Intuitionistic Fuzzy Algebra

CONTENT OF THE TALK ON INTUITIONISTIC FUZZY ALGEBRA 1. What are Intuitionistic fuzzy sets? How they are different from ordinary sets and fuzzy sets ? 2. Some examples of Intuitionistic fuzzy sets 3. Operation on Intuitionistic fuzzy sets ( like Union, Intersection, Complementation , Cartesian product of Intuitionistic fuzzy sets ) 4. (  ,  )-cut set of Intuitionistic fuzzy set and some of their results 5. Intuitionistic Fuzzy Subgroups and their examples 6. Types of Intuitionistic fuzzy subgroups ( like IFNSG , IFASG , IFCSG ) 7. Sum and Product of two Intuitionistic fuzzy subgroups 8. Homomorphism of Intuitionistic fuzzy subgroups 9. Translation of Intuitionistic fuzzy sets and Intuitionistic fuzzy subgroups under the Operators T+ and T10. t- Intuitionistic fuzzy sets and t- Intuitionistic fuzzy subgroups 1. What are Intuitionistic fuzzy sets? Definition Let X be a non-empty set. An Intuitionistic fuzzy set (IFS) A of X is an object of the form A = { < x , A(x) , A(x) > : x X}, where A : X  [0, 1] and A : X  [0, 1] define the degree of membership and degree of non-membership of the element xX respectively and for any x X , we have 0  A(x) + A(x)  1 . Remark (1) When A(x) + A(x) = 1 , i.e. when A(x) = 1 - A(x) . Then A is called fuzzy set. (2) When A is ordinary subset of X , then ( a ) If x  A , then A(x) = 1 and ( b ) If x  A , then A(x) = 0 A(x) = 0 and A(x) = 1. 2. Some Examples of Intuitionistic fuzzy set Example (1) Let G be the Klein 4-group { e , a , b , ab } , where a2 = b2 = e and ab = ba. Define A = { < e , 0.9, 0.1 > , < a , 0.65 ,0.3 > , < b , 0.61, 0.29 > , < ab, 0.6, 0.31 >} be IFS in G. Example (2) Let G = S3 = { i , (12) ,(13) ,(23) ,(123),(132)}be the symmetric group. Consider the functions A : S3  [0, 1] and A : S3  [0, 1] defined by  1 ; if x = i   A ( x)   0 ; if x 2 = i 0.6 ; if x 3 = i  and  0 ; if x = i   A( x)  0.5; if x 2 = i 0.3 ; if x 3 = i  is intuitionistic fuzzy set A of S3 . 2. Operation on Intuitionistic fuzzy sets Definition (3.1) Let A = { < x, A(x), A(x) > : x X} and B = { < x , B(x) , B(x) > : x X} be any two IFS’s of X , then (i) A  B if and only if A(x)  B(x) and A(x)  B(x) for all xX (ii) A = B if and only if A(x) = B(x) and A(x) = B(x) for all xX (iii) AC = { < x, (AC )(x) , (AC )(x) > : x X} , where (AC )(x) = A(x) and (AC )(x) = A(x) for all xX (iv) ฀ A = { < x , A(x) , 1- A(x) > : x X} ( called the Necessity operator on IFS A) (v)  A = { < x , 1- A(x) , A(x) > : x X} ( called the possibility operator on IFS A) (vi) A  B = { < x, (A B )(x) , (A B )(x) > : x X} , where (A B )(x) = Min{ A(x) , B(x)} = A(x)  B(x) and (A B )(x) = Max{ A(x) , B(x) } = A(x)  B(x) (vii) A  B = { < x, (A B )(x) , (A B )(x) > : x X} , where (AB )(x) = Max{ A(x) , B(x)} = A(x) B(x) and (A B )(x) = Min{ A(x) , B(x) } = A(x)  B(x) (viii) A  B = { < ( x , y) , AB(x, y), AB (x, y) > : x  X and y Y} where