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It implies that $x$ and $y$ are the same point ($x = y$). This is the identity of indiscernibles property.

Conditions: $x$ and $y$ are points in the metric space.; $d(x,y) = 0$.

It implies that $x$ and $y$ are the same point ($x = y$). This is the identity of indiscernibles property.

Conditions: $x$ and $y$ are points in the metric space.; $d(x,y) = 0$.

Distance is independent of direction because of the symmetry property of the metric. The formula $d(x,y) = d(y,x)$ shows that swapping the order of the points does not change the distance value.

Conditions: $x$ and $y$ are points in the metric space.

Distance is independent of direction because of the symmetry property of the metric. The formula $d(x,y) = d(y,x)$ shows that swapping the order of the points does not change the distance value.

Conditions: $x$ and $y$ are points in the 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.

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.