is empty set part of every set?? you have a power set of s represented by p(s) and s is { x is integer and either x=<-2 or x>=5} and you have another set d = {{-3 -2 1}, {4}, {6, 7}, {-5, 6, 9}} when you are asked for intersection of p(s) and d in a plain maths question am I meant to include { } the empty set as well since it is a subset of everyset???
The empty set is a subset of every set. But it is not an element of every set. In the question above, how many elements are in set d? Of those elements, which are members of p(s)?
{} is a subset of every set, so it is a subset of s. Therefore, {} is an element of p(s). However, it is clear that {} is not in set d. Therefore, {} is not an element of the intersection of p(s) and d. Hope that answers your question.
The empty set is an actual element of the power set. The empty set is also a subset of the power set. The empty set is not an actual element of d. The empty set is a subset of d. The intersection considers elements of both sets, so the empty set is not an element of the intersection. But, the empty set is a subset of the intersection. The fact that the empty set is an element of the power set means that in addition to the empty set being a subset of the power set, the "set of the empty set" is also a subset of the power set, but this is not true for d. This is the difference.
When speaking about sets, it's tempting to use phrases that don't have precise definitions like "is a part of" and "is contained in" or "contains". This question is a good example of why the more specific phrases "is a subset of" and "is an element of" are needed.