The students' task in learning set theory is to steep themselves in unfamiliar and essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort.
Paul R. Halmos, Naive set theory (adapted slightly).The language of set theory is used throughout mathematics. Many general results involve ‘an integer n’ or ‘a real number a’ and, to start with, set theory notation provides a simple way of asserting for example that n is an integer. However, it turns out that this language is remarkably flexible and powerful and in much mathematics it is indispensable for a proper expression of the ideas involved.
In the second part of this book we introduce the basic vocabulary of the language of sets and functions. The third part of the book will then provide some experience in using this language as we use it to give a precise formulation of the idea of counting, one of the earliest mathematical concepts.
Sets
At this introductory level it is sufficient to define the notion of set as any well-defined collection of objects. We can think of a set as a box containing certain objects. In this section some ways of specifying sets are introduced and also some frequently used notation.
We frequently use a single letter to denote a set. This represents a further stage of mathematical abstraction.
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