R-deficient subset

A fuzzy set B is a maximal mass R-deficient subset of fuzzy set if B is an R-deficient fuzzy set, B c A, and if for all maximal R-deficient fuzzy subsets B of fuzzy set A, the relation m(B ) < m(B) holds. Whereas the fuzzy, maximal mass B-deficient subset B is not necessarily unique for a given fuzzy set A, nevertheless, the total mass m(B) is a unique number for each fuzzy set A. 

A fuzzy set B is called an R-deficient set if B has none of the point symmetry elements of family R. However, by analogy with the case of crisp sets, it takes only infinitesimal distortions to lose a given symmetry element. Consequently, unless further restrictions are applied, the total mass difference between a fuzzy set of a specified symmetry and another fuzzy set that does not have this symmetry can be infinitesimal. As a result, i -deficient fuzzy sets and fuzzy R sets can be almost identical. Nevertheless, the actual symmetry deficiencies of fuzzy continua, such as formal molecular bodies represented by fuzzy clouds of electron densities, can be defined in terms of the deviations from their maximal R subsets and minimal R supersets, defined in subsequent text. 

Alternative fuzzy symmetry deficiency measures are defined in terms of a maximal R subset B, maximal mass R subset B, minimal R superset C, and minimal mass R superset C, of fuzzy set A, discussed in Section IV. If fuzzy set D denotes any one of the R subsets B,B or R supersets C,C of fuzzy set A, D G B, B, C,C, then a fuzzy symmetry deficiency measure... 

The concepts of maximal mass R-subset and minimal mass R-superset provide numerical measures for the evaluation of the degree of symmetry deficiency of various electron density contributions of functional groups. 

The relations involving a fuzzy set A, a maximal mass fuzzy R-subset B of fuzzy set A, and a minimal mass fuzzy R-superset C of fuzzy set A define various measures for symmetry deficiency. 

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