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Mathematics · Ch 1 — Sets
The concept of a set is the bedrock upon which modern mathematics is built. It is so fundamental that nearly every branch of mathematics — from geometry and sequences to probability and calculus — relies on it. When you study relations and functions later in this book, you will be working with sets at every step.
The theory of sets was pioneered by the German mathematician Georg Cantor (1845–1918). Cantor first encountered sets while investigating problems related to trigonometric series. His work was revolutionary: he showed that the idea of a "collection of objects" could be studied with the same rigour as numbers or shapes, and that this study could resolve deep questions about infinity itself.
Before Cantor, mathematicians treated collections of objects informally. Cantor gave sets a precise logical foundation, which allowed mathematics to handle infinite collections without paradox.
This chapter introduces the basic definitions and operations that involve sets. You will learn:
The entire chapter builds from this single opening idea: a set is a well-defined collection of objects. Every result that follows — every property, every operation — is a logical consequence of that definition.
The word well-defined is crucial. A collection is a set only if you can unambiguously decide whether any given object belongs to it. "The collection of tall students" is not a set (tall is subjective). "The collection of students whose height is greater than 180 cm" is a set.
The introduction does not contain any properties, theorems, or derivations. It is a single paragraph that sets the stage. The first numbered properties appear in Section 1.3 (The Empty Set) and Section 1.4 (Finite and Infinite Sets). The first formal theorem appears in Section 1.6 (Power Set).
Do not skip the introduction. Many students jump straight to definitions and formulas, but the historical context and the emphasis on "well-defined" will save you from common mistakes later. When you encounter a question like "Is the collection of all honest people a set?" — the answer is no, because honesty is not well-defined. That distinction comes directly from this opening section.