description/proof of that for map between sets, image of preimage of codomain subset is intersection of codomain subset and map range
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of map.
- The reader admits the proposition that for any map between any sets, the image of the intersection of the intersection of the preimages of any codomain subsets and any domain subset is the intersection of the intersection of the codomain subsets and the image of the domain subset.
Target Context
- The reader will have a description and a proof of the proposition that for any map between any sets, the image of the preimage of any codomain subset is the intersection of the codomain subset and the map range.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(S_1\): \(\in \{\text{ the sets }\}\)
\(S_2\): \(\in \{\text{ the sets }\}\)
\(f\): \(: S_1 \to S_2\)
\(S^`_2\): \(\subset S_2\)
//
Statements:
\(f (f^{-1} (S^`_2)) = S^`_2 \cap f (S_1)\)
//
2: Proof
Whole Strategy: Step 1: apply the proposition that for any map between any sets, the image of the intersection of the intersection of the preimages of any codomain subsets and any domain subset is the intersection of the intersection of the codomain subsets and the image of the domain subset.
Step 1:
For the proposition that for any map between any sets, the image of the intersection of the intersection of the preimages of any codomain subsets and any domain subset is the intersection of the intersection of the codomain subsets and the image of the domain subset, \(\cap_{j \in J} f^{-1} (S_{2, j})\) can be taken to be \(f^{-1} (S^`_2)\) and the domain subset can be taken to be \(S_1\), then \(f (f^{-1} (S^`_2) \cap S_1) = S^`_2 \cap f (S_1)\), but \(f (f^{-1} (S^`_2) \cap S_1) = f (f^{-1} (S^`_2))\).
