IRJET- A Study of Fuzzy Subgroups and Characteristics of Fuzzy Subgroups

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A STUDY OF FUZZY SUBGROUPS AND CHARACTERISTICS OF FUZZY SUBGROUPS Ms. Vigitha Vidyadhar Assistant Professor, Department of Applied Science, Vidya Academy of Science & Technology Technical Campus, Kilimanoor, Trivandrum, Kerala, India ---------------------------------------------------------------------***----------------------------------------------------------------------

Abstract - Fuzzy group theory has many applications in Physics,

chemistry, bio informatics and computer science problems.

Applications of fuzzy set theory have also been established in many areas as engineering, medicine, psychology, economics, robotics, etc In this paper the preliminaries required for the concept of Fuzzy subsets, Fuzzy subgroups and properties of fuzzy groups, introduction to the concept of normal fuzzy groups were investigated.

Key Words: Membership function, Fuzzy sets, Fuzzy Subgroup, Normal fuzzy subgroup

1.

INTRODUCTION

The Mathematical model of Fuzzy the concept was introduced by Lotfi. A.Zadeh(Electrical Engineering and Electronics Research Laboratory University of California Berkeley). He was Benjamin Franklin Medal Winner in Electrical Engineering 2009. He is known as the Father of Fuzzy Logic. In 1965 he laid the foundation of a comprehensive and mathematically sound logic of imprecisely or ambiguously defined sets known as "Fuzzy SETS. The Fuzzy theory has grown into a many faceted field of inquiry and contributing to a wide spectrum of areas ranging from para mathematics to human perception and judgment. Fuzzy sets have been suggested for handling the imprecise real world problems by using truth values ranging between the usual” True" and "False".

For example, consider the collection of all small natural numbers or that of all the students with brilliant academic career or the collection of all tall men etc. All the collections defined above do not constitute sets in the usual mathematical sense. Yet, the fact remains that such imprecisely defined"collections" play an important role in human thinking particularly in the areas of pattern recognition, communication of information etc.

2.

A.

denote higher degrees of set membership. The range of values of membership function is the unit interval [0, 1].

Two distinct notations of membership function of a Fuzzy set A i.

B.

the interval [0, 1] with the value of µA(x) at X representing

the grade of membership of X in A. The set A in X is called the Fuzzy set.

• • • •

The maximal membership of x in A is '1' The minimal membership of x in A is '0'

µA(5) = 0.01 µA(10) = 0.2 µA(100 = 0.95 µA(500) = 1

In the classical set theory, the characteristic function of the crisp set assigns a value of either '1' or '0' to each individual in the universal set, thereby discriminating between members and non-members of the crisp set. This function can be generalized such that the values assigned to the elements of the universal set fall within a specified range and indicate the membership grade of these elements. Such a function is called the Membership function.Larger values

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A={(x,µA(x))|x X}

A Fuzzy set is an extension of a classical set whereas classical set is a special case of a Fuzzy set.

EXAMPLE Let X be the real line R1 and let A be the Fuzzy set of numbers which are much greater than 1. Then µA(0) = 0

Membership Function

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A : X → [0, 1] Fuzzy set

A Fuzzy set A in X is characterized by a membership function µA(x) which associates with each point in X a real number in

PRELIMINARIES

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ii.

µA : X → [0, 1]

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C.

Representation of Fuzzy sets

A Fuzzy set can be represented in following ways: Let X be a classical set with n elements. ie X = {x1, x2, ....xn}.Then the Fuzzy set A can be expressed in one of the following three ways: 1. A = µA(x1)/x1 + µA(x2)/x2 + ....... + µA(xn)/xn 2. A = {(x1, µA(x1)).................} 3. A = {µA(x1), .......µA(xn)} ISO 9001:2008 Certified Journal

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If X is continuous then Fuzzy set A is expressed as A=



Fuzzy variable

αA

Several Fuzzy sets representing linguistic concepts such as low, medium, high and so on are often employed to define states of variable. Such a variable is usually called a Fuzzy variable. For example, temperature within a range [T1, T2] is characterized as a Fuzzy variable.

E.

