Fuzzy Set Theory

What is Fuzzy?

Fuzzy meansnot clear, distinct or precise;not crisp (well defined);blurred (with unclear outline).

Sets Theory

Classical Set: An element either belongs or does not belong to a sets that have been defined.

Fuzzy Set: An element belongs partially or gradually to the sets that have been defined.

Classical Set Vs Fuzzy set theory

Classical Set theory

Classical set theory represents all items elements, A={ a1,a2,a3,…..an}

if elements ai (i=1,2,3,…n) of a set A are subset of universal set X, then set A can be represent for all elements x Є X by its characteristics function,

μA(x) = {thus in classical set theory μA(x) has only

values 0 (false) and 1( true). Such set are called crisp sets

1 if x Є X

0 otherwise

Fuzzy Set Theory

Fuzzy set theory is an extension of classical set theory where element have varying degrees of membership. 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 and 1 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].

Definition

A fuzzy set A, defines in the universal space X, is a function defined in X which assumes values in range [0,1].

A fuzzy set A is written as s set of pairs { x, A(x)} as

A= {{x, A(x)}}, x in the set X.

where x is element of universal space or set X and A(x) is the value of function A for this element.

Example: Set SMALL in set X consisting natural numbers <= 5.

Assume: SMALL(1)=1, SMALL(2)=1, SMALL(3)=0.9, SMALL(4)=0.6, SMALL(5)=0.4

Then set SMALL={ {1,1,},{2, 1},{3,0. 9},{4,0.6}, {5,0.4}}

Fuzzy V/s Crisp set

Is Ram honest? Fuzzy

Is water colourless?

Crisp

Yes

No

Extremely honest(1)

Very honest(0.85

)

Honest at time (0.4)

Extremely dishonest(0

)

Fuzzy operations

Union

The union of two fuzzy sets A and B is a new fuzzy set A U B also on X with membership function defined as

μ A U B (x)= max (μ A (x) ,μ B (x))Example: Let A be the fuzzy set of young people and B be the

fuzzy set of middle-aged people. In discrete form, A=P(x1,0.5), (x2,0.7), (x3,0)} and B={(x1,0.8),( x2,0.2),(

x3,1)}

So find out A U B .

Use formulaμ A U B (x1)= max (μ A (x1) ,μ B (x1))

= max(0.5,0.8) =0.8

μ A U B (x2)= max (μ A (x2) ,μ B (x2))

= max(0.7,0.2) =0.7

μ A U B (x3)= max (μ A (x3) ,μ B (x3))

= max(0,1) =1

So,A U B= {(x1,0.8, x2,0.7, x3,1)}

Intersection

The union of two fuzzy sets A and B is a new fuzzy set A B also on X with membership function defined as

μ A B (x)= min (μ A (x) ,μ B (x))Example: Let A be the fuzzy set of young people and B be the

fuzzy set of middle-aged people. In discrete form, A=P(x1,0.5), (x2,0.7), (x3,0)} and B={(x1,0.8),( x2,0.2),(

x3,1)}

So find out A B .

U

U

U

Use formulaμ A B (x1)= min (μ A (x1) ,μ B (x1))

= max(0.5,0.8) =0.5

μ A B (x2)= min (μ A (x2) ,μ B (x2))

= max(0.7,0.2) =0.2

μ A B (x3)= min (μ A (x3) ,μ B (x3))

= max(0,1) =0

So,A B= {(x1,0.5, x2,0.2, x3,0)}

U

U

U

U

Complement

The complement of a fuzzy set A with membership function defined as

μ A (x)= 1-μ A (x)Example: Let A be the fuzzy set of young people complement

“not young” is defined as Ac. In discrete form, for x1, x2, x3

A=P(x1,0.5), (x2,0.7), (x3,0)}

So find out Ac.

Use formulaμ A (x1)= 1- μ A (x1 )

= 1-0.5=0.5

μ A (x1)= 1- μ A (x1 )= 1-0.7=0.3

μ A (x1)= 1- μ A (x1 )= 1-0=1

So,Ac= {(x1,0.5, x2,0.3, x3,1)}

Product of two fuzzy set

The product of two fuzzy sets A and B is a new fuzzy set A .B also on X with membership function defined as

μ A.B (x)= μ A (x) μ B (x)Example: Let A be the fuzzy set of young people and B be the

fuzzy set of middle-aged people. In discrete form, A=P(x1,0.5), (x2,0.7), (x3,0)} and B={(x1,0.8),( x2,0.2),(

x3,1)}

So find out A.B

Use formulaμ A .B (x1)= μ A (x1).μ B (x1)

= 0.5 . 0.8 =0.040

μ A .B (x2)= μ A (x2).μ B (x2)

= 0.7 . 0.2 =0.014

μ A .B (x3)= μ A (x3).μ B (x3)

= 0 . 1 =0

So,A .B= {(x1,0.040, x2,0.014, x3,0)}

EqualityThe two fuzzy sets A and B is said to be equal(A=B)

if μ A (x) =μ B (x)

Example: A=(x1,0.2), (x2,0.8)}

B={(x1,0.6),( x2,0.8)}

C={(x1,0.2),( x2,0.8)}

μ A (x1) ≠μ B (x1) & μ A (x2) =μ B (x2)

μ A (0.2) ≠μ B (0.6) & μ A (0.8) =μ B (0.8)

so, A≠Bμ A (x1) =μ c (x1) & μ A (x2) =μ c (x2)

μ A (0.2) =μ c (0.2) & μ A (0.8) =μ c (0.8)

so, A=C

Fuzzy Logic

Flexible machine learning techniqueMimicking the logic of human thoughtLogic may have two values and represents

two possible solutionsFuzzy logic is a multi valued logic and allows

intermediate values to be definedProvides an inference mechanism which can

interpret and execute commandsFuzzy systems are suitable for uncertain or

approximate reasoning

Fuzzy Logic

A way to represent variation or imprecision in logic A way to make use of natural language in logic Approximate reasoning Definition of Fuzzy Logic:

A form of knowledge representation suitable fornotions that cannot be defined precisely, but whichdepend upon their contexts.

Superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth - the truth values between "completely true & completely false".

Fuzzy Propositions

A fuzzy proposition is a statement that drives a fuzzy truth value.

Fuzzy Connectives: Fuzzy connectives are used to join simple fuzzy propositions to make compound propositions. Examples of fuzzy connectives are:

Negation(-)Disjunction(v)Conjunction(^)Impication( )

Example