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The first required property is non-negativity. The distance function $d(x,y)$ must always be greater than or equal to zero for any two points $x$ and $y$ in the metric space.

Conditions: $x$ and $y$ are elements of the metric space $A$.; $d$ is the distance function associated with that metric space.

The first required property is non-negativity. The distance function $d(x,y)$ must always be greater than or equal to zero for any two points $x$ and $y$ in the metric space.

Conditions: $x$ and $y$ are elements of the metric space $A$.; $d$ is the distance function associated with that metric space.