Generalizations of Rough Functions in Topological Spaces by Using Pre-Open Sets ()

Amgad S. Salama, Hassan M. Abu-Donia

Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt.

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt.

**DOI: **10.4236/jilsa.2012.42012
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Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt.

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt.

Functions are a means to link or transport from a world to another world may be similarly or completely different from the other world. In this paper we addressed the issue of rough functions and the possibility of transfer it from the real line to the topological abstract view that can be applied to intelligent information systems. The rough function approach has not been studied much specially from a topological point of view. Here we developed a new type of topological generalizations of rough functions with reference to how it is used in medical applications. Considering that the function is in the original a relation can be based on a review of all circular functions from the perspective of relations. Accordingly, the dream that the generalizations of rough functions are transferred to all papers prior to a comprehensive computer application.

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A. Salama and H. Abu-Donia, "Generalizations of Rough Functions in Topological Spaces by Using Pre-Open Sets," *Journal of Intelligent Learning Systems and Applications*, Vol. 4 No. 2, 2012, pp. 127-134. doi: 10.4236/jilsa.2012.42012.

1. Introduction

Rough set theory [1], is an extension of set theory for the study of intelligent systems characterized by inexact, uncertain or insufficient information. Moreover, this theory may serve as a new mathematical tool to soft computing besides fuzzy set theory [2-4], and has been successfully applied in machine learning, information sciences, expert systems, data reduction, and so on. Recently, lots of researchers are interested to generalize this theory in many fields of applications [5-7]. In classical rough set theory, partition or equivalence (indiscernibility) relation is an important and primitive concept. But, partition or equivalence relation is still restrictive for many applications. To study this issue, several interesting and meaningful generalizations to equivalence relation have been proposed in the past, such as tolerance relations [8], topological bases and subbases [9-12]. Particularly, some researchers have used coverings of the universe of discourse for establishing the generalized rough sets by coverings [13]. Others [14-16] combined fuzzy sets with rough sets in a fruitful way by defining rough fuzzy sets and fuzzy rough sets. Furthermore, another group has characterized a measure of roughness of a fuzzy set making use of the concept of rough fuzzy sets [17-19]. They also suggested some possible real world applications of these measures in pattern recognition and image analysis problems. Some results of these generalizations are obtained about rough sets and fuzzy sets in [20-22].

Topological ideas are present not only in almost all areas of today mathematics, for example biochemistry [23] information systems [24] and others for more fields of topology applications [25] and its related links. The subject of topology itself consists of several different branches such as point set topology, algebraic topology and differential topology which have relatively little in common this richness of applications and differentiate between branches of topology implied a difficult to give an accurate definition for topology. The topology concepts like continuity, irresoluteness, compactness, connectedness, convergence, denseness and others are as basic to mathematicians. The topology structure τ on a set X is a general tool for constructing the above concepts. This tool contains many classes of near open sets such as: regular open [26], semi open sets [27], pre-open sets [28], β-open sets [29] and b-open sets [30]. Many authors used the previous types of near open sets to introduce some types of near continuous functions such as: In [28] the concept of pre-continuous functions are introduced. In [31] the concept of α-continuous functions is introduced.

The pair of lower and upper approximation operators is just a pair of interior and closure operators of a topology [32-34]. In [35] the concept of rough functions is introduced. In [35,36] we found the definition of the rough real number. In this paper, we propose to give a further study on rough functions and to introduce some concepts based on rough functions. In the beginning we will study rough sets on the real line.

In Section 2, we will initiate the notion of rough real functions. The aim of Section 3 is to define and study the new notion of “topological pre-rough function”. The main goal of Section 4 is to initiate and study the pre-approximations of a function as a relation. Finally, we aim in Section 5 to define an alternative description of the topological pre-rough functions and topological pre-rough continuity.

A topological space [36] is a pair consisting of a set X and family τ of subsets of X satisfying the following conditions:

1).

2) τ is closed under arbitrary union.

3) τ is closed under finite intersection.

The pair is called a topological space, the elements of X are called points of the space, the subsets of X belonging to are called open set in the space, and the complement of the subsets of X belonging to τ be called closed set in the space; the family τ of open subsets of X is also called a topology for X.

is called τ-closure of a subset.

Evidently, A is the smallest closed subset of X which contains A. Note that A is closed iff.

is called the τ-interior of a subset.

Evidently, is the union of all open subsets of X which containing in A. Note that A is open iff. And is called the τ-boundary of a subset.

Let A be a subset of a topological space. Let, and be closure, interior, and boundary of A respectively. is exact if, otherwise A is rough. It is clear is exact iff.

