Axiom:Strict Ordering Axioms/Formulation 1

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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:

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

These criteria are called the strict ordering axioms.


Also see