190: Subset Is Contained in Map Preimage of Image of Subset

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A description/proof of that subset is contained in map preimage of image of subset

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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
• 3: Note

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Starting Context

• The reader knows a definition of set.
• The reader knows a definition of map.

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Target Context

• The reader will have a description and a proof of the proposition that for any map between sets, any subset is contained in the preimage of the image of the subset.

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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 map, $f:S_1\to S_2$, and any subset, $S_3\subseteq S_1$, such that $S_4=f(S_3)$, $S_3\subseteq f^{-1}(S_4)$.

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2: Proof

For any $p\in S_3$, $f(p)\in S_4$, so, $p\in f^{-1}(S_4)$.

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3: Note

The subset does not necessarily equal the preimage, as $f$ may not be injective and a point not on $S_3$ may map into $S_4$.

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References

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