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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

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Note that geometrically $[0,1]\times [a,5]$ is a rectangle. – André Nicolas Jan 6 '13 at 18:23

$[0,1]\times[a,5]$ 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 $B$,

$$A\times B=\big\{(a,b):a\in A\text{ and }b\in B\big\}\;.$$

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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.

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