= {x/A(x) ≥ α}

+αA = {x/A(x) > α H.

Level Set

Λ(A) = { α / A(x) = α Ƒ or some x

I.

Interval-valued Fuzzy sets

The Support of A

The interval valued Fuzzy sets are defined by the functions of the form µ : X → ε([0, 1]) where ε([0, 1]) denotes the family of all closed intervals of real numbers in [0,1].

Support of A = {x X| A(x) >0}

ε([0, 1])

Core ( ) = C (A˜) = {x X|

ii.

P([0, 1]).

L-Fuzzy sets

Level -n Fuzzy sets are defined by the functions of the form µ : Ƒ(X) → [0, 1] where Ƒ(X) denotes the Fuzzy power set of X. Level-2 Fuzzy sets can be generalized into Level-3 Fuzzy sets by using a universal set whose elements are Level-2 Fuzzy sets. Higher level Fuzzy sets can be obtained recursively in the same way. Their membership Functions have the form µ : Ƒ(X) → Ƒ([0, 1]). i.

Operations of Fuzzy Sets Standard Fuzzy Complement

x X. Standard Fuzzy Union

iii.

The Height of a Fuzzy Set A

h (A) = sup A(x).

A Fuzzy set A is called subnormal when h (A) <1

3.

FUZZY SUBSETS AND FUZZY GROUPS

A. Fuzzy Subsets A Fuzzy subset of X is a function from X into [0, 1]. The set of all Fuzzy subsets of X is called the Fuzzy power set of X and is denoted by ƑP(X) Let µ ƑP(X). Then the set {µ(x) | x X} is called the image of is called the support of µ and is denoted by µ* If µ ƑP(X) then Support of µ = {x X| µ(x) >0}

Consider a universal set X and two Fuzzy sets A and B, defined on X. (A

(x) = 1}

µ and is denoted by µ(X) or Im(µ). The set {x| x X, µ(x) >0},

Given a Fuzzy set A defined on a universal set X, its compliment is another Fuzzy set on X by (x)= 1 - A(x) ∀ ii.

The Core of a Fuzzy Set

A Fuzzy set A is called normal when h (A) = 1.

Level-n Fuzzy sets

F.

J.

K.

Let X be a set and L be a lattice. An L-Fuzzy set of X is characterized by a mappingµ : X → L iii.

X} where Λ denotes the

level set of Fuzzy set A denoted on X.

Types of Fuzzy Sets i.

- Cuts and Strong α -CUTS

Given a Fuzzy set A defined on X and any number α [0, 1] , the α cut, α A and the strong α cut, +αA are the crisp sets

µA(x)/x dx

x

D.

G.

B)(x) = max {A(x), B(x)} ∀ x X

Height of µ = T he cross over point of µ = { x | µ(x) =

Standard Fuzzy Intersection

1 }.

2

µ is said to be normal if height of µ = 1; otherwise µ is called a subnormal Fuzzy subset.

(A ∩ B)(x) = min {A(x), B(x)} ∀x ∀X

B. Fuzzy Subsets and Operations Let Y X and a © 2020, IRJET

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[0, 1]. Define

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www.irjet.net [0, 1]. Let b = iI assertions hold:

a for x  Y   0 x  X | Y 

(x) = 

If Y is a singleton, say {y} then a{y}is called a Fuzzy point (or Fuzzy singleton), and is sometimes denoted by ya. Let 1Ydenote the characteristic function of Y. If S is a set of Fuzzy singletons, then we let foot(S) = {y X | ya S}

a i and c =  iI a i Then the following

1.iI µa i  µ b 2.iI µa i µc Let I be a non empty index set and let collection of nonempty sets.

{Xi | i  I} be a j

Let µ, ν

ƑP(X). If µ(x) ≤ ν(x) ∀x

X , then µ is said to be

contained in ν (or ν contains µ ), and we write µ

ν ( or ν

µ).

Let X denote the Cartesian product of the X i s namely X=

 Xi = {(x )  I | (xi  X , i  I} i i

iI

If µ ν and µ = ν then µ is said to be properly contained in ν ( or ν properly contains µ ) and we write µ ν ( or ν µ ).

