Let I denote the interval [0,oo). For each r (element of) I, define
Ar={(x,y)element of RxR:x^2+y^2=r^2}
Determine (UNION)reI Ar and (INTERSECT)reI Ar
The definition of Ar (Ar={(x,y)element of RxR:x^2+y^2=r^2}) means that for any r>0, Ar contains all points on the circle of radius r centered on the origin.
So how would you determine the Union & Intersection? I'm sure its simple...I'm just not getting it for some reason...as in how to write it out?
Consider that if ALL possible radii are included, than for every point (x,y) there will exist an Ar that contains it.
Also, can you say that every Ar contains different points, so that there does not exist a point (x,y) that exists in Ar for different r?
To determine the union and intersection of all sets Ar for r ∈ I = [0,∞), we need to analyze the properties of these sets.
1. Union (∪): The union of sets is the collection of all elements that appear in any of the sets being united. In this case, we want to find the set that contains all possible points (x, y) that lie within the circles centered at the origin (0,0) with varying radii r (ranging from 0 to infinity).
To find the union (∪reI Ar), observe that:
- For any specific value of r, the set Ar consists of all points (x, y) which satisfy the equation x^2 + y^2 = r^2, representing the circumference of a circle with radius r.
- As r increases from 0 to infinity, the circles increase in size and encompass all points in the plane.
Therefore, the union of all sets Ar for r ∈ I is the entire plane R^2.
2. Intersection (∩): The intersection of sets is the collection of elements that appear in every set being intersected. In this case, we want to find the set that contains all points (x, y) that lie within all circles with radii r (ranging from 0 to infinity).
To find the intersection (∩reI Ar), note that:
- For any specific value of r, the set Ar consists of all points (x, y) which satisfy the equation x^2 + y^2 = r^2.
- As r increases from 0 to infinity, the circles increase in size, but their centers remain fixed at the origin (0,0).
Therefore, the intersection of all sets Ar for r ∈ I is the single point (0,0), as this is the only point that is common to all circles with varying radii.