Let I denote the interval [0,oo). For each r ∈ I, define
A={(x,y)∈ RxR:x^2+y^2=r^2}
B={(x,y)∈ RxR:x^2+y^2<=r^2}
C={(x,y)∈ RxR:x^2+y^2<r^2}
Determine
UA and ∩A
UB and ∩B
UC and ∩C
To determine the sets UA and ∩A, we need to understand the definitions of A and the symbols used.
A={(x,y)∈ RxR:x^2+y^2=r^2} represents a circular shape in the Euclidean plane, where the center of the circle is at the origin (0,0), and the radius is r.
UA denotes the union of all sets A for all values of r in the interval [0,oo). Thus, UA is the collection of all points that lie on any circles centered at the origin with radii ranging from 0 to infinity.
To find UA, we need to consider all possible values of r. We observe that for any given point (x,y) in the plane, there exists a unique r such that (x,y) satisfies the equation x^2 + y^2 = r^2. Therefore, UA is the entire plane, as all points in the plane lie on some circle centered at the origin.
Now, let's consider ∩A, which denotes the intersection of all sets A for all values of r in the interval [0,oo). In this case, we need to find the common points that lie on every circle in A.
If we consider a point (x,y) that lies on every circle in A, it must satisfy the equation x^2 + y^2 = r^2 for all possible values of r. However, there is no such point in the plane that satisfies this condition since the square of the distance between any point and the origin cannot be equal to all possible values of r^2 simultaneously. Therefore, ∩A is an empty set (∅).
Moving on to UB and ∩B, we have the following definitions:
B={(x,y)∈ RxR:x^2+y^2<=r^2} represents a closed disk in the Euclidean plane. It includes all points (including the boundary) within or on the circle centered at the origin with a radius of r.
UB denotes the union of all sets B for all values of r in the interval [0,oo). Thus, UB is the collection of all points that lie inside or on any circles centered at the origin with radii ranging from 0 to infinity.
To find UB, we notice that for any given point (x,y) in the plane, if its distance from the origin (0,0) is less than or equal to r, then it lies on or inside the circle centered at the origin with radius r. Therefore, UB is the entire plane, as it includes all points within or on any of these circles.
Finally, considering ∩B, which denotes the intersection of all sets B for all values of r in the interval [0,oo), we need to find the common points that lie inside or on every circle in B.
If we consider a point (x,y) that lies inside or on every circle in B, it must satisfy the condition x^2 + y^2 <= r^2 for all possible values of r. In other words, it must lie within or on the unit disk (the disk centered at the origin with radius 1). Therefore, ∩B is the closed unit disk.
Lastly, for UC and ∩C, we have the following definitions:
C={(x,y)∈ RxR:x^2+y^2<r^2} represents an open disk in the Euclidean plane. It includes all points inside the circle centered at the origin with a radius of r.
UC denotes the union of all sets C for all values of r in the interval [0,oo). Thus, UC is the collection of all points that lie inside any circles centered at the origin with radii ranging from 0 to infinity.
To find UC, we observe that for any given point (x,y) in the plane, if its distance from the origin (0,0) is less than r, then it lies inside the circle centered at the origin with radius r. Therefore, UC is the set of all points inside the disk bounded by these circles.
Lastly, ∩C, which denotes the intersection of all sets C for all values of r in the interval [0,oo), represents the common points that lie inside every circle in C.
If we consider a point (x,y) that lies inside every circle in C, it must satisfy the condition x^2 + y^2 < r^2 for all possible values of r. In other words, it must lie strictly inside the unit disk (the disk centered at the origin with radius 1). Therefore, ∩C is the open unit disk.
To summarize:
UA: The entire plane RxR
∩A: The empty set (∅)
UB: The entire plane RxR
∩B: The closed unit disk
UC: The set of all points strictly inside the unit disk
∩C: The open unit disk