By Maciej Wygralak
Counting belongs to the main uncomplicated and widespread psychological actions of humans. Its effects are a foundation for coming to a choice in loads of events and dimensions of our lifestyles. This ebook provides a unique method of the complicated and complex case, known as clever counting, within which the gadgets of counting are imprecisely, fuzzily exact.
Formally, this collapses to counting in fuzzy units, interval-valued fuzzy units or I-fuzzy units (Atanassov's intuitionistic fuzzy sets).
The monograph is the 1st one displaying and emphasizing that the offered tools of clever counting are human-consistent: are reflections and formalizations of genuine, human counting systems played below imprecision and, in all probability, incompleteness of data. different purposes of clever counting in a variety of components of clever structures and selection help could be mentioned, too.
The complete presentation is self-contained, systematic, and built with many examples, figures and tables. computing device and data scientists, researchers, engineers and practitioners, utilized mathematicians, and postgraduate scholars attracted to details imprecision are the objective readers.
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18). 3 Main Characteristics of Fuzzy Sets We like to introduce in this section a few auxiliary notions which are useful when analyzing and speaking about fuzzy sets, and which form convenient characteristics of those objects. One very important characteristic, the cardinality of a fuzzy set, will be skipped here. Its study is the subject of Part II. 1 Core, Support, t-Cuts Let A: U 6 [0, 1]. 10). The t-cut set (sharp or not) is thus a set of elements of U whose membership degree in A is sufficiently “large”: is at least equal to t ( t-cut) or is greater than t (sharp t-cut), where t is a given threshold value.
52) ( A 1 AN)( x) dx, if U = [ a, b]. Fuzz1( A) = ~ The next example offers distance-based fuzziness measures. 5. e. d( A, B) 0 [0, 1] for each pair of fuzzy sets. Assume d is such that d( A, AN) = 1 whenever A 0 {0, 1}U and d( B, BN) # d( A, AN) for A #N B. It is a routine task to check that Fuzz( A) = 1! 3. Fuzz( A) now becomes a measure of difficulty in distinguishing between A and its complement AN expressed through distance between A and AN. 54) d p( B, C) = for U = [ a, b] d ú. So, d1( B, C) is just the normed Hamming distance, whereas d2( B, C) collapses to the normed Euclidean distance between B and C.
1 Negations A function ν: [0, 1] 6 [0, 1] is called a negation function or, simply, a negation if ν is nonincreasing, ν(0) = 1 and ν(1) = 0. One says that a negation is strict whenever it is strictly decreasing and continuous. e. ν(ν( a)) = a for each a 0 [0, 1] and, thus, ν B ν = id with id denoting the identity function. 21) νp( a) = ν p( a) = Fig. 59) 46 2 Basic Notions of the Language of Fuzzy Sets Two particular cases of these constructions have to be distinguished, namely the extreme negations ν( = ν0 (known as the Gődel negation) and ν* = ν1 (see the next figure).