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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")

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Bibliography (further reading)

[8055] Hoffmann, D.: “Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise”, Hoffmann, D., 2018

[979] Reinhardt F., Soeder H.: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10

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