Characteristic Sets A & B: Special Name & Importance

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Discussion Overview

The discussion revolves around the properties of two sets, A and B, defined by the condition that either their intersection is empty or one set is a subset of the other. Participants explore whether these sets have a specific name, their significance in mathematics, and potential applications in areas such as graph theory.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that if the intersection of two sets is empty, they are mutually disjoint, and if the intersection equals one of the sets, then the sets are identical.
  • Others question whether the sets A and B being either completely disjoint or one being a subset of the other has a specific name or significance.
  • A participant suggests that the condition could be likened to a physical interpretation, proposing a colloquial name for it, the "GWB condition," referencing a political statement.
  • Another participant expresses interest in the broader implications of such sets, asking if a collection of sets with this property has any particular importance or representation in mathematics.

Areas of Agreement / Disagreement

Participants generally agree on the definitions of disjoint sets and subsets but disagree on whether these sets have a specific name or recognized importance in mathematical literature. The discussion remains unresolved regarding the significance of such sets.

Contextual Notes

Some participants acknowledge that their understanding of set theory is limited, and there may be missing assumptions or definitions regarding the properties of the sets in question.

sutupidmath
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I was wondering if sets A and B, with the following property:

\mbox {Either } A\bigcap B=\emptyset \mbox{ or } A\bigcap B=A.

have a special name. The name per se is not that important, however, what i am asking is whether these are well known/studied sets, or if they are of any special importance/use? If yes, where could i read more abou them?


Thanks
 
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Well, I haven't had a lot of courses on set theory, but if memory serves me right:

If the intersection of two sets is the empty set, then the two sets are said to be mutually disjoint.

If you know the intersection of two sets is equal to either set, then the two sets are the same. It's like the identity operation of intersection.

I studied them as part of my discrete math course, using Epp's Discrete Mathematics with Applications textbook.
 
So, A is either completely outside B or completely inside B? Hm, it might have a name but I haven't heard of one.

Casual Friday said:
If you know the intersection of two sets is equal to either set, then the two sets are the same. It's like the identity operation of intersection.

If by 'either set' you mean 'both sets' then yes.
 
Thanks for your replies, but unfortunately this is not what i was asking for ( i am well aware of the things you pointed out).

The question itself might be a little vague i believe.

Essentially, what i am saying is do these kind of sets show up somewhere? By somewhere i mean say graph theory(someone told me this, but had no more info).

In other words, say, like fibonacci sequences that show up in many places in math, do these kind of sets have 'alike' properties?

Thanks!
 
Just to make this clear. Are you saying that the condition you're placing on these two sets is that they must either be multually exclusive or else one be a subset of the other?
 
If A \bigcap B = \emptyset
then A and B are disjoint.
If A \bigcap B = A
then A is contained in B (A is a (proper) subset of B).

Other than that, I don't think there are any special names.
 
Mark44 said:
Other than that, I don't think there are any special names.

Yes but you can certainly imagine a physical meaning to that combination of set conditions. If I were to give it a name I would call it the "you're either all in or you're all out" condition. Or what was it that George W Bush was saying to (potential) ally nations after 911, "you're either with us or you're against us". Maybe we could call it the GWB relation, or the GWB condition. :)
 
I am not really interested on their 'name' per se, but more on what they could possibly represent.

To be more precise, if we have a collection of sets \{A_i\} such that for any two sets A_i,A_j the property i have described holds, i am wondering what could this collection of sets represent? If it does? That is, are such collection of sets of any particular importance?
 
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