AB(x, y) = Min{A(x), B(y)} and AB (x, y) = Max{A(x),B(y)} Note : Obviously, if A is an ordinary fuzzy set, then ฀ A = A =  A . This shows that IFS are proper extensions of the ordinary fuzzy sets. Definition (3.2) : Let ( X , . ) be a groupoid and A , B be two IFS’s of X . Then the Intuitionistic fuzzy sum and product of A and B are denoted by A + B and AB and are defined as follows : A + B = ( A+B , A+B ) , where   Sup min{A (a ),  B (b)} ; if x = a + b A+B ( x)     0   Inf max{ A (a ),  B (b)} ; if x = a + b  A+B ( x)     For any xX 1 A B( x)  ; otherwise , for all x  X   A B (x) , A B ( x)  , where   Sup  min{ A ( y),  B ( z)}] ; ; 0    A B (x)   and ; otherwise if x = yz if x is not expressible as x = yz  if x = yz  Inf  max{ A ( y),  B ( z)}] ; ; if x is not expressible as x = yz 1    A B (x)   and , 4. (  ,  )-cut set of Intuitionistic fuzzy set and some of their results Definition (4.1): (  ,  ) – Cut Set of Intuitionistic fuzzy set Let A be Intuitionistic fuzzy set of a universe set X . Then (  ,  )-cut set of A is a crisp subset C , (A) of the IFS A is given by C , (A) = { x : x X such that A(x)   , A(x)   } , where  ,   [ 0 , 1 ] with  +   1 . Proposition (4.2) If A and B be two IFS’s of a universe set X , then following holds (i) C , (A)  C  , (A) if    and (ii) C1- , (A)  C  , (A)  C , 1-(A)  (iii) A  B implies C , (A)  C , (B) (iv) C , (A  B) = C , (A)  C , (B) (v) C , (A  B)  C , (A)  C , (B) equality hold if  +  =1 (vi) C , ( A i ) =  C , (A i ) (vii) C0 , 1 (A) = X. (viii) C , ( A  B ) = C , ( A)  C , ( B) (ix) C,(AB) = C,(A) C,(B) (x) C , (A) + C , (B)  C , (A + B) and the equality holds if  +  =1 (xi) C , (A)C , (B)  C , (AB) and the equality holds if  +  =1 5. Intuitionistic Fuzzy Subgroups and their examples Definition (5.1) An IFS A = { < x, A(x), A(x) > : x G} of a group G is said to be intuitionistic fuzzy subgroup of G ( In short IFSG) of G if (i) (ii) (iii) A(xy)  Min {A(x) , A(y) } A(x-1) = A(x) A(xy)  Max {A(x) , A(y) } (iv) A(x-1) = A(x) , for all x , y G Equivalently, an IFS A of a group G is IFSG of G if A(xy -1)  Min{A(x) , A(y)} and A(xy -1)  Max{A(x) , A(y) }holds for all x , y G. Examples of Intuitionistic fuzzy subgroups Example (5.2): Let G = S3 = { i , (12) ,(13) ,(23) ,(123),(132)}be the symmetric group. Consider the IFS A of G defined by  1 ; if x = i   A ( x)   0 ; if x 2 = i 0.6 ; if x 3 = i  and  0 ; if x = i   A( x)  0.5; if x 2 = i 0.3 ; if x 3 = i  Then A is IFSG of group G ( As it satisfies all the conditions of IFSG’s ) Example(5.3). Let G = D3 = < a , b : a 3 = b2 = e , ba = a 2b > be the dihedral group with six elements . Define the IFS A = ( A , A) of D3 by 0.8 0.7 A (x) =  if x  < b > if otherwise and 0.1 if x  < b > 0.2 if otherwise  A (x) =  i.e., A ={ < e, 0.8, 0.1>, < a , 0.7, 0.2>,< a 2 , 0.7, 0.2>,< b, 0.8, 0.1>,< ab, 0.7, 0.1>,< ba , 0.8, 0.1>} It is easy to verify that A is IFSG of