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Subdivisions/Refinement Proofby TheyCallMeMini
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#1
Feb1113, 05:18 PM

P: 5

I understand this isn't a homework area but there is always so much more traffic in this forum rather than the homework. All I'm looking for is a clarification that my ideas to prove both parts are in fact accurate.
1. The problem statement, all variables and given/known data If each of D1 and D2 is a subdivision of [a,b], then... 1. D1 u D2 is a subdivision of [a,b], and 2. D1 u D2 is a refinement of D1. 2. Relevant equations **Definition 1: The statement that D is a subdivision of the interval [a,b] means... 1. D is a finite subset of [a,b], and 2. each of a and b belongs to D. **Definition 2: The statement that K is a refinement of the subdivision D means... 1. K is a subdivision of [a,b], and 2. D is a subset of K. 3. The attempt at a solution My problem is that I've taken a lot of logic courses in the past so when I see the union of two variables I only need to prove that one is actually true. In this particular situation both are true so its obvious but I don't know how to state that fact. For the 2nd part of the proof, wouldn't I just say that D1 is a subset of itself, and its already given that D1 is a subdivision of [a,b]? It just seems too easy... I also had questions about proofs I've already turned in that I did poorly on but I didn't want to flood this place with questions. 


#2
Feb1113, 05:26 PM

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P: 11,617

Consider a simple setup: A={1} B={2} The statement "the set has exactly one element" is true for both A and B, but not for their union C={1,2}. To show that E := D1 u D2 is a subdivision, you have to show that E is a finite subset of [a,b] and a and b are in E. 


#3
Feb1113, 05:33 PM

P: 5

See I was thinking more along the lines if we had x an element of A then its simple enough to just say X is in A u B?
Because it doesn't matter if X is in B since we can just add another set, regardless its still in A. I'm just lost how to incorporate the union into the proof, I guess I'll spend more time on that. Haha. See within 5 minutes I get a reply in this one and not the homework =P. Thank you! 


#4
Feb1113, 05:37 PM

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P: 11,617

Subdivisions/Refinement Proof



#5
Feb1113, 05:48 PM

P: 5

I didn't see the drop down menu for the element, union, and implication arrows. Where are those?



#6
Feb1213, 09:23 AM

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P: 11,617

They are written with the [itex]tags and LaTeX codes. See the FAQ entry for details, or quote my post to see its code.



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