Fuzzy Sets

  • Jaroslav Ramík
  • Milan Vlach
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 41)


A well known fact in the theory of sets is that properties of subsets of a given set X and their mutual relations can be studied by means of their characteristic functions, see, e.g., [11], [32] and [57]. While this may be advantageous in some contexts, we should notice that the notion of a characteristic function is more complex than the notion of a subset. Indeed, the characteristic function χA of a subset A of X is defined by [EQ139-1] Since χA is a function we need not only the underlying set X and its subset A but also one additional set, in this case the set {0, 1} or any other two-element set. Moreover, we also need the notion of Cartesian product because functions are specially structured binary relations, in this case special subsets of X × {0, 1}.


Membership Function Characteristic Function Fuzzy Number Binary Relation Fuzzy Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Jaroslav Ramík
    • 1
  • Milan Vlach
    • 2
    • 3
  1. 1.School of Business AdministrationSilesian UniversityKarvináCzech Republic
  2. 2.School of Information ScienceJapan Advanced Institute of Science and TechnologyIshikawaJapan
  3. 3.Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

Personalised recommendations