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Elements of Logic via Numbers and Sets

D.L. Johnson (Autor)

Buch | Softcover

X, 188 Seiten

1998 | 1998
Springer Berlin (Verlag)
978-3-540-76123-5 (ISBN)

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In mathematics we are interested in why a particular formula is true. Intuition and statistical evidence are insufficient, so we need to construct a formal logical proof. The purpose of this book is to describe why such proofs are important, what they are made of, how to recognize valid ones, how to distinguish different kinds, and how to construct them. This book is written for 1st year students with no previous experience of formulating proofs. Dave Johnson has drawn from his considerable experience to provide a text that concentrates on the most important elements of the subject using clear, simple explanations that require no background knowledge of logic. It gives many useful examples and problems, many with fully-worked solutions at the end of the book. In addition to a comprehensive index, there is also a useful `Dramatis Personae` an index to the many symbols introduced in the text, most of which will be new to students and which will be used throughout their degree programme.

1. Numbers.- 1.1 Arithmetic Progressions.- 1.2 Proof by Contradiction.- 1.3 Proof by Contraposition.- 1.4 Proof by Induction.- 1.5 Inductive Definition.- 1.6 The Well-ordering Principle.- 2. Logic.- 2.1 Propositions.- 2.2 Truth Tables.- 2.3 Syllogisms.- 2.4 Quantifiers.- 3. Sets.- 3.1 Introduction.- 3.2 Operations.- 3.3 Laws.- 3.4 The Power Set.- 4. Relations.- 4.1 Equivalence Relations.- 4.2 Congruences.- 4.3 Number Systems.- 4.4 Orderings.- 5. Maps.- 5.1 Terminology and Notation.- 5.2 Examples.- 5.3 Injections, Surjections and Bijections.- 5.4 Peano's Axioms.- 6. Cardinal Numbers.- 6.1 Cardinal Arithmetic.- 6.2 The Cantor-Schroeder-Bernstein theorem.- 6.3 Countable Sets.- 6.4 Uncountable Sets.- Solutions to Exercises.- Guide to the Literature.- Dramatis Personae.

From the reviews: "It took me a while to appreciate the need for a course intended to introduce mathematics undergraduates to advanced mathematics after they finish the calculus sequence. ... This was the first book I found that seems really close to the core. It is well-written and very well suited to such a course." (James M. Cargal, UMAP Journal, Vol. 31 (1), 2010)

From the reviews:

“It took me a while to appreciate the need for a course intended to introduce mathematics undergraduates to advanced mathematics after they finish the calculus sequence. … This was the first book I found that seems really close to the core. It is well-written and very well suited to such a course.” (James M. Cargal, UMAP Journal, Vol. 31 (1), 2010)

Erscheint lt. Verlag	14.1.1998
Reihe/Serie	Springer Undergraduate Mathematics Series
Zusatzinfo	X, 188 p.
Verlagsort	London
Sprache	englisch
Maße	178 x 235 mm
Gewicht	316 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
Schlagworte	Addition • arithmetic • Cantor • cardinal number • Countable set • Equivalence • Mathematics • Mathematische Logik • Proof • proof by contradiction • Theorem • well-ordering principle • Zahlentheorie
ISBN-10	3-540-76123-3 / 3540761233
ISBN-13	978-3-540-76123-5 / 9783540761235
Zustand	Neuware