(Boolean closure). Show that if are elementary sets, then the union , the intersection , and the set theoretic difference and the symmetric difference are also elementary. If , show that the translate is also an elementary set.
An elementary set is any subset of which is the union of a finite number of boxes. Given that there is no requirement for the boxes to be disjoint, the first part, that is elementary, follows automatically.
Let be the finite union of boxes and the finite union of boxes , then
So we just need to show that is a box (or the null set). Possibilities for are that it is: the null set, , or a proper subset of both and . In this last case let the intervals of intersection on the axes be , the intersection is the box given by the Cartesian product .
Next we prove that is elementary. Let then
Using the previous results for union and intersection it is enough to show that is elementary. If , ; if , ; otherwise we have two boxes which partially intersect. For ease of notation consider and consider the intervals corresponding to box and how they are partitioned by the intersection with . Consider the intersection with partitions this interval into
- the part of the interval which intersects with the corresponding interval of , .
- the lower interval, this is the part of the interval which is less than the interval , or just the lower endpoint of (if the interval overlaps the lower end of the interval).
- the upper interval, this is the part of the interval which is greater than the interval , or just the upper endpoint of (if the interval overlaps the upper end of the interval).
These intervals can be used to build a set of boxes, , is the union of the boxes in . So is elementary.
Given this result, the symmetric difference is the intersection of two elementary sets and is therefore also elementary.
The translation of a box is another box and so the translation of the set is also elementary.