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Topic: Existence of some funcitons on fixed points (Read 428 times) |
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beibeibee
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Existence of some funcitons on fixed points
« on: Oct 6th, 2004, 11:11am » |
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For any natural number n greater that 2, are there functions, say f_1, f_2, …, f_n, such that the class including all the sets {n in N: f_i (f_j(n))=n } for i, j > 0 but i not = j, is exactly a partition of the set of natural numbers?
When n = 2, the answer is Yes. This is easy.  Can u see it?
Then consider generally! It seems very hard. 
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Aryabhatta
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Re: Existence of some funcitons on fixed points
« Reply #1 on: Oct 6th, 2004, 2:26pm » |
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The following seems to work (highlight to view)
Given n > 1.
Let [bbn] be the set of natural numbers.
Split [bbn] into a disjoint union of m = n(n-1) sets C1,...,Cm each set being infinite.
Let S = {(i,j) | i,j [in] {1,2,...,n} and i [ne] j}.
We have a bijection B:S [onetoone] {1,2,...,m}.
Also, for ordered pairs (i,j) and (j,i), we have bijections hij:Ci[onetoone]Cj and hji:Cj[onetoone]Ci such that
hij and hji are inverses of each other.
Define fi as follows.
for each j [ne] i, on CB(i,j), fi coincides with hij
On any other set Ck, fi is the identity.
The partitions which we get are C1,...,Cm
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beibeibee
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Re: Existence of some funcitons on fixed points
« Reply #2 on: Oct 7th, 2004, 6:38am » |
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Re:The following seems to work (highlight to view)
Aha, you are right precisely! It is actually an excise in a class of first order logic.
Thanks for your reply.
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