A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included numerous exercises designed...

This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic...

Now revised and updated, this brisk introduction to functional analysis is intended for advanced undergraduate students, typically final year, who have had some background in real analysis. The author's aim is not just to cover the standard material in a standard way, but to present results of application in contemporary mathematics and to show the relevance of functional analysis to other areas. Unusual topics covered include the geometry of finite-dimensional spaces, invariant subspaces, fixed-point theorems, and the Bishop-Phelps theorem. An outstanding feature is the large number of...

Rigorous yet engaging, Linear Algebra offers a unified treatment of both matrix-oriented and theoretical approaches to the course.

This rigorous undergraduate introduction to dynamical systems is an accessible guide for mathematics students advancing from calculus.

This first textbook on experimental mathematics that gives a comprehensive introduction to the application of computer programs in mathematical innovation.

This concise text clearly presents the material needed for year-long analysis courses for advanced undergraduates or beginning graduates.

With exercises and projects, Exploring Mathematics supports an active approach to the transition to upper-level theoretical math courses.

An introductory probability textbook with the right balance between mathematical precision, probabilistic intuition, and concrete applications.

Concise and modern introduction to differential topology with a hands-on approach and plentiful examples and exercises.