aspect, attractive as a detective novel in its applied aspects, this paper book deserves the attention of both beginners and advanced students in mathematics, logic and computer sciences as well as in social sciences."
---Zentralblatt MATH
"This book could serve as a text and as a reference; it has a different presentation style. ... The book is divided in two parts; Part A is titled Theory and Part B Applications. It has a five and a half page bibliography, and a 13 page extensive index. ... The book provides detailed coverage of topics on Logic and Mathematics. The two part presentation of theory and applications is well thought out. ... the exercises provided are exhaustive." (Saif Terai, The Book Review Column, 2011)