Lemma: Comparing the Elements of Strictly Ordered Sets

In a strictly ordered set $(V,\prec)$, for all elements \(a,b\in V\) exactly one of the following cases holds:

• Either $a \prec b$ ($a$ is smaller than $b$),
• or $a \succ b$ ($a$ is greater than $b$).
• else $a=b$ ($a$ equals $b$).

Because of these three possibilities, the strict order $”\prec”$ is sometimes also called a trichotomous order.

| | | | | created: 2019-02-03 08:53:38 | modified: 2019-02-03 09:34:29 | by: bookofproofs | references: [979], [8055]

1.Proof: (related to "Comparing the Elements of Strictly Ordered Sets")