R-subset

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. 

For any fuzzy R subset B, R superset C, maximal mass fuzzy R subset B, and minimal mass fuzzy R superset C of any fuzzy set A, the relations... 

Here fuzzy set B is a maximal R subset, fuzzy set B is a maximal mass R subset, fuzzy set C is a minimal R superset, and fuzzy set C is a minimal mass R superset of fuzzy set A. 

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... 

A very similar approach is applied to maximum R subsets B A) of a crisp or fuzzy set A. 

An interesting property of maximal R subsets is established by a theorem analogous to the theorem for minimal R supersets. 

Theorem. Each maximal R subset B A) of A can be generated as an intersection respect to some version of the symmetry... 

Proof. Since B ( ) is an R subset of A, there must exist at least one center c and a set of defining parameters P for a version of a symmetry element of type R that qualifies 5 ( ) as an R set. Choose the symmetry operation R that corresponds to this version R. For the intersection of the family A,RA,R A,...,R" A of sets generated using this symmetry operator R,... 

If M is a maximal volume R-subset of T, and N is a minimal volume R-superset of T, then the expressions... 

For any R-subset M, R-superset N, maximal volume R-subset M, and minimal volume R-superset N of any set T the relations... 

If M(x, Mr, Not, and Nr are maximal volume achiral subset, maximal volume R-subset, minimal volume achiral superset, and minimal volume R-superset of a set T, respectively, then... 

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. 

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