G. Definition (5.4) An IFSG A = { < x, A(x), A(x) > : x G} of a group G said to be intuitionistic fuzzy normal subgroup of G ( In short IFNSG) of G if (i) A(xy) = A(yx) (ii) A(xy) = A(yx) , for all x , y G Remark (5.5): It is easy to verify that an IFSG A of a group G is normal if and only if (i) A(g-1 x g) = A(x) (ii) A(g-1 x g) = A(x) and , for all x , g G. Definition (5.6): Intuitionistic fuzzy left and right cosets Let G be a group and A be IFSG of group G . Let x G be a fixed element. Then the set xA ={(g , xA(g), xA(g)): g G} where xA(g) = A(x -1g) and xA(g) = A(x -1g) for all gG is called intuitionistic fuzzy left coset of G determined by A and x similarly , the set Ax = { ( g , Ax(g) , Ax(g) ) : g G } where Ax(g) = A(gx -1) and Ax(g) = A(gx -1) for all g G is called the intuitionistic fuzzy right coset of G determined by A and x . Remark (5.7) : It is clear that if A is intuitionistic fuzzy normal subgroup of G, then the intuitionistic fuzzy left coset and intuitionistic fuzzy right coset of A on G coincide and in this case, we call intuitionistic fuzzy coset instead of intuitionistic fuzzy left or intuitionistic fuzzy right coset . Theorem (5.8) Let A be intuitionistic fuzzy subgroup of a group G and x be any fixed element of G . Then (i) x . C , (A) = C , (xA) (ii) C , (A).x = C , (Ax) for all ,   [0,1] with  +   1 Theorem (5.9) : Let A be intuitionitic fuzzy subgroup of group G . Let x , y be elements of G such that A(x)  A(y) =  and A(x)  A(y) =  . Then (i) xA = yA  x -1y  C , (A) (ii) Ax = Ay  xy -1 C , (A) Example(5.10) : Let G be a group . Then A = { < x , A(x) , A(x) > , x  G : A(x) = A(e) and A(x) = A(e) } is intuitionistic fuzzy normal subgroup of G . Definition (5.11) Let G = D3 = < a , b : a 3 = b2 = e , ba = a 2b > be the dihedral group with six elements . Define the IFS A = ( A , A) of D3 by A ={ < e, 0.8, 0.1>, < a , 0.7, 0.2>, < a 2 , 0.7, 0.2>,< b, 0.8, 0.1>, < ab, 0.7, 0.1>, < ba , 0.8, 0.1>} It is easy to check that A is IFSG of G but A is not IFNSG of G, for A (ab) = 0.7  0.8 = A (ba ) . Theorem (5.12) : If A be an IFS of a group G. Then A is IFSG ( IFNSG ) of G if and only if C , (A) is a subgroup ( normal subgroup) of group G for all  ,   [0 ,1] with  +   1. Theorem (5.13) : If A and B be two IFSG’s ( IFNSG’s) of a group G, then A  B is IFSG ( IFNSG ) of group G. Remark (5.14): Union of two IFSG’s ( IFNSG’s) of a group G need not be IFSG (IFNSG ) of group G. Example (5.15): Consider the Klein four group. G = { e , a , b , ab } , where a2 = e = b2 and ab = ba For 0  i  5 , let ti , si [ 0, 1] such that 1 = t0 > t1 > …> t5 and 0 < s0< s1<…< s5 Define Intuitionistic fuzzy subset A and B as follows: A = { < x, A(x), A(x) > : x G} and B = { < x, B(x), B(x) > : x G} , where A(e) = t1, A(a) = t3 , A(b) = A(ab) = t4 , A(b) = A(ab) = s4, A(a)= s3, A(e)= s1 B(e) = t0, B(a) = t5, B(b) = t2, B(ab) = t5, B(b) = B(ab) = s5, B(a)= s2, B(e)= s0 