Proving Nonempty Fibers of a Map Partition the Domain

In summary, the conversation discusses proving that the nonempty fibers of a map form a partition of the domain. The participants mention the map phi, its pre-image phi^(-1)(t), and the concept of partitioning a set into non-overlapping subsets. They question how to prove that phi^(-1)(t) for t in T constitutes a set of non-overlapping sets that cover S, and whether any two fibers corresponding to different elements of T can intersect nontrivially. The conversation concludes with a suggestion to prove that every element in S lies in a fiber.
  • #1
SNOOTCHIEBOOCHEE
145
0

Homework Statement



Prove that the nonempty fibers of a map form a partition of the domain.

The Attempt at a Solution



Ok so we have some map phi: S -->T

And we want to show that its pre-image phi-1(t) = {s in S | phi(s)=t} forms a partition of the domain.

Im really confused here. I assume that it is talking about that domian of phi which is S (i think) but i have no clue how this preimage forms partitions.
 
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  • #2
any thoughts?
 
  • #3
Doesn't phi^(-1)(t) for t in T constitute a set of non-overlapping sets that cover S? Look up partition.
 
  • #4
Well a partition P of S is a subdivision of S into nonoverlapping subsets

How do you know phi^(-1)(t) for t in T constitute a set of non-overlapping sets that cover S
 
  • #5
Prove it. Can any two fibers that correspond to different elements of T intersect nontrivially? Is there anything in S that doesn't lie in a fiber?
 

FAQ: Proving Nonempty Fibers of a Map Partition the Domain

What does it mean for a map to have nonempty fibers?

A map with nonempty fibers means that for every element in the domain, there is at least one element in the range that maps to it. In other words, the preimage of every element in the range is nonempty.

Why is it important to prove that nonempty fibers partition the domain of a map?

Proving that nonempty fibers partition the domain of a map is important because it ensures that every element in the domain is accounted for and mapped to an element in the range. This is necessary for the map to be well-defined and have a unique output for every input.

What is the significance of partitioning the domain of a map?

Partitioning the domain of a map means that every element in the domain is grouped with other elements that map to the same element in the range. This helps to better understand the structure and relationships within the map.

How can one prove that nonempty fibers partition the domain of a map?

To prove that nonempty fibers partition the domain of a map, one must show that every element in the domain is contained in a fiber and that no two fibers overlap. This can be done by using the definition of nonempty fibers and properties of set theory.

What are the key takeaways when proving nonempty fibers partition the domain of a map?

The key takeaways when proving nonempty fibers partition the domain of a map are that every element in the domain must have at least one element in its fiber, and no two fibers can overlap. Additionally, this proof helps to establish the well-definedness of the map and its relationships between the domain and range.

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