The inclusion relation ∀is a partial ordering on ƑP(X).

i

Let µi  FP(Xi ), for all i  I .Define the fuzzy subset µ of X by µ(x) = iI µi (x i ) x = (x i )i I  X. Thenµ is called the complete direct product of theµi’ s and is denoted by ~

µ=

(µ ∀ν)(x) = µ(x) ∀ ν(x).



i

iI

C. First Decomposition Theorem

(µ ∩ ν)(x) = µ(x) ∀ ν(x). ∀ x X

For every A  FP(X) , Let A =

For any collection, {µi | i I} of Fuzzy subsets of X, where I is a nonempty index set, the least upper bound ∀i∀Iµi and the greatest lower bound ∩i I µi of the S are given by ∀ x X

defined by  Fuzzy union.

A



( x)   .a A( X ) and

[0,1

 A where

αA is

denotes the standard

EXAMPLE (

i∀Iµi)(x)

= ∀i∀Iµi(x)

Consider a Fuzzy set A with the following membership function of triangular shape

(∩i  Iµi)(x) = ∀i∀Iµi(x) respectively. n

 µ (x)  µ ∩µ ∩........∩µ i 1

1

i

2

n

 x - 1 if x  [2, 3]    A(x) = 3  x if x  [2, 3] 0 otherwise   

n

µ (x) i 1

i

= µ1∀µ2∀.........∀µn

i = 1, 2, ..., n.

Let µ ∀ ƑP(X). for a ∀[0, 1] µa is defined as follows : µa= {x ∀X, µ(x) ≥ a} Then µa is called the α cut of µ.

RESULTS

For each   interval  A

= {x  X | A(x)  } = { + 1, 3 - } and the A special Fuzzy set  employed is defined by the

For all µ, ν∀ ƑP(X) 1.

2.

3.

membership function.

µ ∀ν, a ∀[0, 1] ∀µa ∀νa, a ≤ b, a, b ∀[0, 1] ∀µb ∀µa, µ = ν ∀µa = νa, ∀a ∀[0,1]

THEOREM Suppose that

0,1] , the α - cut of A is in this case the closed

=

 if x  [  1, 3 -  ]    0 otherwise

D. Extension Principle

µi | i I  ƑP(X).Then for any a [0, 1] ,

1.

(iI µi )a  iI (µi )a

2.

(iI (µi ) a  (iI µi ) a

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from

X

into

Y,

and

F(µ) FP(Y ) and F-1 ( )  FP(X) by  y  Y -1  v  (x) | x  X, f(x)  y if f  F(µ)(y)=   and 0 otherwise    -1 x  X , F ( )(x) =  (F(x)) Then Ƒ(µ) is called the image

THEOREM L µ FP(X) and {a i | i  I} be a nonempty subset of

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Let F be a function let µ  P(X) and   P(Y ) Define the Fuzzy subsets

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of µ under F and F-1(ν) is called the preimage of ν under Ƒ.

The least upper bound of the empty set is the zero elements.

THEOREM Let F be a function from X into Y and g a function from Y into Z. Then the following assertions hold.

1.

µ1  µ2  f(µ1 )  f(µ2 ) µi  FP(Xi ), for all i  I

f(µ)(x 4 ) = 0.9 f(µ) = {(y1 , .1), (y 2 , 0), (y3 , .5), (y 4 , .9)}

For all

 j  ƒP(Y ), j J, where J is

a nonempty

4.

FUZZY SUBGROUPS

Define the binary operation ”o” on ƒP(G) and the unay operation -1 on FP(G)as follows :  µ,   FP(G) and  x  G

F-1 ( jJ  j ) = ( jJ F-1 ( j )

