Prove that dxy 0 for all pairs of distinct points x and y i

Prove that d(x,y) > 0 for all pairs of distinct points x and y in M.

Note: Use just the axioms in the definition of metric d on set M.

Def. of metric: M x M --> R is a metric on M if Vx, y, z M. (x,y) --> d(x,y)

Axioms:

1) d(x,y) = d(y,x)

2) d(x,z) <= d(x,y) + d(y,z) (Triangle Inequality)

3) d(x,y) = 0 iff x = y

Solution

sol. there is no need to nprove this result because you know the metric d on a set is a distance between two points .since distance is non negative and d(x,y)=0 only when x=y.thus for every pair d(x,y) >0