Category:Axioms/Preordering Axioms
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This category contains axioms related to Preordering Axioms.
Let $\RR \subseteq S \times S$ be a relation on a set $S$.
Definition 1
$\RR$ is a preordering on $S$ if and only if $\RR$ satifies the preordering axioms:
\((1)\) | $:$ | $\RR$ is reflexive | \(\ds \forall a \in S:\) | \(\ds a \mathrel \RR a \) | |||||
\((2)\) | $:$ | $\RR$ is transitive | \(\ds \forall a, b, c \in S:\) | \(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \) |
Definition 2
$\RR$ is a preordering on $S$ if and only if $\RR$ satifies the preordering axioms:
\((1)\) | $:$ | $\RR$ is transitive | \(\ds \RR \circ \RR = \RR \) | ||||||
\((2)\) | $:$ | $\RR$ is reflexive | \(\ds \Delta_S \subseteq \RR \) |
where:
- $\circ$ denotes relation composition
- $\Delta_S$ denotes the diagonal relation on $S$.
Pages in category "Axioms/Preordering Axioms"
The following 3 pages are in this category, out of 3 total.