This is a textbook for an undergraduate mathematics major transition course from technique-based mathematics such as Algebra and Calculus to proof-based mathematics. It motivates the introduction of the formal language of logic and set theory and develops the basics with examples, exercises with solutions and exercises without.
It then moves to a discussion of proof structure and basic proof techniques, including proofs by induction with extensive examples.
An in-depth treatment of relations, particularly equivalence and order relations completes the exposition of the basic language of mathematics. The last chapter treats infinite cardinalities.
An appendix gives some complement on induction and order, and another provides full solutions of the in-text exercises. The primary audience is undergraduate mathematics major, but independent readers interested in mathematics can also use the book for self-study. Originally from France where he obtained his PhD in , he has been teaching at various universities in the US since , teaching a wide spectrum of mathematics courses, from entry level to graduate courses.
A topologist, he has published 35 peer-reviewed research papers, and the book Convergence Foundations of Topology , revisiting classical General Topology from the point of view of his area of research: convergence spaces.
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The Math Department Lounge and two computer rooms are on the fourth floor, in addition to computers in each office. Mathematical life at Penn is very active. It comprises original investigations, discussions, lectures, and teaching at many levels.
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We are deeply committed to superior research in mathematics and the scientific excellence of our faculty is well recognized in the mathematical community. Members of the Department have been elected to the National Academy of Sciences and to similar Academies of foreign countries.
Large segments of current mathematical research are represented in the Department by the active research programs and interests of our faculty. A clearer idea of the exact areas engaging the Department's current interest can be formed from the list of the faculty and their individual areas of research.