A characterisation of bases of triangular fuzzy sets

Fuzzy sets featuring in applications to fuzzy control systems are often required to satisfy specific conditions such as, e.g., convexity or normality. In the same connection, a widespread choice is to work with fuzzy sets whose graphs have triangular shape. The purpose of this paper is to show that the former conditions may be regarded as attempts at approximating the latter choice. Specifically, as our main result we prove that a reasonable set of such conditions suffices to characterise families of triangular fuzzy sets. A second result provides an additional characterisation of such families in terms of properties of the curve that they parametrise.