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, UB, UC, ∩A, ∩B, and ∩C, we need to understand what each of these sets represents and how they are formed.

Let's start with the definitions of A, B, and C:

- A is the set of all points (x, y) in the Cartesian plane such that x^2 + y^2 = r^2 for any value of r in the interval [0, ∞).
- B is the set of all points (x, y) in the Cartesian plane such that x^2 + y^2 ≤ r^2 for any value of r in the interval [0, ∞).
- C is the set of all points (x, y) in the Cartesian plane such that x^2 + y^2 < r^2 for any value of r in the interval [0, ∞).

Now, let's determine the sets UA, UB, and UC:

- UA is the union of all sets A for each r in the interval [0, ∞). In other words, UA is the collection of all points in the Cartesian plane that satisfy the equation x^2 + y^2 = r^2 for any value of r in [0, ∞). Since this equation represents a circle, UA is the union of all circles with radius r, where r ranges from 0 to ∞. Therefore, UA represents the entire Cartesian plane.
- UB is the union of all sets B for each r in the interval [0, ∞). In other words, UB is the collection of all points in the Cartesian plane that satisfy the inequality x^2 + y^2 ≤ r^2 for any value of r in [0, ∞). This set includes all points inside and on the boundary of each circle with radius r, where r ranges from 0 to ∞. Thus, UB represents the set of all points within or on the boundary of any circle with any radius.
- UC is the union of all sets C for each r in the interval [0, ∞). In other words, UC is the collection of all points in the Cartesian plane that satisfy the inequality x^2 + y^2 < r^2 for any value of r in [0, ∞). This set includes all points inside each circle with radius r, where r ranges from 0 to ∞. Therefore, UC represents the set of all points within any circle with any radius.

Lastly, let's determine the sets ∩A, ∩B, and ∩C:

- ∩A denotes the intersection of all sets A for each r in the interval [0, ∞). In other words, ∩A is the collection of all points that satisfy the equation x^2 + y^2 = r^2 for every value of r in [0, ∞). Since no such point exists, ∩A is an empty set.
- ∩B denotes the intersection of all sets B for each r in the interval [0, ∞). In other words, ∩B is the collection of all points that satisfy the inequality x^2 + y^2 ≤ r^2 for every value of r in [0, ∞). This set includes all points within or on the boundary of the smallest circle with radius 0. Therefore, ∩B represents the single point at the origin (0, 0).
- ∩C denotes the intersection of all sets C for each r in the interval [0, ∞). In other words, ∩C is the collection of all points that satisfy the inequality x^2 + y^2 < r^2 for every value of r in [0, ∞). This set includes all points inside the smallest circle with radius 0. Since no such point exists, ∩C is an empty set.

To summarize:

- UA represents the entire Cartesian plane.
- UB represents the set of all points within or on the boundary of any circle with any radius.
- UC represents the set of all points within any circle with any radius.
- ∩A is an empty set.
- ∩B is the single point at the origin (0, 0).
- ∩C is an empty set.