Uncertainty Measure and Reduction in Intuitionistic Fuzzy Covering Approximation Space Feng Tao Mi Ju-Sheng

Basic definitions of IF covering Uncertainty measure of IFSs in an induced IF covering approximation space Reduction of IF covering based approximation space Knowledge Reduction of IF Covering Decision System based on entropy Conclusions

where the functions and denote the degree of membership(namely ) and the degree of nonmembership (namely ) of each element to the set, respectively, and for each. The family of all IF subsets of is denoted by. 1. Basic definitions of IF covering Definition 1 Let be a nonempty and finite fixed set. denotes the family of all crisp subsets of. An IFS is an object having the form Let is the complement set of. is a constant IFS.

Definition 2 Let be a nonempty and finite set called the universe of discourse. If a family of IFSs satisfies the conditions that and,, then is called an IF covering of. If, is an IFS, and, then we call covers.

2. Uncertainty measure of IFSs in an induced IF covering approximation space Definition 3 Suppose is a finite and nonempty universe of discourse, and is an IF covering of. For every, let, then is another IF covering of, which is called the induced IF covering of based on. And for every, if, then. Thus if and, that is,. is the set of all most basic granules and we use to define the IF covering upper and lower approximation operators.

Definition 4 Let be a finite and nonempty universe of discourse, an IF covering of. For any, the induced IF covering upper and lower approximations of w.r.t., denoted by and, are two IFSs and are defined, respectively, as follows:,. Where, ;,. are referred to as induced IF covering upper and lower rough approximation operators w.r.t., respectively. The pair is called an induced IF covering approximation space., if, then is called inner definable; if, then is outer definable. is definable iff.

Theorem 1 Let be an induced IF covering approximation space. Then the induced IF covering upper and lower rough approximation operators w.r.t. satisfy the following properties:, with,

Theorem 2 Suppose is an induced IF covering approximation space, and. (1)If is inner definable, then, and. (2) If is outer definable, then, and. (3) If is definable, then, and. Theorem 3 Suppose is an induced IF covering approximation space. Then is definable iff and whenever and.

Definition 5 Let be a given finite and nonempty universe of discourse,. A real function is referred to as an entropy on, if it satisfies the following properties: (1) iff ; (2) iff, ; (3) When,, if,, or when, if and, then ; (4), is called a intuitionistic fuzzy entropy (IFE, for short) of. For each IFS, let, then E(A) is an entropy on.

Definition 6 Let be an IF covering of, can be considered as the outcomes of an experiment. The information gained by performing the experiment is the expectation. (1) Where denotes the element number of. is referred to as the entropy of the IF covering.

Let and be two IF coverings of,, we define then, is an IFS.

Definition 7 Let and be two IF coverings of. The conditional entropy of given is defined by. (2) measures the uncertainty about the outcomes of the experiment associated with the IF covering given the outcomes of the experiment represented by. Proposition 1 Suppose be a finite and nonempty universe, and are two IF coverings of. Then

Definition 8 Let be a finite and nonempty universe, an IF covering of, for every,, define as:, Where,. is called the degree of membership of w.r.t. based on an IF covering. is called the degree of non-membership of w.r.t. based on an IF covering. is still a IFS.

(1) If is finer than, then and, Proposition 2 Let be a finite and nonempty universe, two IF covering of., (2) If and, then and (3),, (4) If or,,, (5),. Where

Definition 9 Suppose is a finite nonempty universe of discourse, is a IF covering of, is the cardinality of,, define, (3) then is called the IF entropy of with respect to.

Proposition 3 Let be an IF covering of a universe of discourse,, Theorem 4 Suppose is a finite and nonempty universe of discourse, is an IF covering of., if and is definable, then.

3. Reduction of IF covering based approximation space Definition 10 Suppose is a finite and nonempty universe of discourse. Let be an IF covering of, and. If is a union of some IFSs in, then is called a reducible element of, otherwise is an irreducible element of. And if every element of is an irreducible element, we say that is irreducible; otherwise is reducible.( Similarly to Prof. Zhu) Proposition 4 Let be an IF covering of. (1) If is a reducible element of, then is also an IF covering of. (2) is a reducible element of, and, then is a reducible element of if and only if is a reducible element of.

Definition 11 For an IF covering of a universe, the new irreducible IF covering through deleting all reducible elements is called the reduction of IF covering, denoted by. Proposition 5 Let be a finite and nonempty universe of discourse, an IF covering of. If be a reducible element of,then (1), for every. (2), for every. (3),. (4),.

Definition 12 Let be a finite and nonempty universe of discourse, a finite IF covering of, and, then is a superfluous element of. If is an IF covering of satisfying and and none of element in is superfluous, then is called the IF approximation reduction of, denoted by. Theorem 5 Let be a finite and nonempty universe of disco- urse, and a finite IF covering of. Then is a superfluous element of iff, for all. Proposition 6 Let be a finite and nonempty universe of dis- course, a finite IF covering of, and be a reducible element of, then is a superfluous element of.

Proposition 7 Suppose is a finite IF covering of, is a superfluous element, then satisfies one of the following condition: (1) or,. (2),, satisfies, and.

Algorithm 1 Suppose is a finite and nonempty set, is a finite IF covering of. Step 1. For every compute and. end for; Let. Step 2. For i from 1 to r if or, then ; end if; end for; Step 3. Output S.  An algorithm for computing a reduction of an IF covering based approximation space

4. Knowledge Reduction of Consistent IF Covering Decision System based on entropy In an induced IF covering approximation space,, the positive region of w.r.t. is computed by the following formula:. If is an IF covering as an IF decision, then is an IF covering decision system. Hence If for every,, then decision system is called a consistent IF covering decision system. In this section, we let.

Definition 13 Let be a consistent IF covering decision system. For, if for all,, then is called dispensable about in, otherwise is called indispensable. For every IF subcovering, if,, then is a consistent set w.r.t., if every element in is indispensable, i.e., for every,,, then is called an independent about. If is consistent and independent subset, then is a reduction of about. The collection of all the indispensable elements in is called the core of about, denoted by. For every IF subcovering, the conditional entropy of to is

Theorem 6 Let be a consistent IF covering decision system. For every, is dispensable about in iff. Theorem 7 Let be a consistent IF covering decision system and. is a reduction about in if and only if (1) ; (2) is independent about.

Definition 14 Let be a consistent IF cov- ering decision system. For every, is a IF subcovering. We define the significance of the IFS w.r.t. in by Theorem 8 Let be a consistent IF covering decision system. For every, is indispensable about in if and only if. Theorem 9. Proposition 8 Let be a consistent IF cov- ering decision system,,, if, then,.

 Algorithm 2 Reduction of a consistence IF covering decision system Step 1. Let ; Step 2. For each calculate ; if, then. end if ; end for ; Let. Step 3. While, for each calculate ; end for; select, such that, let. end while; Step 4. Output.

5.Conclusions In this paper, we proposed a pair of new definitions of induced IF upper and lower approximation based on an IF covering and discussed their properties. Then, we defined an uncertainty measure of IF sets in an induced IF covering approximation space. The reduction of an IF covering was also studied. Finally, we discuss the reduction of IF covering decision systems using condition entropy. In the future, we will pay more attention to the study of uncertainty in IF covering environments.

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