Understanding Equivalence Relations and Integer Construction

This lecture focuses on the construction of integers through equivalence relations. The instructor begins by discussing the properties of natural numbers and the need for integers to solve equations without solutions in natural numbers. The concept of equivalence relations is introduced, emphasizing reflexivity, symmetry, and transitivity. The instructor illustrates how to define integers as classes of equivalence, where each class represents a unique difference between pairs of natural numbers. The lecture further explores the addition and multiplication of these classes, ensuring that operations are well-defined and independent of the representatives chosen. The properties of these operations are discussed, establishing that the set of integers forms an abelian group under addition. The instructor concludes by highlighting the historical context of these mathematical concepts, particularly their formalization in the 19th century, and sets the stage for future discussions on multiplication and further properties of integers.