# Fuzzy Logic and Fuzzy Modeling - Fuzzy Logic and Fuzzy Modeling Fuzzy Logic: Fuzzy logic deals with

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Fuzzy Logic and Fuzzy Modeling

Fuzzy Logic:

Fuzzy logic deals with fuzzy sets. A fuzzy set or subset is a generalization of an ordinary

or crisp set. A fuzzy subset can be seen as a predicate whose truth values are drawn from

the unit interval , I =[0,1] rather than the set {0,1} as in the case of an ordinary set. Thus

the fuzzy subset has as its underlying logic a multivalued logic. The fuzzy set allows for

the description of concepts in which the boundary between a property and not having a

property is not sharp.

Ex:-

A set of heights forms a fuzzy set and its subsets include heights that can be categorized

as tall, medium and short, here the property is height. All the temperatures in a year can

be clustered into three groups (or fuzzy subsets) that belong to hot, moderate and cold.

Here the property is temperature. In both these examples, the highest value of the

property is taken as unity and the rest lie with in the interval [0, 1].

Membership function:-

Let X be the universe of discourse (the domain of a property). A subset of A of X is

associated with a membership function.

A : x→ [0, 1]

Where A (x) for each x indicates the degree to which x is a member of the set A. I t is

also called the degree of association of x in A.

Depending on the variation of x in the set A, one can choose a particular shape for the

membership function. Some of the shapes are described in the following.

A triangular membership function (MF) is specified by three parameters {a, b, c} as

follows:

Triangular (x; a, b, c) =

0,

,

,

0,

x a

x a a x b

b a

c x b x c

c b

c x

A trapezoidal MF is specified by four parameters {a, b, c, d} as follows:

Trapezoid (a; a, b, c, d) =

0,

,

,

0,

x a

x a a x b

b a

d x c x d

d c

d x

A Gaussian MF is specified by two parameters {c, }

Gaussian (x; c, ) = 1/ 2e 2

x c

A generalized bell MF is specified by three parameters {a, b, c}

Bell (x ; a ,b,c) = 2

1

1

b x c

a

where the parameter b is usually positive.

Fuzzy Set:-

If X is a collection of objects denoted generally by x, then a fuzzy set A in X is defined as

a set of ordered pairs:

A = {[x, A (x) ] | x X }

Where A (x) is the MF for the fuzzy set A. Usually x is referred to as the universe of

discourse or simply the universe.

Ex :

Let X=R+ be the set of possible ages for human beings then the fuzzy set B= “about 50

years old “may be expressed as

Support:

The support of fuzzy set A is the set of all points x in A such that A (x) >0:

Support (A) = {x │ A (x) >0}

Core:

The core of a fuzzy set A is the set of all points x in X such that A (x) =1

Core (A) = {x │ A (x) =1}

Crossover point:

A crossover point of a fuzzy set A is a point x X

At which A (x) = 0.5

Crossover (A) = {x │ A (x) =0.5}

Fuzzy singleton:

A fuzzy set whose support is a single point in x with A (x) = 1 is called fuzzy singleton.

αcut, strong αcut:

The α-cut or α-level set of a fuzzy set A is a crisp set defined by

Aα = {x │ A (x) ≥α}

Strong αcut or strong α-level are defined similarly

Aα’ = {x │ A (x) >α

Normality:

A fuzzy set A is normal if its core is nonempty.

In other words, we can always find a point x X such that A (x) = 1

Convexity:

A fuzzy set is convex if and only if for any x1,x2X and any λ [0,1] , A {λx1 + (1-

λ)x2 } min { A (x1), A (x2) }

Alternatively, A is convex if all its α –level sets are convex.

Fuzzy number:

A fuzzy number A is a fuzzy set in the real line (R) that satisfies the conditions for

normality and convexity.

Fuzzy set operations:

Before introducing the fuzzy set operations, first we shall define the notation of

containment, which plays a central role in fuzzy sets.

