Some operations on intuitionistic fuzzy sets
Let \(X\) be a nonempty fixed set (universe). An Intuitionistic Fuzzy Set (IFS) \(A\) is an object having the form \[ A=\biggl\{ \bigl\langle x, \mu_A(x), \nu_A(x) \bigr\rangle:x\in X\biggr\} \] where the functions \(\mu_A: X\to [0,1]\) and \(\nu_A: X\to[0,1]\) denote the degrees of membership and of non-membership of each element \(x\in X\) to set \(A\), resp., and \(0\leq\mu_A (x)+\nu_A (x)\leq 1\) for each \(x\in X\) [see \textit{K. T. Atanassov}, ``Intuitionistic fuzzy sets'', Fuzzy Sets Syst. 20, 87-96 (1986; Zbl 0631.03040); Intuitionistic fuzzy sets. Theory and applications. Physica-Verlag, Heidelberg (1999; Zbl 0939.03057)]. There are a lot of operations defined over the IFSs. In the present paper some new operations are defined, e.g., for any positive integer \(n\): \[ A^n=\Bigl\{ \biggl\langle x,\bigl(\mu_A (x)\bigr)^n,1- \bigl(1- \nu_A(x) \bigr)^n\biggr \rangle:x \in X\Bigr\}, \] \[ nA= \Bigl\{\biggl \langle x,1-\bigl(1 -\mu_A(x) \bigr)^n, \bigl(\nu_A(x) \bigr)^n\biggr \rangle:x \in X\Bigr\}. \] The basic properties of these operations and relations between them and the already existing operations and modal type operators are studied. For a fixed IFS \(A\), the IFSs \(\text{CON}(A)\) and \(\text{DIL}(A)\) are defined as follows: \[ \text{CON}(A)= \biggl\{\bigl\langle x, \mu_{\text{CON}_{(A)}} (x),\nu_{\text{CON}_{(A)}} (x)\bigr\rangle: x\in X\biggr\}, \] where \(\mu_{\text{CON}_{(A)}} (x)=(\mu_A (x))^2\), \(\nu_{\text{CON}_{(A)}} (x)=1-(1- \nu_A(x))^2\); \[ \text{DIL}(A)= \biggl\{\bigl \langle x,\mu_{\text{DIL}_{(A)}} (x),\nu_{\text{DIL}_{(A)}} (x)\bigr\rangle: x\in X\biggr\}, \] where \(\mu_{\text{DIL}_{(A)}} (x)=(\mu_A (x))^{1\over 2}\), \(\nu_{\text{DIL}_{(A)}} (x)=1-(1-\nu_A (x))^{1 \over 2}\). Basic properties, interpretations and examples of these sets are given.
scientific article; zbMATH DE number 1517325