2023-02-05

186: Maps Composition Preimage Is Composition of Map Preimages in Reverse Order

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A description/proof of that maps composition preimage is composition of map preimages in reverse order

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Starting Context

Target Context


  • The reader will have a description and a proof of the proposition that for any maps composition, the preimage under the composition is the composition of the map preimages in the reverse order.

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: Description


For any sets, S1,S2,...,Sn where 3≤n, any maps, f1:S1→S2,f2:S2→S3,...,fn−1:Sn−1→Sn, and any subset, S⊆Sn, (fn−1∘...∘f2∘f1)−1(S)=f1−1(f2−1(...(fn−1−1(S))...)).

2: Proof


Suppose that n=3. For any element, p∈(f2∘f1)−1(S), f2∘f1(p)∈S, f1(p)∈f2−1(S), p∈f1−1(f2−1(S)). For any p∈f1−1(f2−1(S)), f1(p)∈f2−1(S), f2∘f1(p)∈S, p∈(f2∘f1)−1(S).

Suppose that (fn−1∘...∘f2∘f1)−1(S)=f1−1(f2−1(...(fn−1−1(S))...)) for an n. (fn∘fn−1∘...∘f2∘f1)−1(S)=(fn∘(fn−1∘...∘f2∘f1))−1(S)=(fn−1∘...∘f2∘f1)−1(fn−1(S))=f1−1(f2−1(...(fn−1−1(fn−1(S)))...)).

So, by mathematical induction, the proposition holds for any 3≤n.

References
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