I am given the definition of a ball vs sphere but I can't get an image in my head. I can see a circle for a dotted edge for a ball, and a circle with a 'lined' edge for a sphere. In $\mathbb{R}^n$ are these supposed to be 3D shapes? The words ball and sphere confuse me very much since they mean almost the same in English.
Now the main problem is how to tell if a set is bounded. I've been given the definition of a ball which is when $|x-a|<r$ and sphere which is when $|x-a|=r$. I sort of understand the difference with the dotted line image in my head.
Ive been given the definition $S\subset \mathbb{R}^n$ is bounded if there exists some $r>0$ s.t. $S\subset \beta(r,0)$. I am really confused with this and not understanding the definition very well. Especially the $\beta(r,0)$ notation.
Any help would be great.
Best Answer
Instead of seeing it in terms of balls and spheres it's easier to see $|x-a|\leq r$ as the set of all points $x$ that are at a distance $r$ or lesser from $a$. That way, you can generalize it to more than three dimensions without getting too confused.
Intuitively, a set is bounded if it can be "contained" inside some region. The ball is the simplest way of defining this region, since all it requires is a radius and a center in order to be defined.
So, $\space$ $S = \{ {(x,y) \in \mathbb{R}^{2}} | x = y \}$ is not bounded, since it's an infinite line and thus no conceivable finite ball could entirely contain it. On the other hand, $\space$ $S = \{ {(x,y) \in \mathbb{R}^{2}} | x^{2} + y^{2} = 4 \}$ is a bounded set, because it represents a circle of radius $r=2$ centered at the origin, and any ball $\beta(r,0)$ with $r > 2$ fully contains it. In this case "$\beta(r, 0)$" describes a ball of radius $r$ centered at $(0,0)$.