343: Injective Map Image of Intersection of Sets Is Intersection of Map Images of Sets

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A description/proof of that injective map image of intersection of sets is intersection of map images of sets

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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 for any injective map, the map image of the intersection of any sets is the intersection of the map images of the sets.

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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 $S_2$, any injective map, $f:S_1\to S_2$, and any possibly uncountable number of subsets of $S_1$, $S_{1_{\alpha }}\subseteq S_1$, the map image of the intersection of the subsets, $f({\cap }_{\alpha }{S}_{1_{\alpha }})$, is the intersection of the map images of the subsets, ${\cap }_{\alpha }f(S_{1_{\alpha }})$, which is, $f({\cap }_{\alpha }{S}_{1_{\alpha }})={\cap }_{\alpha }f(S_{1_{\alpha }})$.

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2: Proof

For any element, $p\in f({\cap }_{\alpha }{S}_{1_{\alpha }})$, there is an element, $p^{\prime }\in {\cap }_{\alpha }{S}_{1_{\alpha }}$ such that $p=f(p^{\prime })$, which means that for each $\alpha$, $p^{\prime }\in S_{1_{\alpha }}$. So, $p\in f(S_{1_{\alpha }})$ for each $\alpha$, so, $p\in {\cap }_{\alpha }f(S_{1_{\alpha }})$.

For any element, $p\in {\cap }_{\alpha }f(S_{1_{\alpha }})$, $p\in f(S_{1_{\alpha }})$ for each $\alpha$, so, there is a $p_{\alpha }^{\prime }\in S_{1_{\alpha }}$ for each $\alpha$ such that $p=f(p_{\alpha }^{\prime })$, but as $f$ is injective, all the $p_{\alpha }^{\prime }$s are the same $p^{\prime }\in S_1$, so, $p^{\prime }\in {\cap }_{\alpha }{S}_{1_{\alpha }}$, so, $p\in f({\cap }_{\alpha }{S}_{1_{\alpha }})$.

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3: Note

The map has to be injective for this proposition. Otherwise, see another proposition.

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References

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