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Explanation: Possibilities to Describe Sets, VennDiagrams, List, and SetBuilder Notations
(related to Definition: Set, Set Element, Empty Set)
There are at least three ways we can describe sets:
 Using a Venn diagram, (named after John Venn (18341923)), 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 setbuilder notation. In this case, we also use curly brackets. Inside the brackets, we describe the definite properties of the set elements. Examples of setbuilder notations are
$$\begin{array}{rcl}
A&=&\{x\in A\mid \; x\text{ is an even number}\}\\
B&=&\{y\in B\mid \; y\text{ is an elephant}\}\\
C&=&\{z\in C\mid \; z\text{ is a rightangled triangle.}\}
\end{array} $$
Please note that in the setbuilder 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.
Mentioned in:
Axioms: 1 2
Chapters: 3
Corollaries: 4 5
Explanations: 6
Proofs: 7
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References
Bibliography
 Hoffmann, Dirk W.: "Grenzen der Mathematik  Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
 Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016