Math: Open and Closed Disks and Balls
In order to understand what open and closed disks are, we need to consider the 2D space. Let’s consider a circle, C, centered at (a, b) and having a radius of r.
Closed disk
Let’s now consider a set Y containing all the points enclosed by the circle, C, as well as those lying on C (those lying on the line of the circle). Y is appropriately referred to as the closed disk of radius r centered at (a, b). Essentially, the closed disk is the set of all points (x, y) in the 2D space for which x² +y² ≤ r².
Open disk
The open disk is a set of all points enclosed by the circle, C. The open set excludes the points lying on the line of the circle. Essentially, the open disk is the set of all points (x, y) in the 2D space for which x² +y² < r². This is more appropriately referred to as the open disk of radius r centered at (a, b).
It is clearly safe to say that the open disk is a subset of the closed disk. As such, it could be concluded that an intersection of the open disk and closed disk will result in the open disk. Of course, that is if they both have the same radius and centered at the same coordinates in the 2D space.
This concept of open and closed disks could be extended to the 3D space in which we would be dealing with a sphere. For that, let’s consider a sphere, S, in the 3D space having a radius of R centered at (a, b, c).
Closed ball
A set which contains all points (x, y, z) in the 3D space for which x² +y² +z² ≤ R² is referred to as the closed ball of radius R centered at (a, b, c). This is essentially, the set of all points enclosed by the sphere S, as well as the ones lying on the sphere.
Open ball
As you would have guessed, the set of all points (x, y, z) in the 3D space for which x² +y² +z² < R² is referred to as the open ball of radius R centered at (a, b, c). This set contains all and only the points enclosed by the sphere.
Similar to that of the disks, the open ball could safely be considered as a subset of the closed ball. Also, given a closed ball and an open ball with both having a radius of R and centered at the same coordinate of the 3D space, their intersection will just be the open ball.
Here is our YouTube video on the topic: https://youtu.be/38ZEMGwkdA8
Host: Paul (Math plus Tech); Calculus 2.0
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