Set Theory and its Applications

2. SajidIRanKhan (UW-12-BSc-EE-49)
AtifHafeez (UW-12-BSc-EE-09)
ArslanHaider (UW-12-BSc-EE-43)
Sheryar (UW-12-BSc-EE-47)
KamranHaider (UW-12-BSc-EE-81)
MazharIQbal (UW-12-BSc-EE-101)

3. SET THEORY
AND ITS APPLICATION

4. January 21, 2016
Set Theory
Branch of mathematics that deals
with the properties of sets.

5. 1. Set theory was founded by a single paper in 1874 by Georg Cantor
2. On a Characteristic Property of All Real Algebraic Numbers“
3. Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians
had struggled with the concept of infinity.
4. Modern understanding of infinity began in 1867–71, with Cantor's work on number theory.
5. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking
and culminated in Cantor's 1874 paper.
6. 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
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6. History of Set Theory
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 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

7. 1. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment.
2. 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.
3. 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
4. Set theory is commonly used as a foundational system, although in some areas
category theory is thought to be a preferred foundation.
History
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8. 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)
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9. 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
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10. 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.
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11. 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.
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12. 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}.
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13. 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.
.
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14.  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
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15.  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
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16.  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
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17.  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
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18.  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
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19.  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.
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20.  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
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21. Thank You for your attention
Any Question Please ???