Definition 1.1: A subset A of a topological space is called pro-open if.

The family of all pre-open sets of is denoted by. The complement of pre-open set is preclosed set. The family of preclosed sets is denoted by.

Definition 1.2: A function is said to be pre-continuous if for every [28].

2. Rough Functions on Real Line

Let be the set of non-negative real numbers, and let be a sequence of real numbers defined by such that. The sequence defines the partition of by, where denote open intervals. The sequence is called a categorization of and the ordered pair is an approximation space, where is the equivalence relation associated with.

Let be an approximation space. By in we denote the block of the partition containing x, in particular if, we have, is the closure of with respect to the usual topology on R. Let be an approximation space, by we denote the closed interval for. For any, the seq-lower and the seq-upper approximations of in the approximation space are defined respectively by

The approximations of the closed interval can be understood as the approximations of the real number which are simply the ends of the interval. The number is a rough number if , otherwise it is an exact number.

Example 2.1: Let be the set of all non-negative real numbers, and let be the set of natural numbers to be a sequence in. Then the partition induced by is and hence, is an approximation space. Also, for any number, we have and for any, and, Then every number is a rough number in A.

According to Example 1, we followed the following steps to get the approximations of a number, say. We remark that the required approximations of can be obtained directly in one step by.

Let X and Y be two subsets of, and let and be two approximation spaces, where and P are equivalence relations on X and Y, respectively, is a function. Then we define -lower approximation of as the function, such that for every, and (S, P) -upper approximation of f as the function, such that, for every.

We see that the term in the above definition can be replaced by P only since the approximations of the function f depends only on P.

Let be a real valued function, where X and Y are two subsets of. The function f is called a rough function at a point if and only if and f is called a rough function on X if it is a rough function at every point.

We give the following example to indicate the above notions.

Example 2.2: Let be a real valued function defined by for every. We denote the odd and even integers by O and E, respectively, then and are approximation spaces, where and are partitions of defined by and, then at every point we define E-lower approximation of f by such that

and the E-upper approximation of f by the function such that. For, we have, then

and. Then f is an exact function at, similarly we can prove that f is an exact functional at every odd natural number.

For, then

But

Then is a rough function at, similarly it can be proved that is a rough function at every even natural number.

Also, we notice that is a rough function at every. Then is a rough function at every point or is an even natural number.

Let be a real valued function. Then is called -continuous (roughly continuous) at a point if, where and are approximation spaces.

Let be a real valued function. Then is roughly continuous on if is a roughly continuous at every point.

Example 2.3: According to Example 2, the function is a rough function at but and, then f is not a rough continuous function at the rough number, but at, since and then f is a roughly continuous at, also at every such that x is odd number f is roughly continuous. If x is an even number, then f is not a roughly continuous; hence f is not a roughly continuous function on.

Example 2.4: Let X and Y be subsets of, such that and and the real valued function be defined by, and, and consider the approximation spaces and, where and we define the function by. Then,

, ,

,.

Also for the function such that

. Then, ,

, ,

. Then the function f is P-rough at and f is not P-rough function at.

Now, if, then and we have and

then, i.e., the function f is -roughly continuous at.

If, then and , but then, the function f is -roughly continuous at. Also at we find that f is -roughly continuous, hence f is -roughly continuous on X.

3. Topological Pre-Rough Functions

We purpose to generalize the concept of rough function to topological pre-rough function by using pre-open sets in topological spaces. Let be a topological space and. Then is called the pre minimal set containing the point with respect to pre-open sets in the topology on X.

The principle topology on a set X is the topology has the minimal bases that consists only of minimal open sets at each.

Theorem 3.1: A topology on a set X is principle iff arbitrary intersections of members of are members of [20].

Let be a principle topological space, for any element, we define pre-sequence by the set and by we mean the pre-closure of in.

If is a function between principle spaces and, we define the functions, by

and for every, and, by for all.

Let be a function, where X and Y are principle spaces. The function f is called a topological pre-rough function at the point x in X if and only if, and f is a topological pre-rough function on X if it is a topological pre-rough function at every point x in X.

Example 3.1: Let and be topological spaces, where,

and,. Let be a map defined by, , and. We have the following table (Table 1).

Consequently, for every, hence f is a topological pre-rough function on.

A function is said to be a topological pre-rough continuous at the point if and only if, and it is a topological pre-rough continuous on if it is a topological pre-rough continuous at every point.

Example 3.2: Let and be topological spaces, where and with and

. Let be a map defined by, , and (Table 2).

Consequently, for every, hence f is a topological pre-rough continuous function on X.

