Some Properties of Bipolar Fuzzy Normal HX Subgroup and its Normal Level Sub HX Groups

Some Properties of Bipolar Fuzzy Normal HX Subgroup and its Normal Level Sub HX Groups (GRDJE/ Volume 2 / Issue 1 / 010)

C. Definition 2.3 A fuzzy subset  of X is said to have sup property if, for any subset A of X, there exist a 0  A such that  (a0) = max {  (a) ; aA}.

D. Definition 2.4 A fuzzy subset  of X is said to have inf property if, for any subset A of X, there exist a 0  A such that  (a0) = min {  (a) ; aA}.

E. Definition 2.5 [13] Let G be a finite group. In 2G {}, a non-empty set   2G {} is called a HX group of G, if  is a group with respect to the algebraic operation defined by AB = { ab / a  A and b  B}.

F. Definition 2.6 [16] A mapping f from a HX group 1 to a HX group 2 is said to be a homomorphism if f (XY) = f(X) f(Y) for all X,Y  1.

G. Definition 2.7 [16] A mapping f from a HX group 1 to a HX group 2 ( 1 and 2 are not necessarily commutative) is said to be an antihomomorphism if f (XY) = f(Y) f(X) for all X,Y 1.

H. Definition 2.8 [17] Let  be a non-empty set. A bipolar-valued fuzzy set or bipolar fuzzy set λ in  is an object having the form λ = {A, λ+(A), λ(A) : A  } where λ+ :  → [0,1] and λ :  → [1,0] are mappings. The positive membership degree λ+(A) denotes the satisfaction degree of an element A to the property corresponding to a bipolar-valued fuzzy set λ = {A, λ+(A), λ(A): A   } and the negative membership degree λ(A) denotes the satisfaction degree of an element A to some implicit counter property corresponding to a bipolar-valued fuzzy set λ ={A, λ+(A), λ(A): A  }. If λ+(A)  0 and λ(A) = 0, it is the situation that A is regarded as having only positive satisfaction for λ  = {A, λ+(A), λ(A): A  }. If λ+(A) = 0 and λ(A)  0, it is the situation that A does not satisfy the property of λ = {A, +  λ (A), λ (A) : A  }, but somewhat satisfies the counter property of λ = {A, λ+(A), λ(A): A}.It is possible for an element A to be such that λ+(A)  0 and λ(A)  0 when the membership function of property overlaps that its counter property over some portion of . For the sake of simplicity, we shall use the symbol λ  = ( λ+, λ ) for the bipolar-valued fuzzy set λ = {A, λ+(A), λ(A) : A  }.

I. Definition 2.9 Let f be any function from a set 1 to a set 2, and let λ be any bipolar fuzzy subset of X. Then λ is called f-invariant if f(X) = f(Y) implies λ+(X) = λ+(Y) and λ(X) = λ(Y) ,where X,Y  1.

J. Definition 2.10 [16] Let G1 and G2 be any two groups. Let  = ( +, – ) and φ = (φ +, φ – ) are bipolar fuzzy subsets in G1 and G2 respectively. Let 1  2G1 {} and 2  2G2 {} are HX groups defined on G1 and G2 respectively. let λ = (λ+ , λ– ) and φ = (φ+ , φ– ) are bipolar fuzzy subsets defined on 1 and 2 respectively induced by  and φ. Let f : 1 2 be a mapping , then the image f(λ) of λ is the bipolar fuzzy subset f (λ) = ((f (λ))+ , (f (λ))– ) of 2 defined by for all f(X) = Y 2 ,where X 1

and

also the pre-image f –1(φ ) of φ under f is a bipolar fuzzy subset of 1 defined by (f –1(φ))+ (X) = φ+ (f(X)) , (f –1(φ))– (X) = φ– (f(X)).

K. Definition 2.11 [17] Let  be a bipolar fuzzy subset defined on G. Let   2G {} be a HX group of G. A bipolar fuzzy set λ defined on  is said to be a bipolar fuzzy subgroup induced by  on  or a bipolar fuzzy HX subgroup of  if for X , Y , 1) λ+(XY) ≥ min {λ+(X), λ+(Y)} 2) λ(XY) ≤ max {λ(X), λ(Y)} 3) λ+(X1) = λ+(X) , λ(X1) = λ(X), where λ+(X) = max {+(x) / for all x  X  G} and All rights reserved by www.grdjournals.com

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