Adding Sets: A+B = Union & Intersection

  • Thread starter pivoxa15
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In summary: A+B=AuB+AnB).In summary, the OP wants to say that the sum of two sets includes all of the members of both sets.
  • #1
pivoxa15
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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.
 
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  • #2
pivoxa15 said:
A,B are sets

A + B=AuB + AnB

Does it make sense to add sets? I know union and intersections are possible.

What do you mean that to do? Additive number theory has addition of sets like [itex]X+Y=\{x+y:x\in X,y\in Y \}[/itex] (so that {1, 2, 3} + {10, 40} = {11, 12, 13, 41, 42, 43}). Is that what you want?
 
  • #3
No. I am talking about sets in measure theory.
 
  • #4
pivoxa15 said:
A,B are sets

A + B=AuB + AnB

Does it make sense to add sets? I know union and intersections are possible.

You ccan define "+" to mean anything you want. What is the point of your definition?
 
  • #5
pivoxa15 said:
A,B are sets

A + B=AuB + AnB

Does it make sense to add sets? I know union and intersections are possible.
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? The sum then could not be a set since there are no dupllcates in sets. What kind of object do you want the sum to be, a bag, a.k.a. multiset?
 
  • #6
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 B-A intersection B.
 
  • #7
robert Ihnot said:
I take it that he wants to say: A +B = A union B-A intersection B.
Oh. So symmetric difference (more) then?
 
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  • #8
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.
 
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  • #9
robert Ihnot said:
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.)
Is this what you meant previously? You seem to have just defined this addition to be union. The symmetric difference is "the set of elements belonging to one but not both of two given sets", i.e., "A union B-A intersection B", which I assume you meant as "(A union B) - (A intersection B)", with "-" denoting set difference (A - B = {x | x in A and x not in B}).

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
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!)
 
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  • #11
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+B-A\cap B [/tex]
 
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  • #12
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
CRGreathouse said:
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!)
this form might be better because it works even if measure(a intersect b) is infinite.
 
  • #14
In what sense is that better? It is clearly wrong.
 
  • #15
matt grime said:
In what sense is that better? It is clearly wrong.
It is? Counterexample, please.
 
  • #16
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
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
You need to define the boolean operations properly. You need to use the symmetric difference. Every element is self inverse.
 

1. What is the difference between the union and intersection of two sets?

The union of two sets A and B is the set of all elements that are in either A or B (or both). The intersection of two sets A and B is the set of all elements that are in both A and B.

2. How do you add two sets using the union and intersection method?

To add two sets A and B, we first find the union of A and B, which includes all elements in both sets. Then, we find the intersection of A and B, which includes only the elements that are in both sets. Finally, we add the two resulting sets together to get the final result.

3. Can you add more than two sets using the union and intersection method?

Yes, the union and intersection method can be used to add any number of sets together. We simply need to find the union and intersection of all the sets and then add them together.

4. How is the union and intersection method useful in set theory?

The union and intersection method is useful in set theory because it allows us to combine and compare sets, and to find the relationships between them. It also helps in understanding the properties and operations of sets.

5. What is the commutative property in the union and intersection method?

The commutative property in the union and intersection method states that the order in which we add the sets does not matter. In other words, A+B = B+A for any sets A and B. This is true for both the union and intersection operations.

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