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@ -41,7 +41,7 @@ und eignen sich daher gut dafür sich die Dinge vorzustellen.
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Auch deshalb beschränkt man sich häufig auf diese, wenn man einer Person Mathematik nahe bringen möchte.
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Auch spielt das Rechnen in unserem Alltag an der Supermarktkasse oder beim Zählen eine große Rolle.
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Mathematik zählt wie die [Informatik](/informatik/informatik.md) zu den *Struktur*- bzw. *Formalwissenschaften*.
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Mathematik zählt wie die [Informatik](/docs/informatik/) zu den *Struktur*- bzw. *Formalwissenschaften*.
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Sie nimmt damit eine Sonderrolle unter den Wissenschaften ein und wird manchmal auch als *__die__ exakte* Wissenschaft betrachtet.
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Denn anders, als bei anderen Wissenschaften beruhen die Erkenntnisse auf einem logischen Beweis.
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Ein wahrer logischer Beweis ist prinzipbedingt endgültig und allgemeingültig wahr.
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@ -287,17 +287,17 @@ Die tatsächliche semantische Äquivalenz lässt sich leicht durch aufstellen ei
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:::tip Wichtige semantische Äquivalenzen
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1. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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x \wedge y &\equiv y \wedge x\\
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x \vee y &\equiv y \vee x\\
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x \leftrightarrow y &\equiv y \leftrightarrow x
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\end{aligned}
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&&\text{Kommutativität}
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\end{align*}
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$$
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\end{align*}
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$$
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2. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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x \wedge (y \wedge z) &\equiv (x \wedge y) \wedge z\\
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x \vee (y \vee z) &\equiv (x \vee y) \vee z\\
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@ -305,37 +305,37 @@ $$
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&\equiv (x \leftrightarrow y) \leftrightarrow z
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\end{aligned}
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&&\text{Assoziativität}
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\end{align*}
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$$
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\end{align*}
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$$
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3. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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x \wedge (y \vee z) &\equiv (x \wedge y) \vee (x \wedge z)\\
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x \vee (y \wedge z) &\equiv (x \vee y) \wedge (x \vee z)
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\end{aligned}
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&&\text{Distributivität}
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\end{align*}
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$$
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\end{align*}
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$$
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4. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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x \wedge (x \vee y) &\equiv x\\
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x \vee (x \wedge y) &\equiv x
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\end{aligned}
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&&\text{Absorption}
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\end{align*}
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$$
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\end{align*}
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$$
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5. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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x \wedge x &\equiv x\\
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x \vee x &\equiv x
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\end{aligned}
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&&\text{Idempotenz}
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\end{align*}
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$$
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\end{align*}
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$$
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6. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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\neg(\neg x) &\equiv x\\
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\neg(x \rightarrow y) &\equiv x \wedge \neg y\\
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@ -344,19 +344,19 @@ $$
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\equiv x \leftrightarrow \neg y
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\end{aligned}
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&&\text{Verneinung}
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\end{align*}
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$$
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\end{align*}
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$$
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7. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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\neg(x \wedge y) &\equiv \neg x \vee \neg y\\
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\neg(x \vee y) &\equiv \neg x \wedge \neg y
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\end{aligned}
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&&\text{De Morgan'sche Regeln}
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\end{align*}
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$$
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\end{align*}
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$$
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8. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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x \leftrightarrow y &\equiv (x \rightarrow y) \wedge (y \rightarrow x)\\
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x \rightarrow y &\equiv \neg x \vee y\\
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@ -364,37 +364,37 @@ $$
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x \vee y &\equiv \neg(\neg x \wedge \neg y)
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\end{aligned}
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&&\text{Elimination}
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\end{align*}
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$$
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\end{align*}
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$$
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9. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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x \rightarrow y \equiv \neg y \rightarrow \neg x
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\end{aligned}
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&&\text{Kontraposition}
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\end{align*}
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$$
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\end{align*}
