Do you meanOriginally Posted by TexasGirl
D(x,y) = (0 if x=y), (min(1,d(x,y)) otherwise)
? What you have written down is essentially the discrete metric.
Let (X,d) be a metric space and let D:X*X-->R be defined by:
D(x,y) = (0 if x=y), (max(1,d(x,y)) otherwise)
Prove that (X,D) is a metric space.
I know that I have to show (X,D) is positive definite, symmetric and satisfies the triangle inequality.
The first I am comfortable with. However, I am not sure of how to go about showing symmetry for the x/=y case...same for the triangle inequality. Anybody have any hints or suggestions?
SYMMETRYOriginally Posted by TexasGirl
suppose then
,
but is a metric on so:
,
so:
.
If then
Hence D is symmetric in its arguments (and this works whether or
is used in the definition of ).
RonL