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