267: Minimal Element of Set w.r.t. Inverse of Ordering Is Maximal Element of Set w.r.t. Original Ordering

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A description/proof of that minimal element of set w.r.t. inverse of ordering is maximal element of set w.r.t. original 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 ordering.
• The reader knows a definition of inverse of ordering.
• The reader knows a definition of minimal of set with respect to ordering.
• The reader knows a definition of maximal of set with respect to ordering.

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Target Context

• The reader will have a description and a proof of the proposition that any minimal element of any set with respect to the inverse of any ordering is a maximal element of the set with respect to the original 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 ordering, $R\subseteq S\times S$, any minimal element of $S$, $m\in S$, with respect to $R^{-1}$ is a maximal element of $S$ with respect to $R$.

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2: Proof

There is no $s\in S$ such that $⟨s,m⟩\in R^{-1}$. Then, there is no $s\in S$ such that $⟨m,s⟩\in R$. So, $m$ is a maximal element of $S$ with respect to $R$.

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References

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