Comparability

Topics: Discrete Mathematics - Relation

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(definition)

Let $(A,⪯)$ be a partially ordered set.

We say that the elements of $A$ are comparable if $a⪯b$ or $b⪯a$, given $a,b\in A$.

Examples

Let $X=\{w,x,y,z\}$ and $(\mathcal{P},\subseteq )$ a poset (partially ordered set) ($\mathcal{P}$ is the Power Set).

Note that:

• $\{w,x\}$ and $\{w,x,z\}$ are comparable, since $\{w,x\}\subseteq \{w,x,z\}$
• $\{w,x\}$ and $\{y,z\}$ are not comparable, since $\{w,x\}⊈\{y,z\}$ and $\{y,z\}⊈\{w,x\}$