Set Theory and its Applications

Set Theory and its Applications

1. Sajid IRan Khan ( UW-12-BSc-EE-49) Atif Hafeez ( UW-12-BSc-EE-09) Arslan Haider ( UW-12-BSc-EE-43) Sheryar ( UW-12-BSc-EE-47) Kamran Haider (UW-12-BSc-EE-81) Mazhar IQbal ( UW-12-BSc-EE-101)

2. Set Theory And its Application

3. January 27, 2015 Set Theory Branch of mathematics that deals with the properties of sets.

4. Set theory was founded by a single paper in 1874 by Georg Cantor On a Characteristic Property of All Real Algebraic Numbers“ Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Modern understanding of infinity began in 1867–71, with Cantor's work on number theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper . Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation . Brief history of Set Theory January 27, 2015

5. History of Set Theory January 27, 2015 when it was discovered that Cantorian set theory gave rise to several contradictions antinomies or paradoxes Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves“ which leads to a contradiction since it must be a member of itself, and not a member of itself . In 1899 Cantor had himself posed the question "What is the cardinal numberof the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics

6. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory . The work of analysts such as Henri Lebesguede monstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics Set theory is commonly used as a foundational system, although in some areas category theory is thought to be a preferred foundation . History January 27, 2015

7. Sets : A Set is any well defined collection of “objects.” The elements of a set are the objects in a set. Usually we denote sets with upper-case letters, elements with lower-case letters. The following notation is used to show set membership x ∈A means that x is a member of the set A x ∈A means that x is not a member of the set A. Ways of Describing Sets: List the elements A={1,2,3,4,5,6} Give a verbal description “ A is the set of all integers from 1 to 6, inclusive” Give a mathematical inclusion rule   Set (Basic of Set Theory) January 27, 2015

8. There are different types of sets in set theory. They are listed below: Empty set Singleton set Finite and Infinite set Union of sets Intersection of sets Difference of sets Subset of a set Disjoint sets Equality of two sets Different Types of Sets January 27, 2015

9. Empty Set A set that has no element should be called as Empty set. Another name for Empty set could be Null set and Void set. Number of element in set X is represented as n(X). The empty set is denoted as Φ. Thus, n(Φ) = 0. The cardinality of an empty set is zero since it has no element. Singleton Set A set that has one and only one element should be called as Singleton set. Sometimes, it is known as unit set. The cardinality of singleton is one. If A is a singleton, then we can express it as A = {x : x = A} Example: Set A = {5} is a singleton set. January 27, 2015

10. Finite and Infinite Set: A set that has predetermined number of elements or finite number of elements are said to be Finite set. Like {1 ,2, 3, 4, 5, 6} is a finite set whose cardinality is 6, since it has 6 elements. Otherwise, it is called as infinite set. It may be uncountable or countable. The union of some infinite sets are infinite and the power set of any infinite set is infinite. January 27, 2015

11. Union of Sets: Union of two or else most numbers of sets could be the set of all elements that belongs to every element of all sets. In the union set of two sets, every element is written only once even if they belong to both the sets. This is denoted as ‘∪’. If we have sets A and B, then the union of these two is A U B and called as A union B. Mathematically, we can denote it as A U B = {x : x ∈ A or x∈ B} The union of two sets is always commutative i.e.A U B = B U A. Intersection of Sets: It should be the set of elements that are common in both the sets. Intersection is similar to grouping up the common elements. The symbol should be denoted as ‘∩’. If A and B are two sets, then the intersection is denoted as A ∩ B and called as A intersection B and mathematically, we can write it as A ∩ B ={ x : x ∈ A ∧ x ∈ B }  Difference of Sets: The difference of set A to B should be denoted as A - B. That is, the set of element that are in set A not in set B is A - B = {x: x ∈ A and x ∉ B} And, B - A is the set of all elements of the set B which are in B but not in A i.e. B - A = {x: x ∈ B and x ∉ A}. January 27, 2015

12. Subset of a Set: In set theory, a set P is the subset of any set Q, if the set P is contained in set Q. It means, all the elements of the set P also belongs to the set Q. It is represented as '⊆’ or P ⊆ Q . Disjoint Sets: If two sets A and B should have no common elements or we can say that the intersection of any two sets A and B is the empty set, then these sets are known as disjoint sets i.e. A ∩ B = ϕ . That means, when this condition n (A ∩ B) = 0 is true, then the sets are disjoint sets . Equality of Two Sets: Two sets are said to be equal or identical to each other, if they contain the same elements. When the sets P and Q is said to be equal, if P ⊆ Q and Q ⊆ P, then we will write as P = Q. . January 27, 2015

