Boolean Matrices and Binary Relations

Abstract

Considering the three types of information-driven tasks where graded membership plays a role: classification and data analysis, decision-making problems, and approximate reasoning, Dubois gave the corresponding semantics of the membership grades, expressed in terms of similarity, preference, and uncertainty [1]. For a fuzzy concept ξ in the universe of discourse X, by comparison of the graded membership (Dubois interpretation of membership degree), an “empirical relational membership structure ” $\left<X, R_{\xi } \right>$ is induced [5,6], where R _ξ  ⊆ X×X is a binary relation on X, (x, y) ∈ R _ξ if and only if an observer, an expert, judges that “x belongs to ξ at some extent and the degree of x belonging to ξ is at least as large as that of y. The fundamental measurement of the gradual-set membership function can be formulated as the construction of homomorphisms from an “empirical relational membership structure”, $\left<X, R_{\xi }\right>$, to a “numerical relational membership structure”, $\left<{\{} \mu _{\xi }(x) \ \vert \ x \in X {\}}, \le \right>$.