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We begin BookofProofs with a branch of mathematics dedicated to the set theory.

27Set Theory

Set theory is the systematic study of principles governing sets, which are the most basic mathematical objects. The four basics even a non-mathematician has to know about sets are:

  1. Sets can be completely described (defined) by either listing all their elements or defining the properties, the elements must fulfill.
  2. An object belongs to a set, if it is listed in the set or if it fulfills the defined properties.
  3. Sets do not allow duplicates (taking an element out of the set removes this element from the set completely).
  4. If you remove every element from the set, it becomes empty, (but it still exists, like it was an “empty container”).

Theoretical minimum (in a nutshell)

The study of set theory is highly intuitive and accessible to the undergraduates. However, understanding of the meaning of Zermelo-Fraenkel Axioms might be somewhat demanding at the beginning. It’s definitely worth the effort!

Concepts you will learn in this part of BookofProofs

  • What are the Zermelo-Fraenkel Axioms and how to define sets using them?
  • Which are the set operations and how to “calculate” with sets?
  • How can sets be “compared with each other” and start a fascinating journey into infinite sets using ordinals?
  • How can sets be used to “count things” and learn even more astonishing things about infinite sets, called cardinals?

| | | | Contributors: bookofproofs

651.Basics about Sets

79872.Motivation: Russel's Paradox

813.Zermelo-Fraenkel Set Theory

79864.Set Operations


646.Functions (Maps)

1857.Cardinal Numbers

1128.Ordinal Numbers

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