Part 1: Give an example of sets A and B of real numbers such that A intersect B
ID: 2901997 • Letter: P
Question
Part 1: Give an example of sets A and B of real numbers such that A intersect B is nonempty, sup(A intersect B) < sup(A), and sup(A intersect B) < sup(B)
Part 2: Referring to the previous problem, for sets A and B of real numbers such that A intersect B is nonempty, state and prove a relationship between sup(A), sup(B) and sup(A intersect B).
The problem I have with part 1 is that I don't really see how it could be true for both Sup(A) and Sup(B) to be greater than the sup of the intersection. One of them being greater seems fine. I've made this problem worth a lot of points so don't just answer one part and expect points.
Explanation / Answer
Part 1: A = {1,2,100} B = {1,2,200} AintersectB= {1,2}
supA = 100, supB = 200, sup(A intersect B) = 2. It satisfies all the given conditions.
Part 2: sup(A) > sup(A intersect B);
sup(B) > sup(A intersect B);
this implies, sup(A intersect B) < min(sup(A), sup(B))
Many relations can be found out between these 3, depends on what you want to prove.
Hope this helps!