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Two operators on interval-valued intuitionistic fuzzy sets. II. (English) Zbl 0834.04008

[For Part I see ibid. 47, No. 12, 9–12 (1994; Zbl 0834.04007).]
The interval-valued intuitionistic fuzzy sets (IVIFSs) [the reviewer and G. Gargov, Fuzzy Sets Syst. 31, No. 3, 343–349 (1989; Zbl 0674.03017)]are extensions of the intuitionistic fuzzy sets (IFSs) [the reviewer, ibid. 20, 87–96 (1986; Zbl 0631.03040)]and interval-valued fuzzy sets (IVFSs). Let \(X\neq \emptyset\) be a given universe. An IVIFS in \(X\) is defined by \(\{\langle x, M_A (x), N_A (x)\rangle \mid x\in X\}\) where \(M_A: X\to D[0,1 ]\), \(N_A: X\to D[0,1 ]\) with the condition \(0\leq \sup M_A (x)+ \sup N_A (x)\leq 1\), for every \(x\in X\), and where \(D[0,1 ]\) is the set of closed subintervals of \([0,1 ]\).
The operator \(H_{p,r}: \text{IVIFS} \to \text{IFS}\) is defined by \[ H_{p,r} (A)= \{\langle x, g_p (M_A (x)), g_r (N_A (x)) \rangle\mid x\in X\}, \] where \(0\leq p, r\leq 1\) and \(g_p ([M_L, M_U ])= M_L+ p\cdot (M_U- M_L)\), and its basic properties are studied.

MSC:

03E72 Theory of fuzzy sets, etc.
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