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Spacy Colors II (Posted on 2014-10-26) Difficulty: 5 of 5

  
This is a follow up to a question JLo asked. It has been recast
with subsets instead of colors.

Prove or disprove the following is true for all integers n ≥ 1:

    If Rn is partitioned into n subsets S1, S2, ... , Sn; then
       ∃ i∈In ( |Si| = R+ )

Definitions and Nomenclature

In = { 1,2, ... , n }.

R is the set of real numbers ( complete ordered field ).

R+ = { x∈R | x ≥ 0 }.

Rn = { (x1, x2, ... , xn) | x1, x2, ... , xnR }.

Properties of S1, S2, ... , Sn:

   1) ∀ i∈In ( Si ≠ Φ ),

   2) S1 ∪ S2 ∪ ... ∪ Sn = Rn,

   3) ∀ i,j∈In ( i ≠ j ⇒ Si ∩ Sj = Φ ),

   where Φ denotes the empty set.

If P,Q∈Rn with P = (p1,p2, ... , pn) and Q = (q1,q2, ... , qn), then
   |PQ| = √[Sigma[(pi - qi)2 ; i=1,n]]

If S ⊂ Rn, then
   |S| = { |PQ|∈R+ | P,Q∈S }.
  

See The Solution Submitted by Bractals    
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Comments: ( Back to comment list | You must be logged in to post comments.)
Corrections to The Solution Comment 1 of 1

The following changes should be made to
the Analysis section:
 
   Rule 1: ... K_k "cap" S_i = "empty set"

   Rule 3: ... K_K should be K_k and
               S_i "cup" S_i should be
               S_i "cup" S_j

       After Rule 3: add the line

     "Rules 0-3 applied to the scenario" 

  Posted by Bractals on 2015-06-07 10:25:31
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