394: Composition of Preimage After Map of Subset Contains Argument Set

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A description/proof of that composition of preimage after map of subset contains argument set

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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 map.

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Target Context

• The reader will have a description and a proof of the proposition that for any map, the composition of the preimage after the map of any subset contains the argument set.

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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, $S_3\subseteq S_1$, $S_3\subseteq f^{-1}\circ f(S_3)$.

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2: Proof

For any $p\in S_3$, $f(p)\in f(S_3)$, which means that $p\in f^{-1}\circ f(S_3)$ by the definition of preimage.

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References

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