I'm wondering about this, and if so, what an example would look like. They could easily satisfy the first two properties -- $d(x, y) = d(y, x), d(x, y) = 0 \iff x = y$ -- but I can't think of any instance, at least not using the real numbers, that would satisfy the triangle inequality.
What does that statement say? I cannot understand what meaning. As was stated in reply #2 by definition a metric is a non-negative function. Therefore, you must review the definition of metric.
Here is the standard problem to teach metrics:
If $d$ is a metric on $X\times X$, then $e(x,y)=\dfrac{d(x,y)}{1+d(x,y)}$ is a metric on $X\times X$.
Doing that problem will help you understand the metric concept.