Classical Sets & fuzzy sets

By Ashvini Chaudhari
Pratibha College of commerce and computer
studies Chichwad Pune

Classical Sets and Fuzzy Sets and Fuzzy
relations
• Operations on Classical sets, properties of
classical sets
• Fuzzy set operations, properties of fuzzy sets
• Cardinality operations, and properties of fuzzy
relations.

Classical Set
•A set is defined as a collection of objects,
which share certain characteristics
•A classical set is a collection of distinct
objects -- set of negative integers, set of
persons with height<6 ft, days of the week
etc
• Each individual entity in a set is called a
member or an element of the set.
• The Classical set is defined in such a way
that the Universe of Discourse is split into 2
groups: members and Nonmembers

Classical Set
Tuesday
Wednesday
Saturday
Butter
Days of the week

Classical Set
•A classical set is a container that wholly
includes or wholly excludes any given
element. For example, the set of days of
the week unquestionably includes Tuesday,
Wednesday, and Saturday. It just as
unquestionably excludes butter, liberty,
shoe polish, and so on.
•It was Aristotle who first formulated the Law of
the Excluded Middle, which says X must either
be in set A or in set not-A, ie., Of any subject,
one thing must be either asserted or denied

Defining a Set
There are several ways of defining a set
•A = {2,4,6,8,10}
•A= {x│x is a prime number <20 }
• A= {xi+1 = (xi +1 )/5, i=1 to 10, where x1=1 }
•A= {x│x is an element belonging to P AND Q }
•µ A(x) = 1 if x ∈ A
= 0 if x ∈ A
Here µ A(x) is membership function for set A

•Φ is a null or Empty Set ie., with no elements
•Set consisting of all possible subsets of a given
set A is called a Power Set
P(A)= {x│x ⊆ A }
•For crisp set A and B containing some
elements in universe X, the notations used are
x ∈ A ⇒ x does belong to A
x ∉ A ⇒ x does not belong to A
x ∈ X ⇒ x does belong to universe X

•For classical sets A and B on X we also have
A ⊂ B ⇒ A is completely contained in B
(ie., if x ∈ A then x ∈ B )
A ⊆ B ⇒ A is contained in or equivalent to B
A=B ⇒ A ⊂ B and B ⊂ A

Operations on Classical Sets
Union :
A ∪ B = {x│x ∈ A or x ∈ B }
Intersection :
A ∩ B = {x│x ∈ A and x ∈ B }
Complement :
Ā = {x│x ∉ A , x ∈ X }
Difference:
A-B = A │ B = {x│x ∉ A and x ∉ B }
= A- (A ∩ B ) Ie., All elements in
universe that belong to A but do not belong to B

Properties of Classical Sets
Commutivity :
A ∪ B = B ∪ A ; A ∩ B = B ∩ A
Associatively :
A ∪ ( B ∪ C) = ( A ∪ B ) ∪C
A ∩ ( B ∩ C) = ( A ∩ B) ∩C
Distributivity:
A ∪ ( B ∩ C) = ( A ∪ B ) ∩ ( A ∪ C )
A ∩ ( B ∪ C) = ( A ∩ B) ∪ ( A ∩ C )

Idempotency :
A ∪ A = A ; A ∩ A = A
Transitivity :
If A ⊆ B ⊆ C, then A ⊆ C
Identity:
A ∪ Φ = A ; A ∩ Φ = Φ
A ∪ X = X ; A ∩ X = X

Law of Excluded middle :
A ∪ Ā = X;
DeMorgan’s Law
Law of contradiction :
A ∩ Ā = Φ;

Days of week-end (two valued membership)
Sat
Week-endedness
0
1
Sun Wed Mon Tues

Days of week-end (two valued membership)
Sat
Week-endedness
0
1
Sun Wed Mon Tues
Continous Scale

What is Fuzzy Set ?
• “FUZZY means “vagueness”. Fuzziness occurs when boundary of a
piece of information is not clear-cut.
• Fuzzy set were introduced by Lotfi A Zadeh (1965) as an extension of
classical notion of sets.
• Classical set theory allows the membership of elements in the set in
binary terms, a bivalent condition – an element either belongs or does
not belong to the set.
• Fuzzy set theory permits gradual assessment of membership of
elements in a set, described with the aid of a membership function
valued in the real unit interval [0.1]

Why Fuzzy Set ?
Words like Young, tall, good or high are fuzzy
--- There is no single quantitative value which defines the term ‘young’
--- For some people, age 25 is young, and for others, age 35 is young
--- Concept ‘young’ has no clear boundaries
--- Age 1 is definitely young and age 100 is definitely not young
--- Age 35 has some possibility of being young and usually depends on the
context in which it is being considered

Why Fuzzy Set ?
• In real world, there exists much Fuzzy Knowledge :
• Knowledge that is vague, imprecise, uncertain, ambiguous, inexact or
probabilistic in nature
• Human thinking and reasoning frequently involve fuzzy information,
originating from inherently human concepts. Humans can give
satisfactory answers , which are probably true.
• However, our systems are unable to answer many questions. Reason is ,
most systems are designed based upon classical set theory and two
valued logic which is unable to cope with unreliable and incomplete
information and give expert opinions. .
• We want our systems should also be able to cope with unreliable and
incomplete information and give expert opinions. Fuzzy systems have
been able to provide solutions to many real world problems.
• Fuzzy Set theory is an extension of Classical set theory where elements
have degree of membership