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$$
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10. $$
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\begin{align*}
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\begin{align*}
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\begin{aligned}
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\neg f &\equiv w\\
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\neg w &\equiv f
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\end{aligned}
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&&
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\end{align*}
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$$
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||||
\end{align*}
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$$
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11. $$
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\begin{align*}
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\begin{align*}
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||||
\begin{aligned}
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x \wedge \neg x &\equiv f\\
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x \vee \neg x &\equiv w\\
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x \leftrightarrow \neg x &\equiv f
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\end{aligned}
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&&
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||||
\end{align*}
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$$
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||||
\end{align*}
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$$
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12. $$
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\begin{align*}
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\begin{align*}
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||||
\begin{aligned}
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w \wedge x &\equiv x\\
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f \wedge x &\equiv f\\
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@ -402,10 +402,10 @@ $$
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f \vee x &\equiv x
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\end{aligned}
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&&
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||||
\end{align*}
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$$
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\end{align*}
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$$
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13. $$
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||||
\begin{align*}
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||||
\begin{align*}
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||||
\begin{aligned}
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||||
f \rightarrow x &\equiv w\\
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w \rightarrow x &\equiv x\\
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@ -413,17 +413,17 @@ $$
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||||
x \rightarrow w &\equiv w
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\end{aligned}
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&&
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||||
\end{align*}
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$$
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||||
\end{align*}
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$$
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14. $$
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||||
\begin{align*}
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||||
\begin{align*}
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||||
\begin{aligned}
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f \leftrightarrow x &\equiv \neg x\\
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w \leftrightarrow x &\equiv x
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||||
\end{aligned}
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&&
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||||
\end{align*}
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$$
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||||
\end{align*}
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$$
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:::
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@ -32,8 +32,8 @@ Der Begriff der *Unendlichkeit* wurde dabei von ihm in besonderem Maße geprägt
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Allerdings führte das Cantor'sche Modell der Mengenlehre, auch *naive Mengenlehre* genannt, auch zu *Antinomien* (Widersprüchen).
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[Bertrand Russell](https://de.wikipedia.org/wiki/Bertrand_Russell) entdeckte diese Widersprüche und machte sie publik.
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Die heutige Mathematik beruht auf einer axiomatisierten Mengenlehre und wird auch *Zermelo-Fraenkel-Mengenlehre* genannt.
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Doch ist die naive Mengenlehre ein Teil der ZF-Mengenlehre und daher nach wie vor ein einfacher und guter Zugang.
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Die heutige Mathematik beruht auf einer axiomatisierten Mengenlehre und wird auch *Zermelo-Fraenkel-Mengenlehre mit Auswahlaxiom* genannt.
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Doch ist die naive Mengenlehre ein Teil der ZFC-Mengenlehre (**Z**ermelo, **F**raenkel, Axiom of **C**hoice) und daher nach wie vor ein einfacher und guter Zugang.
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## Naive Mengenlehre nach Cantor
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:::note Menge
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@ -1,19 +1,19 @@
|
||||
// @ts-check
|
||||
// Note: type annotations allow type checking and IDEs autocompletion
|
||||
|
||||
const lightCodeTheme = require('prism-react-renderer/themes/github');
|
||||
const darkCodeTheme = require('prism-react-renderer/themes/dracula');
|
||||
import remarkMath from 'remark-math';
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||||
import rehypeKatex from 'rehype-katex';
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||||
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||||
const math = require('remark-math');
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||||
const katex = require('rehype-katex');
|
||||
const lightCodeTheme = require('prism-react-renderer').themes.github;
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||||
const darkCodeTheme = require('prism-react-renderer').themes.dracula;
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||||
|
||||
/** @type {import('@docusaurus/types').Config} */
|
||||
const config = {
|
||||
export default {
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||||
title: 'Mathematik, Informatik und Programmierung',
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||||
tagline: 'Mathematik und Informatik sind cool',
|
||||
url: 'https://www.niklas-birk.de',
|
||||
baseUrl: '/',
|
||||
onBrokenLinks: 'throw',
|
||||
onBrokenLinks: 'warn',
|
||||
onBrokenMarkdownLinks: 'warn',
|
||||
favicon: 'img/niklas-birk-icon.ico',
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||||
|
||||
@ -38,8 +38,8 @@ const config = {
|
||||
// docs: false,
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||||
docs: {
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||||
sidebarPath: require.resolve('./sidebars.js'),
|
||||
remarkPlugins: [math],
|
||||
rehypePlugins: [katex],
|
||||
remarkPlugins: [remarkMath],
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||||
rehypePlugins: [rehypeKatex],
|
||||
// // Please change this to your repo.
|
||||
// // Remove this to remove the "edit this page" links.
