382: Preimage Under Surjection Is Saturated w.r.t. Surjection

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A description/proof of that preimage under surjection is saturated w.r.t. surjection

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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 saturated subset with respect to map.
• The reader admits the proposition that for any map, the composition of the map after any preimage is identical if and only if the argument set is a subset of the map image.

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Target Context

• The reader will have a description and a proof of the proposition that for any surjection, the preimage of any codomain subset is saturated with respect to the surjection.

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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 sets, $S_1,S_2$, any surjection, $f:S_1\to S_2$, and any subset, $S_3\subseteq S_2$, the preimage, $f^{-1}(S_3)$, is saturated with respect to $f$, which means $f^{-1}\circ f(f^{-1}(S_3))=f^{-1}(S_3)$.

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2: Proof

$f(f^{-1}(S_3))=S_3$, because $f$ is a surjection, by the proposition that for any map, the composition of the map after any preimage is identical if and only if the argument set is a subset of the map image. So, $f^{-1}\circ f(f^{-1}(S_3))=f^{-1}(S_3)$.

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References

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