Introduction to Modern Mathematics

Based on an introductory course in mathematics given by the author at Warsaw University, this book covers the fundamentals of mathematical logic, set theory and abstract algebra, aiming to facilitate the study of modern mathematics. Intended for first year students in mathematics, it may also prove useful to readers who, studying engineering, natural science, social science or the humanities, require a knowledge of abstract mathematics.

The book is divided into three parts. The first part covers the material from set theory which is selected because of its usefulness in the study of other branches of mathematics. The Cantor-Bernstein theorem, the Kuratowski-Zorn lemma on the existence of maximal elements in partially ordered sets, the theorem of transfinite induction, the theorem on definition by transfinite induction and Zermelo's well-ordering theorem are included in this section.

The second part deals with the basics of mathematical logic which can be applied to mathematics, mainly in proving theorems. A formalization of logic is avoided and the exposition is confined to those rules and laws of inference in the propositional and functional calculi which are most frequently used in mathematical reasoning, and to many examples which show their applications in mathematics. The third part explains the basic concepts of abstract algebra, such as subalgebra, homomorphism, isomorphism, congruence, quotient algebra, products of algebras, etc. More sophisticated discussions concern equationally definable classes of algebras and free algebras. The exercises added to each section are intended to illustrate the concepts and methods discussed in the book.