When does a function $d:A\times A \to \mathbb R$ will not be a metric, due to violation of symmetry property

Question

Definition of Metric -- Let, $A$ be a non-empty set. A function $d:A\times A \to \mathbb R$ is said to be a metric on $A$, if the following conditions are satisfied,

1. for all $x,y\in A$, $d(x,y)≥0 \land d(x,y)=0$ if and only if $x=y$ 【 Positiveness Property 】
2. for all $x,y\in A$, $d(x,y)=d(y,x)$ 【 Symmetry Property 】
3. for all $x,y,z\in A$, $d(x,y)≤d(x,z)+d(z,y)$ 【 Triangle Inequality 】

In graduation level, we have questions like,

'Check that the given function $d:A\times A \to \mathbb R$ is a metric on $A$ or not: $d(x,y)=.........$, for $x,y\in A$.'

The questions, which we usually get, holds the symmetry property & which are not metric, they either violate the Positivity property or the triangle inequality. Can you give me such an example where the function $d$ is not a metric, only due to the violation of symmetry property?

Answer 1

As a very simple example, let $A=\{a,b\}$ have two elements and define $d(a,a)=d(b,b)=0$, $d(a,b)=1$ and $d(b,a)=2$.

Answer 2

A more interesting function is $$d(x,y) = \max \{ x-y,0 \}$$ for $x,y$ real.

Answer 3

Take a path-connected directed graph $G$ and let the distance between two nodes on the graph be the length of the shortest path joining them.