Elementary Lectures in Undergraduate Algebra Volume I

This book contains the first twenty (20) lectures of the author on elementary topics that form the foundation of the study of all types of traditional and modern mathematics. These lectures contain every elementary concept and result on basic set theory that are taught to honours students right from the first year of their study and which will always be needed throughout their mathematical career. All the concepts and results considered in these lectures are motivated from examples while the proofs of these results are detailed enough to allow independent study by students. Another feature of this book is the inclusion of Expected Instructional Objectives (at the start of every lecture) which contain the specific and measurable achievements expected of every reader and the Final Questions (at the end of the lecture) which review the learning process. Some of the exercises (numbering more than 450 structured questions (plus 60 tutorials) most of which are supplied with hints for their solutions) consist of PROJECTS which could be undertaken by the reader either as a member of a group or as an individual. Every lecture contains a section called Summary of Lecture where a synopsis of the lecture is done in as few words as possible. The first thirteen (13) lectures of this book on set theory (treating relations and operations in and on sets) may be used by 100 Level students offering elements of mathematics (especially Lectures 1 to 4) by those 200 Level students who are just starting their courses in Real Analysis Linear and Abstract Algebra as well as 300 Level students (offering Metric-space Topology Groups and Rings) and 400 Level students (offering General Topology Algebraic Topology and Commutative Algebra) who need to be reminded where their subject started from. The remaining seven (7) lectures on arithmetic of integers (treating divisibility and congruence in Z) are needed in basic groups and rings taught in 200 Level and they serve as the foundation to the theory of (algebraic) numbers as well as to Groups and Rings for 300 Level students. This book is an attempt at presenting elementary mathematics to all users and readers of mathematics with a view at equipping them with the most often used techniques via examples and reasoning without loosing sight of the importance of the general results and without being too technical and rigid in the presentation of the ideas. It is particularly suitable for self-study and it complements any form of contact that the reader would have with the class.