Some Properties of Distance Functions

The object of this paper is to discuss some properties of distance or metric functions. Given any abstract set of elements R, a non-negative real valued function d(x,y) defined for every pair of elements (x, y) of R is called a metric or a distance function if the following conditions hold :-

(i) d(x, x) = o;

(ii) d(x,y) > o if x ≠ y;

(iii) d(x,y) = d(y, x);

(iv) d(x,y) ≤ d(x, z) + d(z,y)

for any three elements x, y, z of R.