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Defects Of Properties In Mathematics: Quantitative Characterizations
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An introduction to a method of research which can be used in various fields of mathematics. It examines, in a systematic way, the quantitative characterizations of the "deviation from a (given) property", called the "defect of a property", in: set theory; topology; measure theory; real, complex and functional analysis; algebra; geometry; number theory; and fuzzy mathematics. Besides well-known "defects", the text introduces and studies new ones, such as: measures of noncompactness for fuzzy sets; fuzzy and intuitionistic entropies; the defect of (sub, super)additivity; complementarity; monotonicity for set functions; the defect of convexity; monotonicity; differentiability for real functions; the defect of equality for inequalities; the defect of orthogonality for sets and defects of properties for linear operators in normed spaces; defects of properties (commutativity, associativity and more) for binary operations; defects of orthogonality and parallelness in Euclidean and non-Euclidean geometries; defects of integer, perfect, prime and amicable numbers; and the defect of tautology in fuzzy logic.
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