CLASSICAL SETS AND FUZZY

SETSBy 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

Shoe Polish

Tuesday

WednesdaySaturdayLiberty

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 ∈ AHere 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 )

Properties of Classical Sets

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

Properties of Classical Sets

Law of Excluded middle : A ∪ Ā = X;

DeMorgan’s Law

Law of contradiction : A ∩ Ā = Φ;

Days of week-end (two valued membership)

Sat

Wee

k-en

dedn

ess

0

1

Sun Wed Mon Tues

Days of week-end (two valued membership)

Sat

Wee

k-en

dedn

ess

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

LOGICSYSTEM

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

Wee

k-en

dedn

ess

0

1

Sun Wed Mon Tues

0.8

0.5

Days of week-end (multi- valued membership)

Sat

Wee

k-en

dedn

ess

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- μ A(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 ) ;

Thanks