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 Spacy Colors II (Posted on 2014-10-26)

This is a follow up to a question JLo asked. It has been recast

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 No Rating

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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