Abstract

We intend to study a new class of algebraic approximations, called -approximations, and their properties. We have shown that -approximations can be used for applied problems which cannot be modeled by inclusion based approximations. Also, in this work, we studied a subclass of -approximations, called -approximations, and showed that this subclass preserves most of the properties of inclusion based approximations but is not necessarily inclusionbased. The paper concludes by studying some basic operations on -approximations and counting the number of -min functions.

1. Introduction

Uncertainty is often present in real-life applications. Uncertainty in noncrisp sets is characterized by nonempty boundary regions, in which nothing can be said about their elements with certainty. In classical set theory, a subset of a universe induces a partition over that universe. This partition can be interpreted as a knowledge on elements of ; that is, elements in are indiscernible to each other and also the same thing holds for items in . This knowledge may be improved to another partition, for example, , whose items in each partition are indiscernible to each other. In consequence, for a subset of , the problem of whether belongs to or not, with respect to knowledge , may become undecidable; that is, some elements indiscernible to with respect to knowledge may be in , whereas some other indiscernible elements to with knowledge may not belong to . To cope with such uncertainty, some tools were invented such as the Dempster-Shafer theory of evidence [1], theory of fuzzy sets [2–5], and theory of rough sets [6–8]. Rough set theory and fuzzy set theory are two independent approaches for uncertainty. There is a connection between rough set theory and Dempster-Shafer theory. Strictly speaking, lower and upper approximations of rough set theory correspond to the inner and outer reductions from Dempster-Shafer theory [9].

Rough set theory and its generalizations are all based on the inclusion relation [7, 8, 10–15], which is a limitation in approximations. In this work, we introduce a new concept named -approximation set. This concept is independent of inclusion relation and contains rough sets and their generalizations as special cases. We provide some examples of approximations using this new concept, which cannot be obtained by rough set theory.

This paper is organized as follows. The notion of -approximation sets is proposed in Section 2, followed by considering some operations on them. The definition of conditioned rough sets is proposed in Section 3 and the number of such sets is counted. Then we conclude the paper.

2. -Approximation

In this section, with regard to Dempster’s multivalued mappings [16], we propose a new mathematical approach to study approximation spaces and we will show that this concept can be independent of inclusion relations and the rough set and its generalizations are all special cases of this concept.

Definition 1. An -approximation is the quadruple , where and are finite nonempty sets, is a mapping of the form , and is a mapping of the form .
For a nonempty subset of , the upper and lower approximations of are defined as follows: where is the complement of with respect to .

Example 2. Let and be nonempty finite sets and let be a relation from to . We define , where and are defined as The upper approximation of with respect to , , is equal to , the upper approximation of set with respect to in rough set, since Similarly, we can show that is equal to , which is the lower approximation of set with respect to in rough set.

Example 3. The pair , where is a finite nonempty set of vertices and is a collection of subsets of , is called a simple hypergraph if these two conditions hold:(i),(ii) implies .Let us define the -approximation such that , , and is the inclusion function.
A subset of is called a transversal of if and only if for each . It is easy to observe that is a transversal of if and only if .

Remark. For the sake of simplicity, we will use and instead of and , respectively.

2.1. Operations on -Approximation

Definition 4. Suppose is an -approximation. One defines as the complement of .

Proposition 5. Let be an -approximation and its complement. Also suppose that . Then one has

Definition 6. Let be a finite nonempty set. Consider the following: and , . Then one defines the following: where and are arbitrary subsets of .

Definition 7. Let and be two -approximations. One defines and as respectively.

Proposition 8. Let , , and be -approximations. Then Similarly, for , it can be shown that the following relations hold:

3. -Approximations

In this section, we introduce and discuss a condition on which is sufficient for the properties in Proposition 13 to hold. These properties are sometimes vital for many applications.

In the next example we illustrate the fact that it is not necessary for to be the inclusion function in order to satisfy the properties stated in Proposition 13.