Clearly A and B are IFSG of the group G . (i) Now A  B = { < x, (A B )(x) , (A B )(x) > : x G} , where (AB )(x) = Max{ A(x) , B(x)} = A(x) B(x) and (A B )(x) = Min{ A(x) , B(x) } = A(x)  B(x) Here (AB )(e) = t0, (AB )(a)= t3 , (AB )(b) = t2 , (AB )(ab) = t4 (A B )(e) = s0 , (A B )(a) = s2 , (A B )(b) = s4 , (A B )(ab) = s4 C t3 , s4(A) = { x : x G such that A(x)  t3 , A(x)  s4 } = {a , e } C t3 , s4(B) = { x : x G such that B(x)  t3 , B(x)  s4 } = { e } C t3 , s4(A B) ={ x : x G such that AB(x)  t3 ,  AB (x)  s4 } = { x : x G such that A(x) B(x) t3 ,  A (x)  B (x) s4 } ={e,a ,b} Since { e , a , b } is not a subgroup of G i.e. C t3 , s4(A B) is not a subgroup of G and hence A  B is not IFSG of group G. Theorem (5.16) Let A, B be IFSG’s ( IFNSG’s) of group G1 and G2 respectively. Then A B is also IFSG ( IFNSG ) of group G1  G2 . Remark(5.17) If A and B are IFS of group G1 and G2 respectively. If A  B is also IFSG of group G1  G2 , then it is not necessarily that both A and B should be IFSG of G1  G2. Example(5.18): Let G1 = { e1 , a } , where where x2 = y2 = e2 and a2 = e1 and let G2 = { e2 , x , y , xy }, xy = yx . Then G1  G2 = { (e1 , e2) , (e1 , x ), (e1 , y) , (e1 , xy), (a , e2), (a , x), (a , y) , (a , xy) } Let A = { < e1 , 0.7, 0.2 > , < a , 0.6, 0.3 > } and B = {< e2 , 0.9, 0.1 > , < x ,1, 0 >, < y , 0.8, 0.2 >, < xy , 0.7, 0.2 > } be IFS of G1 and G2 respectively. Then A  B = {< (e1 , e2) , 0.7, 0.2 > , < (e1 , x ), 0.7 , 0.2 > , < (e1 , y), 0.7, 0.2 > , < (e1, xy), 0.7, 0.2 > , < (a , e2) ,0.6, 0.3 > , < (a , x), 0.6, 0.3 > ,< (a , y), 0.6, 0.3>, < (a , xy) , 0.6 , 0.3 > } Here A  B is also IFSG of group G1  G2 , where as A is IFSG of G1 but B is not IFSG of G2 as C0.8, 0.2( B) = { x , y } is not a subgroup of G2 . Proposition(5.19) Let A and B be two IFSG’s of group G. Then AB is IFSG of group G if and only if AB = BA Corollary (5.20) Let A be IFSG of group G , then AA = A Definition (5.21) Let A is IFSG of a group G . Then A is called intuitionistic fuzzy abelian subgroup (IFASG) of G if and only if C , ( A) is an abelian subgroup of G for all  ,   (0,1] with 0 <  +   1. Remark (5.22): If G is abelian group, then every IFSG of G is IFASG of G , but converse need not be true Example(5.23): Let G be abelian group and A be any IFSG of G , then C , ( A) being subgroup of G is also abelian , for all  ,   (0,1] with 0 <  +   1 Thus A is IFASG of G. But converse of it is not true . Consider G = S3 = { i , (12) ,(13) ,(23) ,(123),(132)}be the symmetric group. Consider the IFS A of G defined by  1 ; if x = i   A( x)   0 ; if x 2 = i 0.6 ; if x 3 = i  and  0 ; if x = i   A( x)  0.5; if x 2 = i 0.3 ; if x 3 = i  Clearly A is IFSG of group G . Moreover C , ( A) = { i } or { i , (123) ,(132)}are abelian subgroups of G , for all  ,   (0,1] with 0 <  +   1. Hence A is IFASG of G, but G is non-abelian group. Definition(5.24) Let A be IFSG of a group G , then A is called cyclic intuitionistic fuzzy subgroup (CIFSG) of group G , if C , (A) is a cyclic subgroup of G , for all  ,   (0,1] with 0 <  +   1. Remark(5.25): (i) If G be a cyclic group, then every IFSG of G is CIFSG of G , but converse need not be true Proof. Let G = < x > be cyclic group and let A be any IFSG of G then We know that A(xn)  A(xn-1)  A(xn-2) ……..  