Can someone please explain the notation of the set attached in the image?

Question

Can someone please explain the notation of the set attached in the image? I am trying to learn optimization and was reviewing the basic mathematical concepts. Does this mean that B(x) can only take those values of vector components which satisfies the condition mentioned in set. If that is true then shouldnt there be another set for y vector with same conditon. I am not sure how B(x) contains both values of vector x and vector y. In this regard, I am unable to understand the next paragraph.

Answer 1

$B_r({\bf x})$ is the set of vectors (or points) that are a distance less than $r$ away from the vector (or point) $\bf x$.

In particular $B_r$ is not a function that assumes a vector as value, but it assumes a set as value: you plug in a vector $\bf x$, and you get a set of vectors $B_r({\bf x})$ in return.

For example, $B_1(\langle2,3\rangle)$ is the set of all points that are inside the circle with radius $1$ around the center $\langle 2,3\rangle$. For example $\langle 2.5,3\rangle$ and $\langle 2.1,3.3\rangle$ are vectors in the set $B_1(\langle2,3\rangle)$, while the vector $\langle 0.5,3\rangle$ is not (it is too far away) and the vector $\langle 3,3\rangle$ is not as well (it lies exactly on the border of the circle with radius $1$).

Answer 2

You should not think of a the ball as a set of vectors. It is a set of points (you might think of them as "locations.") The "x" and "y" are ways of describing the set of points. Simply put, if you want to describe a ball, you can specify the center of the ball, and the radius of the ball, and then we can know which points are inside the ball.

The "x" is the center, the "r" is the radius. The only use of "y" is instructions on how to tell if a point is in the ball. "If you want to know if a point ('y') is in the ball, measure the distance between the center of the ball ('x') and the point, and if the distance is less than the radius, it is inside, and if not it is outside."

This is a good example of how mathematical notation can make simple things look complicated. On the other hand, mathematicians do such things because it is much more compact, and because it is in some sense much more precise than words. Part of learning math is developing the skill of translating back and forth between the notation and a mental picture.