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relation d'ordre.md Normal file
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up:: [[relation]]
title:: "[[relation réflexive|réflexive]] : $x \mathcal{R} x$", "[[relation antisymétrique|antisymétrie]] : $x\mathcal{R}y \wedge y\mathcal{R}x \implies x=y$", "[[relation transitive|transitive]] : $x\mathcal{R}y \wedge y\mathcal{R}z \implies x\mathcal{R}z$"
#maths/algèbre
---
> [!definition] Relation d'ordre
> Soit $E$ un ensemble
> Soit $\mathcal{R}$ une relation sur $E$
> $\mathcal{R}$ est une **relation d'ordre** ssi :
> - $\forall x \in E, \quad x\mathcal{R}x$ ([[relation réflexive|réflexivité]])
> - $\forall (x, y) \in E^{2}, \quad x\mathcal{R}y \wedge y\mathcal{R}x \implies x = y$ ([[relation antisymétrique|antisymétrie]])
> - $\forall (x, y, z) \in E^{3}, \quad x\mathcal{R}y \wedge y\mathcal{R}z \implies x\mathcal{R}z$ ([[relation transitive|transitivité]])
^definition