*Egypt, Port Said Univ. Math. and Computer Sci. Fuzzy Ideals and Bigranular Computing By A. Salama* Hatem Elagamy *Egypt, Port Said Univ. Math. and Computer Sci.

In 1965, Zadeh generalized the concept of a crisp set by introducing the concept of a fuzzy set. Since the concept of of fuzzy sets corresponds to the physical situation in which there is no precisely defined criterion for membership, fuzzy sets has useful and increasing applications in various fields, including probability theory, artificial intelligence, control systems, biology and economic. Thus developments in abstract mathematics using the idea of fuzzy sets possess sound footing. In accordance with this, fuzzy bitopological spaces were introduced by Chang. After the discovery of fuzzy sets, much attention has been paid to the generalization of basic concepts of classical topology to fuzzy sets and thus developing a theory of fuzzy topology Pi is “computable,” because we can program an ideal computer to output the ith digit for any I. In general, any real number r is computable, if we can program an ideal computer to print it digit by digit, such that any digit we are interested in will eventually be printed. For example, to find the 2^50th digit of r, we start the program and count the output digits. Eventually, we’ll the 2^50th digit will appear. The real numbers we’re familiar with -- pi, e, phi, and the square root of two– are all computable. But in general, are all real numbers computable?

Ideal, is one of the most important notion in general bitopology Ideal, is one of the most important notion in general bitopology. A lot of different kinds of ideals have been introduced and studied by many topologists. Throughout a few last years many types of sets via ideals have been defined and studied by a staff of topologists. As a result of these new sorts of sets, topologists used some of them to construct new forms of topological spaces. This helps us to present several types of functions and investigate some operators which join between the above constructed space. In 1997 Sarker, Abd El-Monsef at el. and Salama extended those ideas from general topology to fuzzy bitopological spaces. As a continuation of the study of fuzzy ideals, this thesis, consists of an introductory chapter and other six chapters,

devoted to: i. Generalize the previous studies, so to define the fuzzy ideal bitopological concepts. Many of its main properties have been discussed. ii. Introduce and study some concepts of fuzzy bitopological spaces in terms of fuzzy ideals and deduce many properties of these spaces. iii. Deduce many types of functions and give the relationships between different fuzzy bitopological spaces via fuzzy ideal, that helps to construct new properties of fuzzy bitopological spaces. . The moral here is that computer programs can be viewed as descriptions of their output. A program that computes pi, for example, can be taken as an unambiguous description of the number pi.

Chapter(1) can be considered as a background for basic material used in this thesis. The moral here is that computer programs can be viewed as descriptions of their output. A program that computes pi, for example, can be taken as an unambiguous description of the number pi.

In Chapter(2), further results on fuzzy ideals bitopological space are given. In Sec. 2.1, we ivestigate and study some properties of fuzzy pairwise local function . In Sec. 2.2, we introduce the basic properties of the concept of a fuzzy bitopological ideal space and investigate some fuzzy concepts that help to construct some new forms of generated fuzzy bitopology. In Sec.2.3, the importance of fuzzy bitopological ideals has been investigated. Moreover, we introduce and study the concepts of resolvable, maximally irresolvable and submaximal in fuzzy bitopological spaces via fuzzy ideals. We study the fuzzy bitopology formed from a given bitopology and the fuzzy ideal of nowhere dense fuzzy sets. So, it is natural to wonder if all real numbers can be described in English. <get answers from audience> Expect: someone will say that you can’t describe an indescribable real, so if he asserts that there is no such thing I wouldn’t be able to supply a counterexample. Response: You’re correct that I cannot show you a specific indescribable number as evidence that they exist. Your argument, however, leaves open the possibility that I could convince that indescribable numbers exist, without showing you a specific one.

Fuzzy P local function t1,2 Fuzzy local function Ideal Fuzzy Ideals t [Sarker] Fuzzy local function (Sarker and Abd Elmonsef t* ) Dense-codense –nowheredense… Fuzzy P local function t1,2 Resolvable – maximally irresolvable Fuzzy bitopological space

In Chapter(3), we introduce and study some different fuzzy pairwise bitopological concepts via fuzzy ideals. In Sec. 3.1, we generalize the notation fuzzy pairwise set and its fuzzy pairwise local function. In Sec. 3.2, we generalize the notion of fuzzy L-open sets and fuzzy L-closed sets due to Abd El-Monsef at el. In Sec.3.3, we generalize the concept of fuzzy L-continuity due to Abd El-Monsef at el. Relatonships between the above new fuzzy notations and other relevant classes are investigated. In Sec. 3.4, fuzzy pairwise quasi PL-openness and fuzzy pairwise quasi PL-continuity are considered as a generalization of considered as a fuzzy pairwise PL-openness and fuzzy pairwise PL-continuity which are known before. Relationships between the above new fuzzy pairwise notions and other relevant classes are investigated. Bonzo has a very natural view on the question. We can take the set of natural numbers, cross off all the naturals, and still be left with an infinite number of odd numbers. So, this means that there are many more natural numbers than even numbers. In other words, the correspondence which maps the even number x to the natural number x, is one-to-one: all the even naturals get mapped to distinct natural numbers. But the correspondence is not onto: all the odd naturals are left over.