Real World Problems
•Uncertainty
•Ambiguity
Imprecision and
Vague Data FUZZY
LOGIC
SYSTEM
Decision
Fuzzy Logic System & Real world Problems

Fuzzy Set Theory
• Fuzzy Set theory is an extension and generalisation of basic concepts of
crisp sets.
--- Fuzzy Logic is derived from fuzzy set theory dealing with reasoning that is
approximate rather than precisely deducted from classical predicate logic
--- Fuzzy logic is capable of handling inherently imprecise concepts
---- Fuzzy logic allows in linguistic form the set membership values to
imprecise concepts like “ slightly”, “quite”, and “very’
---- Fuzzy Set Theory defines Fuzzy operators on Fuzzy Sets
• A fuzzy set is any set that allows its members to have different degree of
membership, called membership Function, in the interval [0,1]
• Vagueness is introduced in Fuzzy set by eliminating sharp boundaries; There is
gradual transition between full membership to non-membership.

Fuzzy Set Theory
• A logic based on the two truth values, True and false , is sometimes
inadequate when describing human reasoning. Fuzzy logic uses the
whole interval between 0 (false) and 1 (true) to describe human
reasoning.
--- A fuzzy set is any set that allows its members to have different degree of
membership, called membership Function, in the interval [0,1]
--- The degree of membership or truth is not the same as probability;
• Fuzzy truth is not likelihood of some event or condition
• Fuzzy truth represents membership in vaguely defined sets
--- Fuzzy Logic is derived from fuzzy set theory dealing with reasoning that is
approximate rather than precisely deducted from classical predicate logic
--- Fuzzy logic is capable of handling inherently imprecise concepts
---- Fuzzy logic allows in linguistic form the set membership values to
imprecise concepts like “ slightly”, “quite”, and “very’
---- Fuzzy Set Theory defines Fuzzy operators on Fuzzy Sets

Days of week-end (multi- valued membership)
Sat
Week-endedness
0
1
Sun Wed Mon Tues
0.8
0.5

Days of week-end (multi- valued membership)
Sat
Week-endedness
0
1
Sun Wed Mon Tues
Continous Scale

With fuzzy Logic , Rules can be written in a
more natural way
Example ; of a rule-
based System without
Fuzzy Logic
----------------------------
• If Temp is 30 deg,
Then switch ON
condenser to 80 %
• If Traffic is 20 Cars,
then extend green
Light by 20 Seconds
Example ; of a rule-
based System with
Fuzzy Logic
----------------------------
• If Temp is HIGH
then switch ON
condenser HIGH
• If Traffic is HEAVY
then extend green
Light LONGER

Fuzzy Set Operations
Union :
μA∪B (x) = max [μA (x), μB(x) ] for all x ∈ U
= μA (x) ∨ μB(x) ;
Here ∨ is the symbol for maximum
Intersection :
μA∩B (x) = min [μA (x), μB(x) ] for all x ∈ U
= μA (x) ∧ μB(x) ;
Here ∧ is the symbol for maximum
Complement :
μ Ā(x) = 1- μA(x)

Fuzzy Set Operations
-- Algebraic and Bounded
Algebraic Sum :
μA+B (x) = μA (x) + μB(x) - μA (x). μB(x) ;
Algebraic Product :
μA.B (x) = μA (x) . μB(x)
Bounded Sum :
μA⊕B (x) = min {1, μA (x) + μB(x) }
Bounded Product :
μA ⊙B (x) = max {0, μA (x) - μB(x) }

Properties of Fuzzy Sets
Properties of Commutivity, Associativity,
Distributivity, idempotency, Identity, transitivity
and Demorgan’s Laws are the same as for
crispy Sets
Fuzzy sets follow the same properties as
crisp sets except for the law of Excluded
middle and law of contradiction
Ie., For Fuzzy Sets
A ∪ Ā ≠ U; A ∩ Ā ≠ Ø;

Fuzzy relations
X ={x1,x2…x n} ; Y ={y1,y2…ym} ;
R(x,y) is the nxm Fuzzy Relation matrix and given
by
A binary relation between 2 fuzzy sets X and Y is
called Binary Fuzzy Relations and is denoted by
R(X,Y)
μR(x1,y1) μR(x1,y2) μR(x1,y3) μR(x1,ym)
μR(x2,y1) μR(x2,y2) μR(x2,y3) μR(x2,ym)
μR(xn,y1) μR(xn,y2) μR(xn,3) μR(xn,,ym)

Example 1
Consider
Universe X= {x1,x2,x3,x4}
Binary Fuzzy Relationship is R(X,X)
Bipartite Graph
Fuzzy Graph

Example 2
Let X= {x1,x2,x3,x4} & Y= {y1,y2,y3,y4}
Relationship from X to Y is R
Fuzzy Graph of Fuzzy
Elements

Fuzzy Relationship Operations
Union :
μR∪S(x,y) = max [μR (x,y), μS(x,y) ]
Intersection :
μR∩S(x,y) = min [μR (x,y), μS(x,y) ]
Complement :
μR(x,y) = 1- μR(x,y)
(R DASH )

Properties of Fuzzy Relations
Properties of Commutivity, Associativity,
Distributivity, idempotency, Identity, transitivity
and Demorgan’s Laws for Fuzzy Relations are
the same as for fuzzy sets
Also
R ∪ Rdash ≠ E (whole Set);
R ∩ Rdash ≠ Ø (Null Set ) ;

Classical Sets & fuzzy sets

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