Metrics cheat sheet

Question: When can a distance function d(x,y) be called metric, pseudo-metric, quasi-metric or semi-metric?

Constraint Metric Pseudo Quasi Semi
Non-negativity: d(x,y) ≥ 0 x x x x
Identity of indiscernibles: d(x,y)=0 ⇒ x=y x x x
Symmetry: d(x,y) = d(y,x) x x x
Triangle inequality: d(x,z) ≤ d(x,y)+d(y,z) x x x

Table derived from Wikipedia article on metric spaces:

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