282: Map Preimage of Whole Range Is Whole Domain

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A description/proof of that map preimage of whole range is whole domain

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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 map.

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Target Context

• The reader will have a description and a proof of the proposition that the preimage of the whole range of any map is the whole domain.

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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, $S_1$ and $S_2$, and any map, $f:S_1\to S_2$, the preimage of the whole range is the whole domain, which is $f^{-1}(S_2)=S_1$.

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2: Proof

Suppose $p\in f^{-1}(S_2)$. $p\in S_1$. Suppose $p\in S_1$. $f(p)\in S_2$, so $p\in f^{-1}(S_2)$.

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3: Note

The point is that the map does not have to be surjective.

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References

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