325: Intersection of Set of Transitive Relations Is Transitive

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A description/proof of that intersection of set of transitive relations is transitive

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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 transitive relation.

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Target Context

• The reader will have a description and a proof of the proposition that the intersection of any set of any possibly uncountable number of transitive relations is transitive.

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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 of any possibly uncountable number of transitive relations, $S=\{R_{\alpha }|\alpha \in A\}$ where $A$ is a possibly uncountable indexes set, the intersection, ${\cap }_{\alpha \in A}{R}_{\alpha }$, is transitive.

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2: Proof

For any $⟨s_1,s_2⟩,⟨s_2,s_3⟩\in {\cap }_{\alpha \in A}{R}_{\alpha }$, $⟨s_1,s_2⟩,⟨s_2,s_3⟩\in R_{\alpha }$ for each $\alpha$. As each $R_{\alpha }$ is transitive, $⟨s_1,s_3⟩\in R_{\alpha }$ for each $\alpha$, so, $⟨s_1,s_3⟩\in {\cap }_{\alpha \in A}{R}_{\alpha }$.

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References

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