By M. Wygralak
In fresh years, an impetuous improvement of recent, unconventional theories, equipment, thoughts and applied sciences in computing device and knowledge sciences, platforms research, decision-making and keep an eye on, professional platforms, information modelling, engineering, and so on. , led to a substantial raise of curiosity in enough mathematical description and research of gadgets, phenomena, and tactics that are imprecise or vague by means of their very nature. Classical two-valued common sense and the comparable thought of a collection, including its mathematical results, are then usually insufficient or inadequate formal instruments, and will even develop into dead for purposes due to their (too) express personality: 'true - false', 'belongs - doesn't belong', 'is - is not', 'black - white', '0 - 1', and so on. the reason for this is that one replaces classical good judgment by means of quite a few kinds of many-valued logics and, nonetheless, extra common notions are brought rather than or beside that of a collection. allow us to point out, for example, fuzzy units and by-product suggestions, flou units and twofold fuzzy units, that have been created for various reasons in addition to utilizing detailed formal and casual motivations. a type of numerical details touching on of 'how many' components these gadgets are composed appears to be like one of many easiest and extra very important sorts of information regarding them. To get it, one wishes an appropriate concept of cardinality and, in addition, feasible to calculate with such cardinalities. regrettably, neither fuzzy units nor the opposite nonclassical techniques were outfitted with a passable (nonclassical) cardinality theory.