As per Wikipedia, a distance metric, metric, or distance function, “is a function that defines a distance between each pair of elements of a set.”

A distance metric $d(\cdot )$ requires the following four axioms to be true for all elements $x$, $y$, and $z$ in a given set.

• Non-negativity: $d(x,y)\ge 0$ – The distance must always be greater than zero.
• Identity of indiscernibles: $d(x,y)=0\iff x=y$ – The distance must be zero for two elements that are the same (i.e. indiscernible from each other).
• Symmetry: $d(x,y)=d(y,x)$ – The distances must be the same, no matter which order the parameters are given.
• Triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$ – For three elements in the set, the sum of the distances for any two pairs must be greater than the distance for the remaining pair.