**Explanation**: Possibilities to Describe Sets, Venn-Diagrams, List, and Set-Builder Notations

There are at least three ways we can describe sets:

- Using a
**Venn diagram**, we can represent a set with as a circle and its elements as dots, for example:

- Using a
**list notation**(sometimes also called the**roster notation**), where we write the set $A$ listing all its elements in curly brackets, for instance: $$A=\{a,b,c,d\}.$$ - Using a
**set-builder notation**. In this case, we also use curly brackets. Inside the brackets, we describe the**definite properties**of the set elements. Examples of set-builder notations are

$$\begin{array}{rcl}

A&=&\{x\in A\mid \; x\text{ is an even number}\}\\

B&=&\{y\in B\mid \; y\text{ is an elephant}\}\\

C&=&\{z\in C\mid \; z\text{ is a right-angled triangle.}\}

\end{array}

$$

Please note that in the set-builder notation, we always have to make use of properties, which have to be introduced already. Otherwise, we risk that our set definition will contain some ambiguities.

| | | | created: 2014-04-27 13:49:57 | modified: 2018-12-09 12:11:21 | by: *bookofproofs* | references: [656], [7838]

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[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

[7838] **Kohar, Richard**: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016

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