|
I have this problem: consider the two sets $A$ and $B$ $$A=[0,1]\times [a,5]$$ and $$B=\{(x,y):x^2+y^2<1\}$$ What are the values of $a$ that guarantee the existence of a hyperplane that separates $A$ from $B$. Given a chosen value of $a$, find one of those hyperplanes. My main problem is axiomatics: how do I read: $A=[0,1]\times[a,5]$, what's with the $\times$? Thank you |
||||
|
|
$[0,1]\times[a,5]$ is the Cartesian product of the intervals $[0,1]$ and $[a,5]$: $$[0,1]\times[a,5]=\big\{(x,y):0\le x\le 1\text{ and }a\le y\le 5\big\}\;.$$ More generally, for any sets $A$ and $B$, $$A\times B=\big\{(a,b):a\in A\text{ and }b\in B\big\}\;.$$ |
|||
|
|
|
The $\times$ stands for cartesian product, i.e. $X\times Y=\{(x,y)\mid x\in X, y\in Y\}$. Whether ordered pairs $(x,y)$ are considered a basic notion or are themselves defined (e.g. as Kurtowsky pairs) usually does not matter. See alo here. |
|||
|
|