Example 9. For arbitrary subsets , , define the following: It can be verified that this function is not the same as the inclusion function, but the properties of Proposition 13 hold for the -approximation with arbitrary chosen , , and .

The reason why in the above example satisfies the properties of Proposition 13 is that its function meets the -min condition introduced below.

Definition 10 (-min condition). Let be an -approximation. One says is a function in class if it satisfies for arbitrary nonempty subsets , , and of . One says an -approximation is an -approximation if belongs to the class.

Remark 11. The inclusion function does belong to the class but there are other noninclusion functions in this class as well.

Example 12. For arbitrary subsets , , and of , define the following: It is easy to check that is a noninclusion member of the class.

Proposition 13. Let be an -approximation. For all , and , the following hold: (1) implies that, for all , , (2), (3), (4), (5) implies that , (6) implies that , (7), (8), (9), (10).

Proof. (1) For the first property, note that so . Hence, for all , we have , which implies that .
(2) For this property, we have which implies that By a similar argument for , it can be shown that By combining these inequalities,
(3) Consider the following:
(4) Consider the following:
(5) Consider the following: which implies that .
(6) Consider the following:
(7) Consider the following:
(8) Consider the following: The proof of properties and is entirely straightforward.

In the next example, we show that, in -rough sets, it is not always the case that , although this property always holds in Pawlak’s rough sets.

Example 14. Suppose is an -approximation, where , and .
In this case , while so .

3.1. Cardinality of the Class

Definition 15. Let be a nonempty finite set. A function is said to be minimizing if, for each ,

Lemma 16. Let be a minimizing function. For each , , if , then .

Proof. Since , , so by definition which implies that .

Lemma 17. Let be an -approximation and let . One labels the nonempty subsets of as . Then there exist minimizing functions of the form such that, for every and , .

Proof. It is straightforward.

Lemma 17 leads us towards counting the number and finding the structure of minimizing s.

Definition 18. Let be a minimizing function. A nonempty subset of the set is called an atom of if and only if and, for each proper nonempty subset of such as , .

Proposition 19. Let be a minimizing function and and two nonidentical atoms of . Then .

Proof. Let ; then, since and are nonidentical atoms, and . Suppose that . By Definition 15, , so is a proper subset of and and which is a contradiction with Definition 18.

Proposition 20. Let be a minimizing function and the set of all atoms of . Then, for a subset of , if and only if there exists such that .

Proof. It is obvious that if , then at least one of the subsets of is an atom. On the other hand, let be an atom of ; then, by Lemma 16, .

Proposition 21. Let be a minimizing function, the set of all atoms of , and . Then, for each , is an atom of .

Proof. Let and be two different atoms of . Define and ; then, by Proposition 20, and, by Proposition 19, . is minimizing so ; hence is an atom of .

By previous propositions, it is clear that we have either no atoms, exactly one atom, or an atom per element.

Proposition 22. Define as the set of all minimizing functions of the form , where . Then the total number of elements in is equal to .

Proof. To count the number of elements in , we consider these three cases for each . (i) does not have any atoms: in this case, is determined uniquely. ( for each nonempty .)(ii)Each unary subset of forms an atom: in this case, is determined uniquely too. ( for each nonempty .)(iii)There is exactly one atom: in this case, we can choose different atoms (excluding ) giving us different s. In case , this case is a repetition of the previous case.

Proposition 23. Let be a nonempty finite set of size . The total number of different functions which belong to the -min class is equal to .

Proof. This number can be obtained using the multiplication principle, Lemma 17, and Proposition 22.

4. Conclusion

In this paper, we proposed a new class of algebraic approximation, called -approximation sets. Corresponding to the problem under consideration, we can define the elements of -approximation set for obtaining the approximations. Moreover, we investigated the properties of a subclass of -approximation sets, -approximation sets. We have shown that this subclass preserves most of the properties of inclusion based approximations but is not necessarily inclusion based. Finally, we have considered some basic operations on -approximation sets and counted the number of functions which have the property.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.