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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.

The video explains the triangle inequality using a detour analogy. It compares going straight from point $x$ to point $z$ with taking a detour through an intermediate point $y$.

Conditions: $x, y, z$ are points in the metric space.; $y$ is an arbitrary intermediate point.

The video explains the triangle inequality using a detour analogy. It compares going straight from point $x$ to point $z$ with taking a detour through an intermediate point $y$.

Conditions: $x, y, z$ are points in the metric space.; $y$ is an arbitrary intermediate point.

The video uses the analogy of commuting from work to home. Going straight home represents the direct distance $d(x,z)$.

Conditions: $x$ is work, $z$ is home, and $y$ is the shop.; The path is a journey in a metric space.

The video uses the analogy of commuting from work to home. Going straight home represents the direct distance $d(x,z)$.

Conditions: $x$ is work, $z$ is home, and $y$ is the shop.; The path is a journey in a metric space.

The four required properties are: 1. Non-negativity ($d(x,y) \ge 0$), 2.

Conditions: $x, y, z$ are arbitrary elements of the set $A$.; $d$ is the distance function of the metric space.

The four required properties are: 1. Non-negativity ($d(x,y) \ge 0$), 2.

Conditions: $x, y, z$ are arbitrary elements of the set $A$.; $d$ is the distance function of 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.