How to know if an intersection contains another intersection?

Question

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

• I = i₁ i₂ ... i_n
• J = j₁ j₂ ... j_n
• nor I nor J are empty, therefore no_x or j_x the saint

I know the answer is ternary, it can be True, False or Undefined... As a matter of fact, I already know part of the answer, but since I am a single limited mind, Maybe there’s still some case I’m not able to visualize. Maybe someone with a more solid mathematical foundation can put an end to this.

Edition 1: Available operations

Sets are not enumerable. It is possible to use operators between sets. In my interface I have defined Contains(other), Intersects(other) and IsEmpty(), each operation can return a third value in addition to Yes and Not, that is Undefined... but for the tests carried out between the sets ix and jx we can assume a boolean response, because the undefined value would simply be propagated. There are also operations that return sets Intersection(other) and Union(other).

In short:

• it is not possible to list
• there are test operations each returning Yes, Not and Undefined:
  • Contains(other)
  • Intersects(other)
  • IsEmpty()
• there are operations that return other sets
  • Intersection(other)
  • Union(other)

Answer 1

Here’s what I believe is 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 j_n

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

  1. example with 3 sets forming I and 3 forming J

     =>

  2. example with 3 sets forming I and 2 forming J

     =>

  3. example with 2 sets forming I and 3 forming J

     =>

• it’s not enough that all the j_n are contained in one of the_n. See a counter-example:

  =>

  How is it possible to see all the j_n are contained in any i_n, meanwhile I does not contain J

• the negative does not guarantee that I does not contain J. See a counter-example:

  =>

  As it is possible to see, not all i_n contains some j_n, meanwhile Icontains J even so. In the case presented above

I does not contain J for sure

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

• any i_n no intersection with any j_n

  If any i_x is disjoint from any j_k, as I is a subset of that i_x and J is a subset of that j_k, then I and J are also disjoint.

  

  In this example the sets outlined in red are disjoint, and therefore the yellow intersection and the blue intersection are also disjoint.

• The negative does not guarantee that I contains J:

  

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

Undefined

The cases that cannot be certain are indefinite. It is possible that there are more cases where it is possible to be sure, both positive and negative. But I have no way to prove this mathematically.