Can someone possibly explain this to me, I'm having difficulties visualizing it:
For each $r\in\mathbb{R}$, let $A_r=\{(x, y, z)\in\mathbb{R}^3 : x+y+z=r\}$. How can I tell if this is a partition of $\mathbb{R}^3$?
EDIT: Okay, so I pretty much understand this problem, but what if it were like:
For each $r\in\mathbb{R}$, let $A_r=\{(x, y, z)\in\mathbb{R}^3 : x^2+y^2+z^2=r^2\}$. Is it correct to think that it would not be a partition because $A_r$ and $A_{-r}$ would have a value in common?