Insimul potentia

In mathematica, le insimul potentia^[1] es le insimul de omne subinsimules de un certe insimul. Le insimul de potentia de ${\displaystyle A}$ es habitualmente notate como ${\displaystyle {\mathfrak {P}}(A)}$, ${\displaystyle {\mathcal {P}}(A)}$, ${\displaystyle \wp (A)}$, o ${\displaystyle 2^{A}}$. In vice de insimul potentia, on alsi pote dicer insimul del subinsimules. Le numero cardinal del insimul potentia del insimul ${\displaystyle A}$ es ${\displaystyle 2}$ elevate al numero cardinal de ${\displaystyle A}$: ${\displaystyle |{\mathcal {P}}(A)|=2^{|A|}}$ o, in altere maniera de scriber, ${\displaystyle {\big |}2^{A}{\big |}=2^{|A|}}$.

Exemplos

• ${\displaystyle {\mathcal {P}}(\{1\})={\big \{}\emptyset$ ;\{1\}{\big \}}} con ${\displaystyle |{\mathcal {P}}(\{1\})|=2^{1}=2}$.
• ${\displaystyle {\mathcal {P}}(\{1;2\})={\big \{}\emptyset$ ;\{1\};\{2\};\{1;2\}{\big \}}} con ${\displaystyle |{\mathcal {P}}(\{1;2\})|=2^{2}=4}$.
• ${\displaystyle {\mathcal {P}}(\{1;2;3\})={\big \{}\emptyset$ ;\{1\};\{2\};\{3\};\{1;2\};\{1;3\};\{2;3\};\{1;2;3\}{\big \}}} con ${\displaystyle |{\mathcal {P}}(\{1;2;3\})|=2^{3}=8}$.
• ${\displaystyle {\mathcal {P}}(\{1;2;3;4\})={\big \{}\emptyset$ ;\{1\};\{2\};\{3\};\{4\};\{1;2\};\{1;3\};\{1;4\},\{2;3\};\{2;4\};\{3;4\};\{1;2;3\};\{1;2;4\};\{1;3;4\};\{2;3;4\};\{1;2;3;4\}{\big \}}} con ${\displaystyle |{\mathcal {P}}(\{1;2;3;4\})|=2^{4}=16}$.

Vide etiam

• Producto cartesian

Referentias

1. Derivation (in ordine alphabetic): (ca) Conjunt de les parts || (de) Potenzmenge || (en) Power set || (es) Conjunto potencia || (fr) Ensemble des parties d'un ensemble || (it) Insieme delle parti || (pt) Conjunto de partes || (ro) || (ru) Множество всех подмножеств