266: Inverse of Partial Ordering Is Partial Ordering

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A description/proof of that inverse of partial ordering is partial ordering

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of set.
• The reader knows a definition of partial ordering.
• The reader knows a definition of inverse of ordering.

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Target Context

• The reader will have a description and a proof of the proposition that the inverse of any partial ordering is a partial ordering.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

For any set, $S$, and any partial ordering, $R\subseteq S\times S$, the inverse of $R$, $R^{-1}$, is a partial ordering.

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2: Proof

$⟨s_1,s_1⟩\notin R^{-1}$, because $⟨s_1,s_1⟩\notin R$.

If $⟨s_1,s_2⟩\in R^{-1}$ and $⟨s_2,s_3⟩\in R^{-1}$, $⟨s_1,s_3⟩\in R^{-1}$, because if $⟨s_2,s_1⟩\in R$ and $⟨s_3,s_2⟩\in R$, $⟨s_3,s_1⟩\in R$.

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References

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