An introduction to Propositional Logic and Set Theory: Foundations of Mathematical Reasoning. Worked Examples and Exercises (Understanding Calculus Book 1)

This book is especially designed for students who are finishing high school and wish to build a strong foundation in mathematics before starting college or university. It is also suitable for those already enrolled in higher education who continue to face challenges in understanding more advanced mathematical concepts. Often, these difficulties stem from gaps in foundational knowledge, which can create resistance to learning more complex topics.

To address these gaps, the book is written in clear and accessible language, making the learning experience approachable—even for students who may not particularly enjoy mathematics. However, simplicity of expression does not mean a lack of rigor. The content maintains academic depth and precision, ensuring that students gain a solid understanding of the key concepts.
The book is organized into three chapters, each addressing essential building blocks for the study of Calculus (or Mathematical Analysis). We begin with Propositional Logic , which introduces the language and structure of mathematical reasoning, as well as the fundamentals of constructing proofs. The second chapter covers Set Theory , where we develop the core concepts that underpin definitions central to Calculus, such as that of a function. The final chapter presents the Real Number System , explored through an axiomatic approach that highlights its foundational role in analysis.

Each concept is accompanied by illustrative examples, and every chapter includes a variety of exercises designed to reinforce understanding. Answers to the exercises are provided at the end of each chapter. The exercises vary in difficulty—from basic to moderate—with the goal of supporting comprehension rather than testing mathematical ability.

By the end of this book, students will have developed the logical and conceptual foundation necessary to confidently approach key topics in Calculus, including functions, limits, continuity, derivatives, and integrals—all of which will be addressed in future volumes by the author.