A(x2)  A(x) and A(xn)  A(xn-1)  A(xn-2) ……..  A(x2)  A(x) holds for all nN Therefore if xm  C , (A) for some mN , then xm , xm+1 , xm+2 ,……  C , (A) i.e. C , (A) = < x -1> , which is a cyclic subgroup of G, for all  ,   (0,1] with 0 <  +   1. Hence A is CIFSG of G. Converse need not be true : For example see example(5.23) A is cyclic intuitionistic fuzzy subgroup of G , but G is not cyclic. (ii) Every CIFSG of a group G is IFASG , but converse need not be true see example(5.23) 6. Homomorphism of Intuitionistic fuzzy subgroups Definition (6.1): Let X and Y be two non-empty sets and f : X  Y be a mapping . Let A and B be IFS’s of X and Y respectively . Then the image of A under the map f is denoted by f (A) and is defined as  f  A  ( y)   f  A  ( y) ,  f  A ( y)  , where 1 1    {  A ( x) : x  f (y)} and  {  A ( x) : x  f (y)}  f  A  ( y)   0 ; otherwise 1 ; otherwise      f  A  ( y)    {  ( x) : x  f  i.e. f  A  ( y)     A ( 0, 1) 1 (y)} , {  A( x) : x  f 1 (y)} ;  otherwise Also the pre-image of B under f is denoted by f -1( B) and is defined as f 1  B   x   (  f 1 B ( x) ,  f 1 B ( x)) where f 1 f 1  B ( x)   B ( f ( x)) and  B x   ( B ( f ( x)) ,  B ( f ( x)) ) f 1  B ( x) =  B ( f ( x)) i.e. Proposition (6.2) : Let f : X  Y be a mapping. Then the following holds (i) f  C ,  ( A)   C ,  ( f ( A )) ,  A  IFS( X ) (ii) f 1  C ,  ( B)  = C ,  ( f 1 ( B )) ,  B  IFS(Y) Theorem(6.3) : Let f : G1  G2 be surjective homomorphism. Let A be IFSG (IFNSG) { IFASG} [CIFSG ] of group G1. Then f (A) is IFSG (IFNSG) {IFASG}[CIFSG] of group G2 . Theorem (6.4) : Let f : G1  G2 be homomorphism of group G1 into a group G2. Let B be IFSG ( IFNSG) { IFASG}[CIFSG ] of group G2. Then f -1(B) is IFSG (IFNSG) { IFASG} [CIFSG ] of group G1 . Theorem(6.5) : Let G be a group and A be IFNSG of group G . Then there exists a natural homomorphism f : G  G/A defined by f (x) = xA ; for all x G Theorem(6.6) : Let A= { < x , A(x) , A(x) > , x  G such that. A(x) = A(e) , A(x) = A(e)} be IFNSG of group G and B be an IFS of group G and f : G  G/A be natural homomorphism defined by f (x) = xA , for all x G , then f -1( f (B)) = A  B 7. Translation of Intuitionistic fuzzy sets and Intuitionistic fuzzy subgroups under the Operators T+ and T- ) Definition(7.1) Let A = (A , A) be an IFS of X and   [ 0, 1]. We define T +(A)(x) = (T +(x),  T + (x)) and T -(A)(x) = (T -(x),  T - (x)) , where T +(x) = Min{ A(x) +  , 1} ,  T + (x) = Max{ A(x) -  , 0 } and T -(x) = Max { A(x) -  , 0} ,  T - (x) = Min{ A(x) +  , 1 } T+(A) and T-(A) are respectively called the - up and - down intuitionistic fuzzy operators of A. We shall call T + and T - as the intuitionistic fuzzy operator. Results (7.2) : The following results can be easily verified from definition (i) T0+(A) = T0 -(A) = A (ii) T1 +(A) = 1 (iii) T1 -(A) = 0 Remark (7.3) It can be easily checked that if A is IFS of X , then both T +(A) and T -(A) are IFS of X. In other words 0  T +(x) +  T +(x)  1 and for all x X 0  T -(x) +  T - (x)  1 , Example(7.4): Let X = {1, , 2 }. Let A = {< 1, 0.3, 0.4 > , <  ,0.1, 0.25 > , < 2 ,0.5, 0.3 >} be an IFS of X Take  = 0.2, then T +(A) = {{< 1, 0.5, 0.2 > , <  ,0.3, 0.05 > , < 2 ,0.7, 0.1 >} T -(A) = {{< 1, 0.1, 0.6 > , <  ,0, 0.45 > , < 2 ,0.3, 0.5 >} and Proposition(7.5) For any IFS A of X and   [0,1] , we have T +(Ac) = ( T -(A))c (ii) T - (Ac) = ( T +(A))c Remark (7,6) In general, for any given   [0,1], the operators T + and T - donot commute each other. In other words T -(T +(A)) and T +(T - (A)) may be different from A . Example(7.7) Let X = {1, , 2 }.Let A = {< 1, 0.3, 0.4 > , <  ,0.1, 0.25 > , < 2 ,0.5, 0.3 >} be an IFS of X Take  = 0.3, then T +(A) = {{< 1, 0.6, 0.1 > , <  ,0.4, 0 > , < 2 ,0.8, 0 >} And T -(A) = {{< 1, 0, 0.7 > , <  ,0, 0.55 > , < 2 ,0.2, 0.6 >} Now T -(T +(A)) = { < x , *T -(x), * T - (x) > ; x  X } , where *T -(x) = Max { T +(x) -  , 0 } and * T - (x) = Min {  T + (x) +  , 1 } Therefore T -(T +(A)) = {< 1, 0.3, 0.4 > , <  ,0, 0.3 > , < 2 ,0.5, 0.3 >}  A And T +(T - (A)) = { < x , **T +(x), ** T + **T +(x) = Min { T -(x) +  , 1 } and (x) > ; x  X } , where ** T + (x) = Max {  T - (x) -  , 0 } Therefore T +(T - (A)) = {< 1, 0.3, 0.4 > , <  ,0.3, 0.25 > , < 2 ,0.5, 0.3 >}  A It may be checked that if we take  = 0.2, then T -(T +(A)) = A Theorem (7.8): For any IFS A of X and   [0,1] (a) T -(T +(A)) = A    Min { 1 – p , q } , where and q = Min { A(x) : x X } and q = Max { A(x) : x X }. (b) T +(T - (A)) = A    Min { p , 1 - q }, where Proposition(7.9): For any IFS A of X and  ,   [0,1]  T(   )  A  if  + <1 (a) T  (T   A )  T  (T   A )    1 if  +  1 p = Max { A(x) : x X } p = Min { A(x) : x X }   T(   )  A  if  + <1 (b) T  (T   A )  T  (T   A )   if  +  1  0 Remark (7.10): For any IFS’s A and B of X with A  B , we have T +(A)  T +(B) and T - (A)  T - (B) , for all   [0,1] Translation of intuitionistic fuzzy groupsHere we study the action of T + and T - on IFSG of a group G. We prove that these operators takes on IFSG to an IFSG and preserve some properties of Intuitionistic fuzzy group. Theorem(7.11): If A is IFSG (IFNSG) of a group G, then T +(A) and T -(A) are IFSG (IFNSG) of G, for all   [0,1]. Remark(7.12) : If T +(A) or T -(A) is IFSG of group G , for a particular value of   [ 0,1] , then it cannot be deduced that A is IFSG of group G. Example(7.13): Let G be the Klein 4-group { e , a , b , ab } , where a2 = b2 = e and ab = ba. Define A = { < e , 0.9, 0.1 > , < a , 0.65 ,0.3 > , < b , 0.61, 0.29 > , < ab, 0.6, 0.31 >} be IFS in G. Take  = 0.4 , then T +(A) = { < e ,1 , 0 > , < a ,1 , 0 > , < b, 1 , 0 > , < ab , 1 , 0 >} = 1 Clearly , T +(A) is an IFSG of G , however A is not IFSG of G. Proposition (7.14) Let A be IFS of a group G such that T +(A) be IFSG (IFNSG) of G , for some   [0,1] with  < Min { 1- p , q } , then A is IFSG (IFNSG) of G , where p = Max { A( x ) : x G – GA } and q = Min { A( x ) : x G –GA } and GA = { x  G ; A( x ) = A( e ) and A( x ) = A( e ) } is a subgroup of G. Proposition (7.15) Let A be IFS of a group G such that T -(A) be IFSG (IFNSG) of G , for some   [0,1] with  < Min { 1- p* , q*}, then A is IFSG (IFNSG) of G , where p* = Max { A( x ) : x G –GA} and q* = Min { A( x ) : x G –GA } and GA = { x  G ; A(x) = A(e) and A(x) = A(e)} is a subgroup of G. Proposition (7.16): For any IFSG of a group G and x  G ,   [0,1], we have (i) (T +(A))x = T +(Ax) (ii) x(T +(A)) = T +(xA) (iii) (T -(A))x = T -(Ax) (iv) x(T -(A)) = T -(xA) Now , we give an example to show that translate of intuitionistic fuzzy abelian subgroups need not be intuitionistic fuzzy abelian subgroups Example (7.17): Let G = S3 = { i , (12) ,(13) ,(23) ,(123),(132)}be the symmetric group. Consider the IFS A of G defined by  1 ; if x = i   A ( x)   0 ; if x 2 = i 0.6 ; if x 3 = i  and  0 ; if x = i   A( x)  0.5; if x 2 = i 0.3 ; if x 3 = i  Clearly A is IFSG of group G . Moreover C , ( A) = { i } or { i , (123) ,(132)}are abelian subgroups of G , for all  ,   (0,1] with 0 <  +   1 Thus A is IFASG of G. Now if we take  = 0.5 , then T +(A) = { < i , 1 , 0 > , < (12) , 0.5 , 0 > , < (13) , 0.5 , 0 > , < (23) , 0.5 , 0 >, < (123) , 1 , 0 > , < (132) , 1 , 0 > } and T -(A) = { < i , 0.5 , 0.5 > , < (12) , 0 , 1 > , < (13) , 0 , 1 > , < (23) , 0 , 1 >, < (123) , 0.1 , 0.8 > , < (132) , 0.1 , 0.8 > } When we take  = 0.5 and  = 0.2 , then C , ( T +(A) ) = S3 , which is not abelian Hence T +(A) is not IFASG of S3 . However C , ( T -(A) ) is either empty set or { i, (123) ,(132)} for all  ,   (0,1] with 0 <  +   1 Therefore T -(A) is IFASG of S3. 8. t - Intuitionistic fuzzy subset and t - Intuitionistic fuzzy subgroup Definition (8.1). Let A be an IFS of a group G. Let t  [0,1]. Then the IFS At of G is called the t- intuitionistic fuzzy subset of G w.r.t. IFS A and is defined as At  ( At , At ) , where  At ( g )  min{ A( g ), t} and  At ( g )  max{ A( g ), 1-t}, for all g  G Remark (8.2) : Clearly, A1 = A and A0 = 0 , where 0 (x) = (0, 1) Re sult. (8.3). Let At  (  At , At ) and Bt  (  Bt , Bt ) be two t-IFS of a group G. Then ( A B)t = At  Bt . Definition (8.4). Let A be an IFS of a group G. Let t  [0,1]. Then A is called t- intuitionistic fuzzy subgroup ( In short t- IFSG ) ( t – intuitionistic fuzzy normal subgroup ) (In short tIFNSG) of G if At is IFSG (IFNSG) of G. Proposition(8.5) If A is IFSG (IFNSG) of