Aufgabe:
Text erkannt:
b) Which of the following relations is well-founded? Give a justification of why or why not.
i) A relation that is irreflexive, asymmetric, and finite.
ii) The relation \( R=\{(n-1, n) \mid n \in \mathbb{N}\} \).
iii) The relation \( R R \), where \( \mathrm{R} \) is defined as above.
iv) The partial order on \( \mathbb{N} \times \mathbb{N} \) where \( (a, b) \preceq(c, d) \) iff \( \max (a, b) \leq \max (c, d) \).
v) The divisibility order on \( \mathbb{N} \), i.e. \( R=\{(a, b) \mid a, b \in \mathbb{Z} \) and \( \exists c(c \in \mathbb{Z} \wedge a \cdot c=b)\} \).
vi) The relation \( \left\{(u a b v, u b a v) \mid u, v \in\{a, b, c\}^{*}\right\} \); that is, two words \( w \) and \( w^{\prime} \) are related if we can obtain \( w^{\prime} \) from \( w \) by replacing one occurrence of the subword \( a b \) with \( b a \).
Problem/Ansatz:
kann mir hier wer weiterhelfen ob diese Relationen well founded sind und warum?