(µo )(x) =  { µ(y)   (z) | y, z  G, yz = x}

F-1 ( jJ  j ) =  jJ F-1 ( j ) and

and µ -1 (x) = µ(x -1 ) µo

Therefore

the product of µ and  and

1  2  F-1 (1 )  F-1 ( 2) 1 , 2  FP(Y ) f -1 (f(µ))  µ  FP(X).In particular, if f is an

injection, then f (f(µ)) = µ µ  FP(X) f -1(f(µ)) = µ ∀ µ ∀ƑP(X) This means µ  f(µ) is an injection from ƑP(X) into ƑP(Y ) and ν  f-1(ν) is a surjection from ƑP(Y ) onto ƑP(X) -1

f(f ( ))   ,   FP(Y ). In particular ,if f is a surjection ,then f(f -1 ( )) =  ,    FP(Y ) and therefore µ  f(µ) is a surjection from ƑP(X) onto ƑP(Y ) and ν  f-1(ν) is an injection from ƑP(Y ) into 4.

f(µ)(x 2 ) = 0 f(µ)(x 3 ) = 0.5

index set,

3.

f(µ)(x) =  {µ(x) | x  X, f(x) = y}iff f -1(y)  0 f(µ)(x1 ) = 0.1

Forall

f(iI µi ) = iI f(µi ) andsoµ1 , µ2  ƒP(X) 2.

Let µ = {(x1 , .1), (x 2 , 0), (x 3 , .5), (x 4 , .9)}

-1

µ -1is the inverse of µ THEOREM

Let µ,  , µi  FP(G), i  I. Let a =  {µ(x) | x  G}

.

Then the following assertions hold:

1. (µo )(x) =  yG {µ(y )   (y -1 x)}

ƑP(X)

  yG {µ(y-1x)   (y )}x  G

5.

f(µ)    µ  f ( )  µ  FP(X) and   FP(Y)

6.

-1 3. (µoa g(f(µ)) = (gof)(µ)  µ  g(f(µ)) = (gof)(µ) µ  FP(X) and f -1(gy-1)(x) ( ))== µ(xy (gof) -1  ( )x, y  G -1 4. (µ )  1 = µ   FP(Z )

-1

1.(a y oµ)(x) = µ(y -1x)  x, y  G

5. µ  µ - 1

EXAMPLE Let X = {x1, x2, x3, x4} and Y = {y1, y2, y3, y4} and f : ƑP(X) → ƑP(Y ) be defined by f(x1 ) = f(x 2 ) = y1 , f(x 3 ) = y3 , f(x 4 ) = y4 ,ν = {(y1, .2), (y2, .4), (y3, .5), (y4, 1)} Then

 µ -1  µ  µ = µ -1  µ(x)  µ(x -1 ) x  G  µ(x -1 )  µ(x) x  G  µ(x)µ = µ(x -1 ) x  G

f -1( )(x1 ) =  (f(x1 ) =  (y1 ) = .2

6. µ    µ -1 

-1 1

-1

f -1( )(x 2 ) =  (f(x 2 ) =  (y 2 ) = .2

7. (iI µ i ) = iI (µi )

f -1( )(x 3 ) =  (f(x 3 ) =  (y3 ) = .5

8.( iI µ i ) -1 = iI µi

f -1( )(x 4 ) =  (f(x 4 ) =  (y 4 ) = 1

9.(µo ) -1 =  -1 o µ -1

-1

f -1( ) = {(x1 , .2), (x 2 , .2), (x 3 , .5), (x 4 , 1)} © 2020, IRJET

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A. Fuzzy Subgroup of G

f(µ)(-1) =  {µ(x) | x  f -1 (-1)} =  {µ(x) | f(x) = - 1} = 1/2

Let µ FP(G).Then µ is called a fuzzy subgroup of G if 1.

µ(xy) = µ(x)  µ(y),x, y  G and

Therefore f(µ) is a fuzzy subgroup on H.

µ(x -1 ) = µ(x),  x  G

2. denoted by F(G), the set of all fuzzy groups of G .

If

µ F(G).Let

Where

*  x  G |  ( x)   (e),

Define ν on H by ν(1) = 1 and ν(-1) = 1/2

* denotes support of  .

f

EXAMPLE Consider the group G = {1, - 1, i, - i} under the usual multiplication Define µ on G by

  1 if x  1    1  µ(x) =  if x  1  2  1   3 if x  i or x  -i  

1

1 when x is even    (v)( x)  v( f ( x)) 1   2 when x is odd   

If x and y are even , then x+y is even .Then

f -1 ( )(x + y) = 1 , f -1 ( )(x) = 1and f -1 ( )(y) = 1. f -1 ( )(x + y) = 1 = 1  1 = f -1 ( )(x)  f -1 ( )(y).

