Ordered Sum of Sets: M_1+M_2

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In summary, the ordered sum M_1+M_2 is the set M_1\cupM_2 with the ordering defined as: if a,b \epsilon M_1 or a,b \epsilon M_2 then order them as they would be in the original orderings. If a \epsilon M_1 and b \epsilon M_2 then a<b. However, if you can somehow resolve this case, for example if the original wording gives one some excuse to declare that M_1's ordering takes precedence, then I think (a < b ?) will always be unambiguous.
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Office_Shredder
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If [tex]M_1[/tex] and [tex]M_2[/tex] are ordered sets, the ordered sum [tex]M_1+M_2[/tex] is the set [tex]M_1\cupM_2[/tex] with the ordering defined as:

If [tex]a,b \epsilon M_1[/tex] or [tex]a,b \epsilon M_2[/tex] then order them as they would be in the original orderings. If [tex]a \epsilon M_1[/tex] and [tex]b \epsilon M_2[/tex] then [tex]a<b[/tex]

The question then is if [tex]a \epsilon M_1[/tex] and [tex]a \epsilon M_2[/tex], then we get [tex]a< a[/tex] which is impossible. In general, it seems you'll get a is less than and greater than some elements, which means [tex]M_1+M_2[/tex] isn't really ordered at all

(I use epsilon as the 'element of' symbol as I couldn't find a more appropriate one in the latex pdfs)
 
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Office_Shredder said:
If [tex]a,b \epsilon M_1[/tex] or [tex]a,b \epsilon M_2[/tex] then order them as they would be in the original orderings. If [tex]a \epsilon M_1[/tex] and [tex]b \epsilon M_2[/tex] then [tex]a<b[/tex]

The question then is if [tex]a \epsilon M_1[/tex] and [tex]a \epsilon M_2[/tex], then we get [tex]a< a[/tex] which is impossible.

It seems like there's a loophole here since if [tex]a \epsilon M_1[/tex] and [tex]a \epsilon M_2[/tex], then you should be interpreting the ordering there under the first clause, not the second clause (since [tex]a,a \epsilon M_1[/tex]).

Alternately how is "[tex]M_1_2[/tex]" defined? If this is a union, then shouldn't the a = a case never come up since something cannot be a member of a set "more than once"?

Otherwise maybe whoever you're getting this from just made a mistake in their wording...

In general, it seems you'll get a is less than and greater than some elements, which means [tex]M_1+M_2[/tex] isn't really ordered at all

Well, this again comes back to the wording being kind of confusing. Let's say you have a, b where [tex]a,b \epsilon M_1[/tex] and also [tex]a,b \epsilon M_2[/tex]. And let's say by [tex]M_1[/tex]'s ordering a < b, and by [tex]M_2[/tex]'s ordering b < a. What do you do here?

However if you can somehow resolve this case, for example if the original wording gives one some excuse to declare that [tex]M_1[/tex]'s ordering takes precedence, then I think (a < b ?) will always be unambiguous.
 
  • #3
Right, I see what you're talking about with the case [tex]a,a \epsilon M_1[/tex] cutting you off from 'seeing' the [tex]a \epsilon M_1[/tex] and [tex]a \epsilon m_2[/tex] case. But you still have trouble if [tex]b \epsilon M_1[/tex], [tex]c \epsilon M_2[/tex] and a<b in [tex]M_1[/tex] c<a in [tex]M_2[/tex] then a<b<c<a which means it's not transitive.

I have to apologize, when I wrote [tex]M_{12}[/tex] it was just poorly writing [tex]M_1 \cup M_2[/tex] so it didn't come out right. But even though a can only be an element of that set once, the way the ordering is defined it still works out fishily, unless it's modified to be if a is in [tex]M_2[/tex] and not [tex]M_1[/tex] then it gets ordered as if it was in [tex]M_2[/tex].

EDIT: This isn't well defined, as if [tex]M_1=M_2=N[/tex] then the order type of [tex]\omega + \omega = \omega[/tex] which certainly isn't true if [tex]M_1=N M_2=Z_-[/tex] where the negative integers are ordered by their absolute value
 
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1. What is the "Ordered Sum of Sets"?

The "Ordered Sum of Sets" is a mathematical operation that combines two sets, M1 and M2, in a specific order. It is denoted as M1+M2 and results in a new set that contains all the elements of M1 and M2 in a specific order.

2. How is the "Ordered Sum of Sets" calculated?

The "Ordered Sum of Sets" is calculated by first arranging the elements of M1 and M2 in a specific order, and then combining them to form a new set. The order of the elements in the new set is determined by the specific order in which the elements of M1 and M2 were arranged.

3. What is the difference between "Ordered Sum of Sets" and "Union of Sets"?

The "Union of Sets" combines two sets without any specific order, resulting in a set that contains all the elements of both sets. On the other hand, the "Ordered Sum of Sets" combines two sets in a specific order, resulting in a new set with a specific arrangement of elements.

4. Can the "Ordered Sum of Sets" be applied to more than two sets?

Yes, the "Ordered Sum of Sets" can be applied to more than two sets. When combining more than two sets, the operation is performed in a stepwise manner, starting with the first two sets and then adding the third set to the result, and so on. The final result will be a new set with all the elements of the original sets arranged in a specific order.

5. Are there any specific rules or properties for the "Ordered Sum of Sets"?

Yes, there are some rules and properties for the "Ordered Sum of Sets" operation. For example, the operation is commutative, meaning the order of the sets can be switched without changing the result. Also, the operation is associative, meaning the grouping of sets does not affect the result. Additionally, the identity element for the "Ordered Sum of Sets" is the empty set, and the inverse element for each set is itself.

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