4. The Pre-Approximations of Functions

A function f from X to is a relation from to that satisfies:

1).

2) If and, then.

If, we say is a function on. A function is completely represented by its graph .

The concept of rough relations is defined by using a certain type of relation products. The following proposition

Table 1. and for some subsets of X.

Table 2. topological pre-rough continuous function on X.

will simplify the process of getting via and.

Theorem 4.1: Let and be a pre-approximation spaces. Then we have .

Proof: Since for any and, we have, iff and

. Let. Then we have

Hence.

Let be any function, where and are pre-approximation spaces, such that and are equivalence relations on and respectively. We define the equivalence relation such that

is a partition of for the function we define the pre-approximations

A function is said to be roughly in the pre-approximation space, where

and are pre-approximation spaces and, if

otherwise f is pre-exact function.

Example 4.1: Let and and consider the function defined by

.

Consider the partitions and. Then

is a partition of.

Then and

Therefore the function f is a rough function such that.

For the function, we observe that in general and are not functions from into. We point that, the process of defining an pre-approximations on such that and are functions is an open question to be solved in our next work.

Theorem 4.2: For every function such that and are selective preapproximation spaces then f is an pre-exact function.

Proof: Since in any selective pre-approximation space, then then f is an preexact function.

Example 4.2: Let and. Consider the function, defined by = and consider the partitions

and. Then

is a partition of.

Then and , then f is an pre-exact function.

For a function such that and are selective pre-approximation spaces then 1) If f is a one-to-one function then also both and.

2) If f is onto function then also both and.

3) If f is a pre-continuous function then also both and.

No function such that and are not selective approximation spaces is pre-exact, and f is not a constant function.

5. An Alternative Description of Topological Pre-Rough Functions

Let and be any topological spaces, the function, can be considered as a relation of and if is a basis of and is a basis of, then is a basis of the topology on. In the topology

we define and for the function f. Let be a function, where and, are topological spaces, the function f is called a topological pre-rough function in iff otherwise, f is an preexact function in.

Example 5.1: Let and be any topological spaces where,

, ,

Consider and are basis of and respectively. Let, and are mappings defined by

,

and.

Then and

Then f is a pre-rough function in. Also, and

We call g is a function not defined from pre-lower and from upper. Finally, for the constant function h, we have, and h is an pre-exact function. In fact, h is the only exact function in.

According to Example 1, we have the following:

1) The function f is continuous, but and are not functioning, hence we cannot say that or f is pre-continuous.

2) The function h is always precontinuous function, and it is an pre-exact function, hence and is pre-continuous functions.

6. Experiments and Evaluations

This section shows the effectiveness of using pre-rough functions for extracting new data from multi-valued information systems.

In this section, we briefly describe the Rheumatic Fever datasets mentioned in [37] as a topological application of rough functions. As mentioned in [39] rheumatic fever is a very common disease and it has many symptoms differs from patient to another though the diagnosis is the same. So, we obtained the following example on four rheumatic fever patients. All patients are between 9-12 years old with a history of Arthurian began from age 3-5 years. This disease has many symptoms and it is usually started in young age and still with the patient along his life.

Table 3 in [37] introduced the seven patients characterized by 8 symptoms (attributes) using them to decide the diagnosis for each patient (decision attribute). Where the attributes are satisfied in Table 2 in [37].

We recall and sell it here Table 3.

If we defined the following mapping on Table 3:

:

,

From the relation where a is an element of the power set of the set of condition attributes. The the following classes and are two subbases of two topologies on U such that . Then according to Table 3 we have the following couples of topologies:

,

According to the mapping and using each one of the above topologies we can deduce that the decision topology can be given by:

.

Now we can construct a familiar system of Table 3 contains only the pre-rough images constructed using the terminology of pre-rough functions. This system can be the reduction system of Table 3 and it given in Table 4.

This means that we can remove the conditional attribute without any loss of information.

7. Conclusions

We conclude that the emergence of topology and its operators [38,39] in the construction of some rough set concepts will help to get rich results that yields a lot of logical statements which discover hidden relations between data and moreover, probably help in producing

Table 3. Multi-valued information system of [37].

Table 4. Reduced System.

accurate programs. These topological operators will play an essential role in data mining and knowledge discovery in databases. In this paper, we give an overview of several dissipated results on the pre-rough functions. More specifically, we attempt to show: usefulness of this new concept in a calculus of rough functions.

The future application of this work will be useful in many fields such as Fuzzy Expert Systems [40] by generalizations of rough functions for fuzzy rough functions. It also is useful in knowledge discovery methods [41].

Conflicts of Interest

The authors declare no conflicts of interest.

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