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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.

The video uses absolute value because simple subtraction can yield a negative result depending on the order of the operands. Distance must be non-negative ($d(x,y) \ge 0$).

Conditions: The points are real numbers on a number line.; The distance function must satisfy the non-negativity property.

The video uses absolute value because simple subtraction can yield a negative result depending on the order of the operands. Distance must be non-negative ($d(x,y) \ge 0$).

Conditions: The points are real numbers on a number line.; The distance function must satisfy the non-negativity property.