L-open set Fuzzy P L-open set Ideal Fuzzy Ideals [Sarker] (Abd Elmonsef) Fuzzy P L-open set Fuzzy P L-cont. function Fuzzy quasi P L-cont. function FPL- open F quasi PL-open FPB-open

In Chapter (4), we study the concepts of fuzzy pairwise compactness and separation axioms in terms of fuzzy ideals. In Sec. 4.1 and Sec.4.2, we define and study some new various types of fuzzy pairwise compactness with respect to fuzzy ideals namely fuzzy pairwise PL-compact, fuzzy pairwise PL*-compact spaces. In Sec.4.3 and Sec. 4.4, we introduce and study a new class of fuzzy pairwise compactness in terms of fuzzy ideals namely, fuzzy pairwise quasi PL-compact and Fuzzy pairwise semi P-compact which contains the class of fuzzy pairwise PL-compact space. Bonzo’s intuitive argument is very natural, but it’s not a proof. Odette has a better idea. In fact, the evens and the naturals DO have the same cardinality, and there is a one-to-one onto correspondence from the naturals to the even naturals which proves it. We map 0 to 0, 1 to 2, 2 to 4 and so forth, mapping the natural number x to the even natural 2*x. It’s easy to see that this correspondence is one-to-one – every natural gets mapped to a distinct even, and onto—every even is mapped from some natural.

Fuzzy L-compact Ideal Fuzzy Ideals I-compact [Sarker] (Abd Elmonsef) Fuzzy P L-compact Fuzzy P L* -compact Fuzzy quasi P L-compact Fuzzy P semi L-compact Fuzzy PF-nearly compact Fuzzy PF-almost compact

PFL- compact PF-semi compact PF-compactt PF- nearly compact PF- almost compact PFL- compact

In Chapter (5), we study the concepts of fuzzy pairwise separation axioms in terms of fuzzy ideals. In sec. 5.1, we introduce and study some new fuzzy pairwise separation Axiom’s with respect to fuzzy ideals, namely fuzzy pairwise PF*-. The relation between these axioms and some other fuzzy pairwise separation axioms are studied. Some properties of fuzzy pairwise PF*- spaces are also investigated. The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh. Subsequently, Chang defined the notion of fuzzy bitopological. Since then various aspects of general topology were investigated and carried out in fuzzy sense by several authors of this field. The local properties of a fuzzy bitopological space, which may also be in cretin cases the properties of the whole space, are important field for study in fuzzy bitopological space by introducing the notion of fuzzy ideal and fuzzy local function . In 2001, Abd El-Monsef et al. defined and studied the notion of fuzzy L-open set in fuzzy topological space. In sec.5.2. We generalize the concept of L-Hausdorff space due to Salama, we introduce and characterize the notion of fuzzy pairwise L-Hausdorff spaces which is a generlization of some fuzzy concepts by using a fuzzy pairwise L-open sets. Bonzo’s intuitive argument is very natural, but it’s not a proof. Odette has a better idea. In fact, the evens and the naturals DO have the same cardinality, and there is a one-to-one onto correspondence from the naturals to the even naturals which proves it. We map 0 to 0, 1 to 2, 2 to 4 and so forth, mapping the natural number x to the even natural 2*x. It’s easy to see that this correspondence is one-to-one – every natural gets mapped to a distinct even, and onto—every even is mapped from some natural.

Fuzzy P L-Hausdorff L-Hausdorff (salama) Fuzzy L-separation Hausdorff Fuzzy separation Fuzzy L-separation (Abd Elmonsef) Fuzzy P L-separation Hausdorff L-Hausdorff (salama) Fuzzy P L-Hausdorff

In Chapter (6) In this chapter we introduce and study some applications on fuzzy sets via Matlab program and relationships between different fuzzy concepts. Bonzo’s intuitive argument is very natural, but it’s not a proof. Odette has a better idea. In fact, the evens and the naturals DO have the same cardinality, and there is a one-to-one onto correspondence from the naturals to the even naturals which proves it. We map 0 to 0, 1 to 2, 2 to 4 and so forth, mapping the natural number x to the even natural 2*x. It’s easy to see that this correspondence is one-to-one – every natural gets mapped to a distinct even, and onto—every even is mapped from some natural.

On Fuzzy Sets

a = newfis('tipper'); a = addvar(a,'input','service',[0 10]); a = addmf(a,'input',1,'poor','gaussmf',[1.5 0]); a = addmf(a,'input',1,'good','gaussmf',[1.5 5]); a = addmf(a,'input',1,'excellent','gaussmf',[1.5 10]);plotmf(a,'input',1)

fismat = readfis('tipper');getfis(fismat) returns Name = tipper Type = mamdani NumInputs = 2 InLabels = service food NumOutputs = 1 OutLabels = tip NumRules = 3 AndMethod = min OrMethod = max ImpMethod = min AggMethod = max DefuzzMethod = centroidans =tipper

  لَا يُكَلِّفُ اللَّهُ نَفْسًا إِلَّا وُسْعَهَا لَهَا مَا كَسَبَتْ وَعَلَيْهَا مَا اكْتَسَبَتْ رَبَّنَا لَا تُؤَاخِذْنَا إِنْ نَسِينَا أَوْ أَخْطَأْنَا رَبَّنَا وَلَا تَحْمِلْ عَلَيْنَا إِصْرًا كَمَا حَمَلْتَهُ عَلَى الَّذِينَ مِنْ قَبْلِنَا رَبَّنَا وَلَا تُحَمِّلْنَا مَا لَا طَاقَةَ لَنَا بِهِ وَاعْفُ عَنَّا وَاغْفِرْ لَنَا وَارْحَمْنَا أَنْتَ مَوْلَانَا فَانْصُرْنَا عَلَى الْقَوْمِ الْكَافِرِينَ

Ahmed Salama and Hatem elagamy Oh, Thank You Ahmed Salama and Hatem elagamy