|
||||
// // editUrl:
|
||||
@ -118,21 +118,21 @@ const config = {
|
||||
items: [
|
||||
{
|
||||
label: 'Twitter',
|
||||
href: 'https://twitter.com/SirNik_OdTW',
|
||||
href: 'https://twitter.com/theoremofnik',
|
||||
},
|
||||
{
|
||||
label: 'Mastodon',
|
||||
href: 'https://mathstodon.xyz/web/@SirNik',
|
||||
},
|
||||
{
|
||||
label: 'Twitch',
|
||||
href: 'https://www.twitch.tv/theoremofnik',
|
||||
},
|
||||
],
|
||||
},
|
||||
{
|
||||
title: 'More',
|
||||
title: 'Mehr von mir',
|
||||
items: [
|
||||
// {
|
||||
// label: 'Blog',
|
||||
// to: '/blog',
|
||||
// },
|
||||
{
|
||||
label: 'Website',
|
||||
href: 'https://www.niklas-birk.de',
|
||||
@ -143,6 +143,15 @@ const config = {
|
||||
},
|
||||
],
|
||||
},
|
||||
{
|
||||
title: 'Legal',
|
||||
items: [
|
||||
{
|
||||
label: 'Impressum',
|
||||
href: 'https://www.niklas-birk.de/impressum.html',
|
||||
},
|
||||
],
|
||||
},
|
||||
],
|
||||
copyright: `Copyright © ${new Date().getFullYear()} Niklas Birk. Built with Docusaurus.`,
|
||||
},
|
||||
@ -162,5 +171,3 @@ const config = {
|
||||
},
|
||||
],
|
||||
};
|
||||
|
||||
module.exports = config;
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||||
|
||||
19871
package-lock.json
generated
19871
package-lock.json
generated
File diff suppressed because it is too large
Load Diff
27
package.json
27
package.json
@ -1,6 +1,6 @@
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||||
{
|
||||
"name": "niklas-birk-blog",
|
||||
"version": "0.0.0",
|
||||
"version": "0.1.1",
|
||||
"private": true,
|
||||
"scripts": {
|
||||
"docusaurus": "docusaurus",
|
||||
@ -14,19 +14,20 @@
|
||||
"write-heading-ids": "docusaurus write-heading-ids"
|
||||
},
|
||||
"dependencies": {
|
||||
"@docusaurus/core": "^2.2.0",
|
||||
"@docusaurus/preset-classic": "^2.2.0",
|
||||
"@mdx-js/react": "^1.6.22",
|
||||
"clsx": "^1.2.1",
|
||||
"hast-util-is-element": "^1.1.0",
|
||||
"prism-react-renderer": "^1.3.5",
|
||||
"react": "^17.0.2",
|
||||
"react-dom": "^17.0.2",
|
||||
"rehype-katex": "^5.0.0",
|
||||
"remark-math": "^3.0.1"
|
||||
"@docusaurus/core": "^3.4.0",
|
||||
"@docusaurus/preset-classic": "^3.4.0",
|
||||
"@mdx-js/react": "^3.0.1",
|
||||
"clsx": "^2.1.1",
|
||||
"hast-util-is-element": "^3.0.0",
|
||||
"prism-react-renderer": "^2.3.1",
|
||||
"react": "^18.3.1",
|
||||
"react-dom": "^18.3.1",
|
||||
"rehype-katex": "^7.0.0",
|
||||
"remark-math": "^6.0.0"
|
||||
},
|
||||
"devDependencies": {
|
||||
"@docusaurus/module-type-aliases": "^2.1.0"
|
||||
"@docusaurus/module-type-aliases": "^3.4.0",
|
||||
"@docusaurus/types": "3.4.0"
|
||||
},
|
||||
"browserslist": {
|
||||
"production": [
|
||||
@ -41,6 +42,6 @@
|
||||
]
|
||||
},
|
||||
"engines": {
|
||||
"node": ">=16.14"
|
||||
"node": ">=18.0"
|
||||
}
|
||||
}
|
||||
|
||||
@ -24,9 +24,9 @@ const FeatureList = [
|
||||
Verarbeitung befasst.</p>
|
||||
<p>
|
||||
<q><cite>In der Informatik geht es genau so wenig um Computer, wie in der Astronomie um
|
||||
Teleskope.</cite></q>
|
||||
</p>
|
||||
Teleskope.</cite></q><br />
|
||||
- Edsger Wybe Dijkstra
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||||
</p>
|
||||
</>
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||||
),
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||||
},
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||||
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||||
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Before Width: | Height: | Size: 6.7 KiB After Width: | Height: | Size: 16 KiB |
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Reference in New Issue
Block a user