How do you know if an intersection contains another intersection?

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If I and J are two sets formed by intersections, how to know if I contains J , knowing that:

  • n
  • ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ sub <
  • nor I nor J are empty, therefore no x or j x are

I know the answer is ternary, can be True, False or Indefinite ... to tell you the truth I already know part of the answer, but since I am only one limited mind, there may still be some missing case that I can not visualize. Perhaps someone with a more solid mathematical base can put an end to it.

Issue 1: Available Operations

The sets are not enumerable. You can use operators between sets. In my interface I have defined I , J and Contains(other) , and each operation can return a third value besides Yes and No , which is Undefined ... but for the tests made between sets Intersects(other) and IsEmpty() we can assume a Boolean answer, because the undefined value would simply be propagated. There are also operations that return sets ix and jx .

In short:

  • Can not enumerate
  • There are test operations each returning No , No and undefined :
    • Intersection(other)
    • Union(other)
    • Contains(other)
  • There are operations that return other sets
    • Intersects(other)
    • IsEmpty()
asked by anonymous 22.03.2015 / 20:28

1 answer

3

Here's what I believe to be the answer, or at least part of it:

I contains J for sure

One of the ways to make sure I contains J is below (maybe there are more conditions, I'm not sure)

  • all i n contain at least one of the n

    Note that the first intersection contains the second intersection in all cases presented below.

  • example with 3 sets forming I and 3 forming J

    =>

  • example with 3 sets forming I and 2 forming J

    =>

  • example with 2 sets forming I and 3 forming J

    =>

  • It is not enough for all j n to be contained in one of i n . Here is a counterexample:

    =>

    Since it is possible to see all j n are contained in some n , however, I does not contain%

  • Negative does not guarantee that J does not contain I . Here is a counterexample:

    =>

    As you can see, not all n contains some j n , however J contains I anyway. In the case presented above

J does not contain I for sure

I only know of one way to ensure that J does not contain I :

  • n has no intersection with any n

    If any x is disjoint of any j k , as J is subset of that x and I is subset of this j k , then J and I are also disjoint.

    Inthisexamplethesetsoutlinedinredaredisjoint,sotheyellowintersectionandtheblueintersectionarealsodisjoint.

  • ThenegativeofthisdoesnotguaranteethatJcontainsI:

    Although there is an intersection between all i and all j, J does not contain I ... J intercepts I , but does not contain%. The yellow intersection does not contain the blue intersection.

Undefined

Cases that you can not be sure of are undefined. There may be more cases where you can be sure, both positive and negative. But I can not prove it mathematically.

    
22.03.2015 / 20:28
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