Axiom:Preordering Axioms/Formulation 1

Definition

Let $\RR \subseteq S \times S$ be a relation on a set $S$.

$\RR$ is a preordering on $S$ if and only if $\RR$ satifies the 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 $

These criteria are called the preordering axioms.

\((1)\)	$:$		$\RR$ is reflexive	\(\ds \forall a \in S:\)	\(\ds a \mathrel \RR a \)
\((2)\)	$:$		$\RR$ is transitive	\(\ds \forall a, b, c \in S:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \)