# Distance metric

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: $$d(x, y) \geq 0$$ – The distance must always be greater than zero.
• Identity of indiscernibles: $$d(x, y) = 0 \Leftrightarrow x = y$$ – 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: $$d(x,z) \leq d(x,y) + d(y,z)$$ – 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.