In: Advanced Math
*(4) (a) Prove that if p=(x,y) is in the set where y<x and if r=distance from p to the line y=x then the ball about p of radius r does not intersect with the line y=x.
(b) Prove that the set where y<c is an open set.
Justify your answer
1A. Suppose
de notes the distance between the points
and
.
r = distance from the point p to the line y=x .
There exist
such that
.
Let
be any point on the line
.
Now ,
The point
does not lie inside the ball centered at p and radius r .
As
be arbitrary point on the line
. Hence the line
does not intersect the ball centered at p and radius r .
1B. Let
.
We will prove that the set B is open .
Let ,
then
.
.
If we choose
Then the ball centered at
and radius
entirely contained in
.
(x,y) is an interior point of B .
As (x,y ) is arbitrary so every point of B is an interior point of B .
Hence the set B is open .