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:

- Sets can be completely described (defined) by either listing all their elements or defining the properties, the elements must fulfill.
- An object belongs to a set, if it is listed in the set or if it fulfills the defined properties.
- Sets do not allow duplicates (taking an element out of the set removes this element from the set completely).
- 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

## 635.Relations

## 646.Functions (Maps)

## 1857.Cardinal Numbers

## 1128.Ordinal Numbers

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