If x is even and y is odd, then x+y is odd. Then

f -1 ( )(x + y) = 1/2, f -1 ( )(x) = 1, and f -1 ( )(y) = 1/2

EXAMPLE Let G = (R,+) . Define µ on G

f -1 ( )(x + y) = 1/2  1.

1 if x  0    µ(x) = 0.7 if x  Q - {0} 0.3 if x  R - Q   

1 = f -1 ( )(x)  f -1 ( )(y) If 2

x is odd and y is odd, then x+y is even. Then

f -1 ( )(x + y) = 1, f -1 ( )(x) = 1/2, and f -1 ( )(y) = 1/2. f -1 ( )(x + y) = f -1 ( )(x)  f -1 ( )(y).Hence x, y  G

f -1 ( )(x + y) = f -1 ( )(x)  f -1 ( )(y)

Then µ is a fuzzy group EXAMPLE

f -1 ( ) is a Fuzzy subgroup of G

Let G = (Z,+) and H = {1,-1 }, is a subgroup under

LEMMA

multiplication. Define f : G → H by

1 when x is even  f (x)    - 1 when x is odd 

Let µ  F(G). Then  x  G 1. µ(e) = µ(x) 2. µ(x) = µ(x - 1)

Then f is a subgroup homomorphism.Define µ on G by

1 when x is even     (x)   1   2 when x is odd   

THEOREM

If µ is a fuzzy subgroup of G and if x, y  G with µ(x)  µ(y) then µ(xy) = µ(x)  µ(y).

Then µ is a fuzzy subgroup on G. We have

f(µ)(1) =  {µ(x) | x  f (1)}   {µ(x) | f(x) = 1}  1 -1

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PROPOSITION

THEOREM

If G i and G j are fuzzy subgroups, then

Let µ  FP(G).

Then µ  F(G) if and only if satisfies the following conditions : (1) µ o µ  µ; (2) µ -1  µ

1.G i  G j is also a fuzzy subgroup if i < j 2.G i  G j is also a fuzzy subgroup if i < j 6. HOMOMORPHISMS OF GROUPS

Let f : G  G 1 be a homomorphism of groups.

THEOREM

Let

µ,   F(G).

and

only

Then

µ o   F(G)

For any fuzzy set µ  G 1 . Define a

if

if µo  =  o µ.

fuzzy set µ f  G byµf (x) = µ(f(x)) for all x  G

THEOREM

PROPOSITION

Let µ  F(G) and H be a group.

Suppose that f is a homomorphism of G into Let G and G1 be groups and f be a homomorphism from G H. Then f(µ)  F(H).

onto G1, if

THEOREM

Let H be a group and v  F(H). Let f be a

1. if µ is a fuzzy group of G1 then µf is a fuzzy group of G

homomorphism of G into H. Then f -1 (v)  F(G).

2. if µf is a fuzzy group of G the µ is a fuzzy group of G1.

THEOREM

7. ABELIAN FUZZY SUBGROUPS

Let { µi | i  I }  F(G). Then iI i  F(G)

THEOREM

5.

Let µ  FP(G).

SOME PROPERTIES OF FUZZYSUBGROUP

Then the following assertions are equivalent : (1) µ(yx) = µ(xy) x, y  G; Then, µ is called an Abelian fuzzy subset of G.

Let S a fuzzy subgroup of µ iff ^

µ ^ (xy -1 ) = µ ^ (x)  µ ^ (y) for all x, y  S S

S

S

(2) µ(xyx -1 ) = µ(y)  x, y  G PROPOSITION

(3) µ(xyx -1 )  µ(y)  x, y  G. (4) µ(xyx -1 )  µ(y)  x, y  G. (5) µo = oµ   FP(G).

If µ is a fuzzy subgroup, then G m = {(x, µ G (x) m | x  G} is fuzzy subgroup.

COROLLARY

The fuzzysubgroup G n is a fuzzysubgroup of G m , if m  n.

