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In this chapter, we introduce the student to the world of sets. Actually, only a little bit of it, the part that is needed to get going.
After giving a rough intuitive idea of what sets are, we present the basic relations between them: inclusion, identity, proper inclusion and exclusion. We describe two common ways of identifying sets and pause to look more closely at the empty set. We then define some basic operations for forming new sets out of old ones: intersection, union, difference and complement. These are often called Boolean operations, after George Boole, who first studied them systematically in the middle of the nineteenth century.
Up to this point, the material is all ‘flat’ set theory, in the sense that it does not look at what we can do when the elements of sets are themselves sets. However, we need to go a little beyond flatland. In particular, we need to generalize the notions of intersection and union to cover arbitrary collections of sets and introduce the very important concept of the power set of a set, that is, the set of all its subsets.
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For a very gentle introduction to sets, which nevertheless takes the reader up to an outline of the Zermelo-Fraenkel axioms for set theory, see:
Bloch ED (2011) Proofs and fundamentals: a first course in abstract mathematics, 2nd edn. Springer, New York, chapter 3
A classic of beautiful exposition, but short on exercises (readers should instead verify the claims made in the text):
Halmos PR (2001) Naive set theory, new edn. Springer, New York, chapters 1–5, 9
The present material is covered with lots of exercises in:
Lipschutz S (1998) Set theory and related topics, Schaum’s outline series. McGraw Hill, New York, chapters 1–2 and 5.1–5.3
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