I'm trying to solve a few examples with intervals. The problem is I'm confused, and I couldn't find much info on the internet that could help me. I'll list the exercises and share my thoughts and answers.
- $conv\mathbb{N}$
Well, it seems to me that $[1,\infty]$ is the smallest convex set that covers it.
- $conv\{(x,y)\in\mathbb{R}^2|x^2+y^2=1\}$
$\{(x,y)\in\mathbb{R}^2|x^2+y^2\le1\}$ - as above.
- $\mathbb{N}+[0,1]$
This one I'm confused by. Is it an ordinary union, that is $[0,1)\cup\mathbb{N}$? Or do I add the interval to every element in $\mathbb{N}$, making it $\{[1,2],[2,3],[3,4],...\}$? Or is it something entirely different?
$[0,1]^2+\{(0,1),(1,0),(1,1)\}$
$[0,1]^2+\{(t,t)|t\in[0,1]\}$
Here, I'm not sure what $[0,1]^2$ represents. Is it just a multiplication, i.e. $[0,1]*[0,1]=[0,1]$? Or is it a unit square? Or something different? As in the previous case, I don't know how to go about adding these.
- $[-1,1]^2+\{(x,y)\in\mathbb{R}^2|x^2+y^2\le1\}$
Well, it's a more complicated version of the previous ones and I don't know how to approach it.
I'd appreciate if someone could answer the questions and/or point me to some literature that explains it in a clear fashion.
A + B = { a + b : a in A, b in B } is accepted notation.
For 4 and 5, $[0,1]^2$ is the unit square and + vector addition.
Similar for 6.
Refer to the source of these problems for clues about notation.