13. The theory of fuzzy logic is based on the notion of relative graded membership, as inspired by the processes of human perception and cognition. Lotfi A. Zadeh published his first famous research paper on fuzzy sets in 1965. Fuzzy logic can deal with information arising from computational perception and cognition, that is, uncertain, imprecise, vague, partially true, or without sharp boundaries. Fuzzy logic allows for the inclusion of vague human assessments in computing problems. Also, it provides an effective means for conflict resolution of multiple criteria and better assessment of options. New computing methods based on fuzzy logic can be used in the development of intelligent systems for decision making, identification, pattern recognition, optimization, and control. Set theory of fuzzy Logic January 27, 2015

14. Fuzzy logic is extremely useful for many people involved in research and development including engineers (electrical, mechanical, civil, chemical, aerospace, agricultural, biomedical, computer, environmental, geological, industrial, and mechatronics ), mathematicians, computer software developers and researchers, natural scientists (biology, chemistry, earth science, and physics), medical researchers, social scientists (economics, management, political science, and psychology), public policy analysts, business analysts, and jurists. Indeed, the applications of fuzzy logic, once thought to be an obscure mathematical curiosity, can be found in many engineering and scientific works. Fuzzy logic has been used in numerous applications such as facial pattern recognition, air conditioners, washing machines, vacuum cleaners, antiskid braking systems, transmission systems, control of subway systems and unmanned helicopters, knowledge-based systems for multi objective optimization of power systems, weather forecasting systems, models for new product pricing or project risk assessment, medical diagnosis and treatment plans, and stock trading. Fuzzy logic has been successfully used in numerous fields such as control systems engineering, image processing, power engineering, industrial automation, robotics, consumer electronics, and optimization. This branch of mathematics has instilled new life into scientific fields that have been dormant for a long time Set theory of fuzzy Logic January 27, 2015

15. This paper focuses on the comprehensive review of the literature on applications of rough set theory in Civil Engineering. The relationships between rough set theory and other mathematical methods, such as conventional statistical methods, fuzzy sets, and evidence theory, are briefly introduced. The applications of rough set theory in Civil Engineering are discussed in structure engineering, pavement engineering, traffic engineering, transportation engineering, land management, and water resources management. The great majority of civil engineering applications of rough set theory are based on the rule induction. Recent works on integration of rough set theory with other computing paradigms such as genetic algorithm, fuzzy logic, evidence theory are presented . The Applications of Rough Set Theory in Civil Engineering January 27, 2015

16. The characterization of rock masses is one of the integral aspects of rock engineering. Over the years, many classification systems have been developed for characterization and design purposes in mining and civil engineering practices. However , the strength and weak points of such rating-based classifications have always been questionable. Such classification systems assign quantifiable values to predefined classified geotechnical parameters of rock mass. This results in subjective uncertainties, leading to the misuse of such classifications in practical applications. Fuzzy set theory is an effective tool to overcome such uncertainties by using membership functions and an inference system. APPLICATION OF FUZZY SET THEORY TO ROCK ENGINEERING January 27, 2015

17. Set theory is a mathematical abstract concerned with the grouping of sets of numbers that have commonality. For example, all even numbers make up a set, and all odd numbers comprise a set. All numbers that end in zero make up a set of numbers that can be divided by 10. Using and comparing sets enables the creation of theories and rules that have practically unlimited scope, whether in mathematics or applied to areas such as business. Applied to business operations, set theory can assist in planning and operations. Every element of business can be grouped into at least one set such as accounting, management, operations, production and sales. Within those sets are other sets. Sets and Set Theory in Business Operations January 27, 2015

18. In operations, for example, there are sets of warehouse operations, sales operations and administrative operations. In some cases, sets intersect -- as sales operations can intersect the operations set and the sales set.. A study published in the "International Journal of Mathematical Models and Methods in Applied Sciences" used rough-set theory to study historical bank failures . The study was conducted to assist in predicting a potential failure of Turkish banks. Since every economy depends on the health of its banks, rough-set theory is useful. Using rough-set theory, banks can form an early warning system of potential bank failures, improving financial management. January 27, 2015

19. One of the primary features of mathematics is its transformation of the workings of nature into symbolic form. After systematically examining, manipulating and analyzing these abstract symbols, hopefully one may obtain a deeper understanding of the world . The generality of set theory leads to few direct practical applications. Instead, precisely because of its generality, portions of the theory are used in developing the algebra of groups, rings, and fields, as well as, in developing a logical basis for calculus, geometry, and topology. These branches of mathematics are all applied extensively in the fields of physics, chemistry, biology, and electrical and computer engineering. CONCLUSION January 27, 2015

20. Thank You for your attention Any Question Please ???