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, (named after John Venn (1834-1923)), we can represent a set with as a circle and its elements as dots, for example:

venn0

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.

Axioms: 1 2
Chapters: 3
Corollaries: 4 5
Explanations: 6
Proofs: 7

Thank you to the contributors under CC BY-SA 4.0!

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