Axiom:Strict Ordering Axioms/Formulation 2

Definition

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

$\RR$ is a strict ordering on $S$ if and only if $\RR$ satisfies the axioms:

$(1)$	$:$		Antireflexivity	$\ds \forall a \in S:$	$\ds \neg \paren {a \mathrel \RR a} $
$(2)$	$:$		Transitivity	$\ds \forall a, b, c \in S:$	$\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \implies a \mathrel \RR c $

These criteria are called the strict ordering axioms.

\((1)\)	$:$		Antireflexivity	\(\ds \forall a \in S:\)	\(\ds \neg \paren {a \mathrel \RR a} \)
\((2)\)	$:$		Transitivity	\(\ds \forall a, b, c \in S:\)	\(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \implies a \mathrel \RR c \)