Order Relation

Topics: Discrete Mathematics - Relation

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(definition)

A relation $R$ on a set $A$ is said to be an order relation if $R$ is reflexive, antisymmetric and transitive.

If $(a,b)\in R$, then we will denote the relation by writing $a⪯b$.

Examples

• $(\mathbb{N},\le )$ is a well-ordered set.
• $(\mathbb{R},\le )$ is an ordered set.
• $(A,\subseteq )$ is an ordered set ($A$ is a set).
• $(\mathcal{P}(A),\subseteq )$ is an ordered set ($\mathcal{P}$ is the Power Set of $A$)

When we define an order relation on a given set $A$, we say that $A$ is a partially ordered set. When every pair in $A$ is comparable, then $⪯$ is a total order relation and $A$ is a totally ordered set.