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⊆ L := Σ^* \backslash L$ + +**Beweis:** +- wähle beliebiges, festes $u ∈ N$ +- für jedes $v ∈ N$ gilt: $u$ und $v$ sind rechtsäquivalent +- insbesondere: $u = u * ε ∈ L ↔ v * ε = v ∈ L$ +- also entweder sind _alle_ solche Wörter $v$ aus $L$ (wenn $u ∈ L$) +- oder es sind alle solche Wörter $v$ nicht aus $L$ (wenn $u \not ∈ L$) + +#### Mitgegangen-Mitgefangen +> Sei $L$ eine Sprache und $u,v ∈ Σ^*$ Wörter aus der gleichen Nerode-Klasse. +> Dann gilt für alle $s ∈ Σ^*$: $N(u*s)=N(v*s)$ + +**Beweis:** +- zu zeigen: $(u*s, v*s) ∈ R_{L(A)}$ + - d.h. $u*s$ und $v*s$ sind [rechtsäquivalent](#rechts-quivalenz) +- sei dazu $w ∈ Σ^*$ beliebig, dann gilt: + - ![image_782.png](image_782.png) + + +#### Gemeinsame Reise +> Sei $A = (Σ, Q, q_s, Q_A, δ)$ ein DEA. +> Betrachte zwei Worte $u,v ∈ Σ^*$ die $A$ in den gleichen Zustand überführen. +> Dann sind $u$ und $v$ rechtsäquivalent bzgl. $L(A)$ + +![image_783.png](image_783.png) + + + ## Der Satz von Myhill-Nerode ## Reaping the Fruits! ## Minimierungsalgorithmus