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Hyperplane

For both 1 and 2, could you tell me whether or not there is a hyperplane that strictly separates the given sets A,B. If there is, find one. If there is not, prove so please.

1) A={(x,y):abs(x) + abs(y) <=1}, B={(1,1)}

2) A={(x,y):xy >= 4}, B={(x,y):x^2+y^2 <= 1}

where abs = absolute value

Solution Preview

The two problems are all in x-y plane. So a hyperplane is actually a line.

(1) Yes.
We can select the line: x+y=1.5
For element (1,1) in B, we know 1+1=2>1.5
But for any point (x,y) in A, we know |x|+|y|<=1, then we have ...

Solution Summary

This is a proof regarding hyperplanes with given characteristics.

$2.19