## Cardinalities of Fuzzy SetsCounting is one of the basic elementary mathematical activities. It comes with two complementary aspects: to determine the number of elements of a set - and to create an ordering between the objects of counting just by counting them over. For finite sets of objects these two aspects are realized by the same type of num bers: the natural numbers. That these complementary aspects of the counting pro cess may need different kinds of numbers becomes apparent if one extends the process of counting to infinite sets. As general tools to determine numbers of elements the cardinals have been created in set theory, and set theorists have in parallel created the ordinals to count over any set of objects. For both types of numbers it is not only counting they are used for, it is also the strongly related process of calculation - especially addition and, derived from it, multiplication and even exponentiation - which is based upon these numbers. For fuzzy sets the idea of counting, in both aspects, looses its naive foundation: because it is to a large extent founded upon of the idea that there is a clear distinc tion between those objects which have to be counted - and those ones which have to be neglected for the particular counting process. |

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### Contents

Triangular Operations and Negations | 1 |

11 Triangular Norms and Conorms | 2 |

12 Negations | 4 |

13 Associated TrianguIar Operations | 5 |

14 Archimedean Triangular Operations | 8 |

15 Induced Negations and Complementary Triangular Operations | 14 |

16 Implications Induced by Triangular Norms | 19 |

Fuzzy Sets | 23 |

411 The Corresponding Equipotency Relation | 70 |

412 Inequalities | 76 |

413 Arithmetical Operations | 84 |

4132 Subtraction | 97 |

4133 Multiplication | 98 |

4134 Division | 113 |

42 Generalized FLCounts | 124 |

421 Equipotencies and Inequalities | 126 |

22 Operations on Fuzzy Sets | 27 |

23 Generalized Operations | 29 |

24 Other Elements of the Language of Fuzzy Sets | 31 |

25 Towards Cardinalities of Fuzzy Sets | 34 |

Scalar Cardinalities of Fuzzy Sets | 45 |

32 Cardinality Patterns | 48 |

33 Valuation Property and Subadditivity | 53 |

34 Cartesian Product Rule and Complementarity | 56 |

35 On the Fulfilment of a Group of the Properties | 60 |

Generalized Cardinals with Triangular Norms | 67 |

422 Addition and Other Arithmetical Operations | 131 |

43 Generalized FECounts | 143 |

431 The Height of a Generalized FECount | 147 |

432 Singular Fuzzy Sets | 152 |

433 Equipotencies Inequalities and Arithmetical Questions | 164 |

List of Symbols | 181 |

185 | |

193 | |

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### Common terms and phrases

AeFCS AeFFS analogous Assume automorphism BeFFS bijection binary operations cancellation law cardinal number cardinalities of fuzzy cardinality pattern cartesian product rule CeFFS classical-like collapses complementarity rule completes the proof condition Consequently core(A Corollary 4.4 counterexample defined definition denote elements equal equipotency relation equivalence Example extension principle f-cut sets fceN FGCounts finite finite sets formulated fulfilled function fuzziness measure GOTTWALD Hence implies inequality instance isomorphic Let teAtn Lukasiewicz many-valued logic means membership values Moreover multiset nonincreasing nonnegative integers nonsingular nonstrict Archimedean t-norms notation ordinal sum PeGFG PeGFL,v respectively satisfied scalar cardinalities Section 2.5 sentential calculus sequence sigma count singular fuzzy set strict negation strict t-norms strong negation subadditivity property Subsection supp supp(A t-conorm t-norm teStn t-operations teStn Theorem 1.1 thesis tion triangular norm Triangular Operations unique valuation property veSng virtue of Theorem whereas WYGRALAK yeGFG ZADEH zero divisors