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Does this make sense?by pivoxa15
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#1
Aug507, 07:14 AM

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A,B are sets
A + B=AuB + AnB Does it make sense to add sets? I know union and intersections are possible. 


#2
Aug507, 09:01 AM

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#3
Aug507, 03:38 PM

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No. I am talking about sets in measure theory.



#4
Aug507, 04:02 PM

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Does this make sense?



#5
Aug507, 04:03 PM

PF Gold
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#6
Aug507, 04:43 PM

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honestrosewater: I take that to mean that you want an element that is in both A and B to show up twice in the sum of A and B?
I take it that he wants to say: A +B = A union BA intersection B. 


#7
Aug507, 06:14 PM

PF Gold
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#8
Aug507, 07:44 PM

PF Gold
P: 1,059

If we take the sets {1,2,3} + {2,3,4} = {1,2,3,4}= A U B, which for n=1 to 4 is the whole set. Thus [tex]A\cup B+A\cap B =A\cup B [/tex] (I don't think measure theory has any effect on that.)
However if we thought of these as collections, then we would have: {1,2,3}+{2,3,4} = {1,2,2,3,3,4} (From Wikipedia: When two or more collections are combined into a single collection, the number of objects in the single collection is the sum of the number of objects in the original collections. ) This is easier to follow if we were thinking of collections of furniture like lamps, rugs, etc. So I believe that you are correct about the symmetric difference of sets. 


#9
Aug507, 09:47 PM

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The original definition, "A + B=AuB + AnB" appears to be circular since the symbol that it is defining is used in the definition, so who knows. Normally, when you add two things, the result includes, in a loose sense, all of what you started with. For sets, this would seem to simply be union, but I assume the OP had something more than union in mind. You at least don't usually lose, or subtract, things when you add, so I assume the OP was thinking that the sum of two sets should include everything that was in those sets in some way that union doesn't, i.e., by including any duplicates. 


#10
Aug507, 10:08 PM

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Since you say measure theoretic, perhaps you mean measure(a) + measure(b) = measure(a union b) + measure (a intersect b)? (for finitely additive measures, of course!)



#11
Aug607, 12:59 AM

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Yes, you are right. Measure is a mathematical concept, so we can use the plus or minus sign. So that in general: [tex] A\cup B = A+BA\cap B [/tex]



#12
Aug607, 05:52 AM

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But that equation doesn't say anything about measure! Do you mean having first defined A+ B as [tex] A\cup B + A\cap B [/tex]. Of course, as has been pointed out, that is just equal to [tex] A\cup B[/itex]
It WOULD make sense if you would do what people have been asking you to do and write the "measure": [tex] measure(A\cup B) = measure(A)+ measure(B)measure(A\cap B) [/tex] 


#13
Aug607, 09:04 AM

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#14
Aug607, 04:25 PM

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In what sense is that better? It is clearly wrong.



#16
Aug607, 06:38 PM

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I think I had read your post the wrong way round (i.e. so that it agreed with the wrong assertion that m(A+B)=m(A)+m(B)+M(AnB). Sorry.)



#17
Aug607, 10:04 PM

P: 1,572

Yeah, I was agreeing with m(A)+m(B) = m(A u B) + m(A n B)...just a tad better than the other way around, m(A)+m(B)  m(A n B) = m(A u B) as that's not quite true if m(A n B) is infinite. I wasn't agreeing with the other formulations.
The idea of "adding" sets though... How could addition be defined so that additive inverses might exist? 


#18
Aug707, 12:37 AM

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You need to define the boolean operations properly. You need to use the symmetric difference. Every element is self inverse.



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