(4.1) Let E,F be nonempty subsets of R, E is compact and F is closed. Then there exist (e, f) ∈ E ×F such that d(E,F ) = d(e, f). proof Let α = d(E,F ) = inf(x,y)∈E×F d(x, y) be the distance between the two sets. Let > 0, and let x0 be a given element of E. Since E is compact, it is in particular bounded, and there exists r > 0 such that E ⊂ B(r, x0). Now consider the closed ball B(r + α+ , x0)...

We discuss some optimization problems for fuzzy variable functions and show two interesting results. First result is related to conditions for existence of global optimal solutions for some general class of optimization problems on E, the space of compacts, upper semicontinuous and normal fuzzy sets of R. The second result is about preservation of local optimality points, via Zadeh’s principle ...

We present new tools for fuzzy arithmetic with fuzzy numbers, based on the parametric representation of fuzzy numbers and new fuzzy operations, the generalized difference and the generalized division of fuzzy numbers. The new operations are described in terms of the parametric LR and LU representations of fuzzy numbers and the corresponding algorithms are described. An application to the soluti...

In this paper, we apply the concept of fuzzy set to ideals and closed ideals in B-algebras. The notion of a fuzzy closed ideal of a B-algebra is introduced, and some related properties are investigated. Also, the product of fuzzy B-algebra is investigated. Keywords—B-algebras, fuzzy sets, fuzzy ideals, homomorphism, fuzzy closed ideals, homomorphism, equivalence relation, level cut, product of ...

The purpose of this paper is to define and study some new types hesitant fuzzy open sets namely, α -open, preopen, semiopen, b-open β -open in topological space. Some properties the relationships between these are investigated. Furthermore, them subspace introduced.