Metric (mathematics)

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In mathematics a set is said to have a metric if there is a notion of distance on the set that fits our intuition about how distance should behave. More formally, an operation on a set A to the real numbers is said to be a metric if for all and , we have the following four properties.

  1. d(x, y) ≥ 0     (non-negativity)
  2. d(x, y) = 0   if and only if   x = y     (identity)
  3. d(x, y) = d(y, x)     (symmetry)
  4. d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality).

Intuitively, property 1 says that distance cannot be negative and two elements of the set are zero distance away from each other if and only if they are the same element. Property 2 says that the distance from x to y is the same as the distance from y to x. Property three says that the distance going from x to y to z is at least the distance to go from x to z. This third property is known as the "triangle inequality."