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Now define µ2 on G by

8. NORMAL FUZZY SUBGROUPS Let µ  F(G). Then µ is called a normal fuzzy subgroup of G if it is an Abelian fuzzy subset of G. Let N F(G) denote the set of all normal fuzzy subgroups of G. If µ,   F(G) and there exits u  G such that µ(x) =  (uxu -1 )  x  G, then µ and

 are called conjugate fuzzy subgroups

  1 when x  1    1  1 ( x)   when x  1  4  1   5 when x  ior  i   

and µ =  u , where  n (x) =  (uxu -1 )  x  G.

Then µ1 is a fuzzy subgroup of G such that

Also 1G and 1{e} are normal fuzzy subgroup of G.

µ1  µ. µ1(xyx-1 ) = µ1 (y)  µ(x),  x, y  G Hence µ1µ

If G is a commutative group, every fuzzy subgroup of G is normal.

A fuzzy subgroup of G is normal if and only if µ = µ z z  G.

THEOREM

If µ  N F(G) and   F(G), then µ   is a normal fuzzy subgroup of .

THEOREM

Supposeµ  F(G).

Let N(µ) = { x | x  G, µ(xy) = µ(yx), y  G}.

9. CONCLUSION

Then N(µ) is a subgroup of G.

A. Normal Fuzzy Subgroup Of The Fuzzy Subgroup ν

Let µ,   F(G) and µ  .

Then µ is called a normal fuzzy subgroup of the fuzzy subgroup  , denoted by µ , if µ(xyx-1 )  µ(y)   (x) x, y  G EXAMPLE Let G = {1, - 1, i, - i} G1 = {1, - 1, i, - i} G2 = {1, - 1} G3 = {1} Then G = G1  G2  G3 = {1}.

Also G2  G1, G3 G2 and G1| G2 and G2 | G3 are abelian.

G is abelian.

Define µ on G by

  1 when x  1    1   ( x)   when x  1  2  1   3 when x  ior  i   

BIBLOGRAPHY [1] Akgul,M. Some properties of fuzzy groups,J. Math Anal. Appl,(1988) 93-100 [2] Anthony, J.M and Sherwood,H, Fuzzy groups rede_ned, J.Math.Anal.Appl. 69,14-130,(1979).

[3] Asaad,M., Groups and fuzzy groups, FSS60 39 (1991) 323-328

µ is a fuzzy group on G. © 2020, IRJET

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This project includes the preliminaries required for the concept of Fuzzy subsets and Fuzzy subgroups and properties of fuzzy groups, introduction to the concept of normal fuzzy groups. Fuzzy sets are sets whose elements have degree of membership. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition - an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1].

This work focused on the study of fuzzy subsets and fuzzy subgroups. This helps to investigate the concept of fuzzy groups and obtain some results. Fuzzy group theory has many applications in Physics, chemistry, bioinformatics and computer science problems. Applications of fuzzy set theory have also been established in many areas as engineering, medicine, psychology, economics, robotics, etc. Thus in this modern society the importance of fuzzy set is realized and developments on this topic are increasing day by day. A study on the structure of the collection of fuzzy subgroups will be a prosperous venture. This project has made a humble beginning in this direction.

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[4] Azriel.Rosenfeld, John Mordeson, Kiran.R.Bhutani,Fuzzy Group Theory, Stud Fuzz 182, 1-39 (2005) [5] Azriel. Rosenfeld, Fuzzy groups, J.Math.Anal.Appl, 35, 512-517, (1971) [6] George J.Klir and Bo Yuan, Fuzzy sets and fuzzy logic theory and applications published by Prentice-Hall of india private limited,New Delhi, Third edition (2000) [7] John.B.Fraleigh A First Course in Abstract Algebra, Addision Wesely, Third edition, (1999) [8] John.N.Moderson and D.S Malik, Fuzzy Commutative Algebra World Scienti_c Publishing, (1998) [9] Nanda, S and Das, N.R Fuzzy Mathematical Concept [10] Zadeh, L.A. [ 1965 b], "Fuzzy sets".Information and Control, 8(3), pp. 338-353 (1965)

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