330: Composition of Map After Preimage Is Contained in Argument Set

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A description/proof of that composition of map after preimage is contained in argument set

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Topics

About: set
About: map

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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, the composition of the map after any preimage is contained in 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, $S_3\subseteq S_2$, $f\circ f^{-1}(S_3)\subseteq S_3$.

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2: Proof

For any $p\in f\circ f^{-1}(S_3)$, $p=f\circ f^{-1}(p^{\prime })$ for a point, $p^{\prime }\in S_3$, but $p=f\circ f^{-1}(p^{\prime })=p^{\prime }$, so, $p\in S_3$.

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3: Note

It is not necessarily $f\circ f^{-1}(S_3)=S_3$, as for which, there is another proposition.

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References

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