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

  • Let $(V,\prec)$ be a strictly ordered set.
  • By definition, $”\prec”$ is a total order. Therefore it is ensured that at least one of the possibilities $a \prec b,$ $a \succ b,$ or $a=b$ is fullfilled. It remains to be shown that exactly one is possible:
    • The possibilities $a \prec b$ and $a=b$ cannot occur simultaneously, because $”\prec”$ is irreflexive, by definition.
    • The same argument holds for the possibilities $a \succ b$ and $a=b$ occuring simultaneously.
    • Neither can the possibilities $a \succ b$ and $a \prec b$ occur simultaneously, because $”\prec”$ is asymmetric, by definition.
q.e.d

| | | | created: 2019-02-03 09:34:07 | modified: 2019-02-03 09:34:48 | by: bookofproofs | references: [979], [8055]


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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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