Containment or subset:

A fuzzy set A is contained in set B (or, equivalent A is α subset of B) if and only if A

(x) ≤ B (x) for all. In symbols,

A B A (x) ≤ B (x)

Union (disjunction):

The union of two fuzzy sets A and B is a fuzzy set C , written as C=A B or C=a or B,

whose MF is related to those of A and B by

C (x)= max{ A (x), B (x)}= A (x) B (x) it is the “smallest” fuzzy set

containing both A and B.

Intersection (conjunction):

The interconnection of two fuzzy sets A and B is a fuzzy set C written as C=A B or C=

A and B whose MF is related to those of A and B by

C (x) = min { A (x), B (x)}= A (x) B (x)

It is the “largest” fuzzy set, which is contained in both A and B.

Complement (negation):

The complement of a fuzzy set A is denoted by A

1 AA

MFs of two Dimensions:

Some times it is necessary to use MFs with two inputs, each on a different universe of

discourse. MFs of this kind are generally referred to as two-dimensional MFs, where as

ordinary MFs are referred to as one-dimensional MFs. One natural way to extend one-

dimensional MF to two-dimensional ones is via cylindrical extension.

Cylindrical extension of one-dimensional fuzzy sets:

If A is a fuzzy set in X, then its cylindrical extension in X x Y is a fuzzy set c(A) defined

by

C (A) = A XxY

/(x,y)

Usually, A is referred to as a base set. The operation of projection on the other hand

decreases the dimension of a given (multidimensional) MF.

Projection of fuzzy sets:

Let R be a two dimensional fuzzy set on XxY. Then the projection of R onto X and Y are

defined as

xR = [max y R x

(x,y)]/x

yR = [max x R y

(x,y)]/y

respectively.

T and S operators:

The four conditions: commutativity, associativity, monotonicity and respective identities

have been used to characterize the T and S operators which in turn define the general

class of intersection and union.

An operator T: [ 0,1] x [ 0,1] [ 0,1 ]

Is called a t-norm operator if

(1) T (a, b) = T (b, a) Commutativity

(2) T( a,T( b,c)) = T(T ( a,b ),c) Associativity

(3) T (a, b) ≥ T (c,d) if a ≥ c and b ≥ d Monotonicity

(4) T(a,1) = a One identity

To see that the T reduces to the crisp intersection we note that (4) implies T(0,1) =0 and

T (1,1) =1.

Condition (1) implies T(0,1) = T(1,0) = 0. Finally, this fact along with condition (3)

implies that T (0,0) = 0.

We note that the Min and product operators are examples of these t-norm operators. Part

of the uniqueness of the Min operator as a choice for the implementation of the

intersection operator is based on the fact that the Min operator is the largest of possible t-

norms, T(a,b) ≤ Min (a,b).

t-co norm or s operator:

An operator s: [0,1] x [0,1] [ 0,1]

Is called a t-conorm operator if

(1) s(a,b)= s(b,a) Commutativity

(2) s(a, s(b,c)) = s(s(a,b),c) Associativity

(3) s(a,b) ≥ s(c,d) if a ≥ c and b ≥ d Monotonicity

(4) s(a,0) = a Zero identity

It can be seen that the above conditions imply

s(1,1) = s(1,0) = s(0,1) = 1

s(0,0) = 0

We note that the Max and the a + b – ab operators are examples of these t- conorm

operators. Another example of the t- conorm is the bounded sum Min [1, a+b]. part of the

uniqueness of the max operator as a choice for the implementation of the union operator

is based on the fact that the Max operator is the smallest of all the t-co norms, for all s:

Max(a,b) ≤ s(a,b).

It should be noted that the only distinction between the T and S operators is in conditions

(4) and (4’). These conditions can be seen as the defining characteristics of the respective

operators. Essentially condition (4) implies that the smallest argument is the most

influential in the formula

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