\Huge 
 \textcolor{violet}{杊}
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Evolution

X \ / \ 集 \\ Set
\textcolor{violet}{\mathcal{T}} \ / \ \textcolor{violet}{拓} \\ Topology
\textcolor{violet}{X} \ / \ \textcolor{violet}{拓间} \\ Topological \ space

Definition

Definition…






Origin





Urprung…





Calligraphy





Dialects





Details






Details
















Diagram





Phonology





Details








Details








Gallery





Notes





Properties





Details








Details








Details








Functoriality





Examples











As an example…





Example:











Theorems











Considering…





Statment \ 1 \implies Statment \ 2











Pathologies





Applications





Taxonomy





    

  • Set -> 
    • 




  • Collection -> 
    • 




  • Relation -> 
    • 





















\Huge 
\textcolor{gray}{\mathcal{P}}





DeutschPotenzmenge
EnglishPower set
EspañolConjunto potencia
中文幂集
日本ベキ集合
РусскийБулеан / Множество всех подмножеств
العربيةمجموعة القوى
한국어멱집합









Evolution







X \ / \ Set







Definition





The power set of a set is the set of all subsets.





\textcolor{gray}{\mathcal{P}}(X) := \set{ S \subseteq X} \qquad \empty, X \in \textcolor{gray}{\mathcal{P}}(X)





Origin





Origin








Calligraphy





Dialects





Set of all subsets














Diagram





Gallery





Notes





Properties





Cardinality

Since the power set is itself a set we can use the cardinality.







Details








Details








Functoriality





Examples











For a set with only two elements the power set can be written explicitly as well:





A = \set{ a , b } \qquad
\textcolor{gray}{\mathcal{P}}(A) =
 \set{ \empty , \{ a\} ,\{ b\} , A}











Theorems











For a finite set the power sets cardinality can be explicitly calulated.





\# X = n \implies \# \textcolor{gray}{\mathcal{P}}(X) = 2^n





Proof














Pathologies





Applications





Taxonomy





    

  • Set -> 
    • 




  • Collection -> 
    • 




  • Relation -> 
    • 





















\Huge 
\textcolor{violet}{Manifold}





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English
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Evolution







X \ / \ Set





\textcolor{violet}{\mathcal{T}} \ / \ Topology





\textcolor{violet}{X} \ / \  Topological \ space







Definition





Definition…










Origin





Origin








Calligraphy





Dialects





Details






Details
















Diagram





Gallery





Notes





Properties





Details








Details








Details








Details








Functoriality





Examples











As an example…





Example:











Theorems











Considering…





Statment \ 1 \implies Statment \ 2











Pathologies





Applications





Taxonomy





    

  • Set -> 
    • 




  • Collection -> 
    • 




  • Relation -> 
    • 

















Relatives





Sources