Proving Nonempty Fibers of a Map Partition the Domain

[SNOOTCHIEBOOCHEE]

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.

 

[SNOOTCHIEBOOCHEE]

any thoughts?

 

[Dick]

Doesn't phi^(-1)(t) for t in T constitute a set of non-overlapping sets that cover S? Look up partition.

 

[SNOOTCHIEBOOCHEE]

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

 

[morphism]

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?