Cheat sheet

Non-strict orders: ≤ The symbol ≤ denotes a generalization of “less than or equal”, and it defines either a partial or total ordering over a set P (in the table below a,b ∈ P): Constraint (Non-strict) partial order (Non-strict) total order Reflexivity: a ≤ a x x Antisymmetry: if a ≤ b and b ≤ […]

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