a group G, then A is also t- IFSG (t-IFNSG) of G. Remark (8.6). The converse of above proposition (8.5) need not be true Example (8.7). Let G = { e , a , b , ab }, where a 2 = b2 = e and ab = ba be the Klein four group. Define the IFS A = {< e , 0.1 , 0.1> , < a , 0.3 , 0.3> , < b , 0.3 , 0.4> , < ab , 0.2 , 0.4> } of G. Clearly , A is not IFSG of G . Take t = 0.05 . Then A(x) > t for all x G A(x) < 1- t for all x G also  A ( x)  min{ A( x), t}  t  A ( x)  max{ A( x), 1-t}  1  t , for all x  G and t t Therefore,  At ( xy)  min{ At ( x),  At ( y)} and  At ( xy)  max{ At ( x),  At ( y)} hold Further, as a 1  a , b1  b , (ab) 1  (ab). So  At ( x1 )   At ( x) and  At ( x1 )   At ( x) hold Hence A is t- IFSG of G. Example(8.8). Let G = D3 = < a , b : a 3 = b2 = e , ba = a 2b > be the dihedral group with six elements . Define the IFSG A = ( A , A) of D3 by 0.8 0.7 A (x) =  if x  < b > if otherwise and 0.1 if x  < b > 0.2 if otherwise  A (x) =  i.e., A ={ < e, 0.8, 0.1>, < a , 0.7, 0.2>,< a 2 , 0.7, 0.2>,< b, 0.8, 0.1>,< ab, 0.7, 0.1>,< ba , 0.8, 0.1>} Note that A is not IFNSG of G , for A (ab) = 0.7  0.8 = A (ba ) . Now, take t = 0.6 , we get min{ A( x1 g ), t} = t = min{ A( gx1 ), t} max{ A( x1 g ), 1- t} = 1- t = max{ A( gx1 ), 1- t} , for all x , g  G. i.e xA ( g )   A x ( g )  t and  xA ( g )   A x ( g )  1  t , for all x  G t t Hence A is t- IFNSG of G. t t and i. Proposition(8.9). Let A be a IFS of group G such that A(x-1) = A(x) and A(x-1) = A(x) hold for all x G. Let t < min{ p , 1- q } , where p = min{ A(x) : for all xG} and q = max{ A(x) : for all xG}. Then A is t-IFSG of G. Proposition(8.10). Intersection of two t-IFSG’s of a group G is also t-IFSG of G. Corollary(8.11). Intersection of a family of t-IFSG’s of a group G is again t-IFSG of G. Theorem(8.12) : Let f : G1  G2 be surjective homomorphism and A be (t-IFSG) [t-IFNSG] of group G1. Then f (A) is (t-IFSG) [t-IFNSG] of group G2 . Theorem (8.13) : Let f : G1  G2 be homomorphism of group G1 into a group G2. Let B be (t-IFSG) [t-IFNSG] of group G2. Then f -1(B) is (t-IFSG) [t-IFNSG] of group G1 . Proposition(8.14) Let G/ At denote the collection of all t – intuitionistic fuzzy cosets of an t-IFNSG A of G. i.e. G/ At = { At x : xG }. Then the binary operations  defined on the set G/ At as follows: At x  At y  At xy , for all x , y  G is a well defined operation and it form a group under the operations . Proposition(8.15) A mapping f : G  G/ At , where G is a group and G/ At is the set of all t-intuitionistic fuzzy cosets of the t-IFNSG A of G defined by f (x) = At x , is an onto homomorphism with ker f = N ( = Ct ,1-t(A)), where t [0,1] ) REFERENCES 1. P.K. Sharma “(, )-Cut of Intuitionistic fuzzy groups” International Mathematics Forum , Vol. 6, 2011, no. 53, 2605-2614 2. P.K. Sharma “Homomorphism of Intuitionistic fuzzy groups ” International Mathematics Forum , Vol. 6, 2011, no. 64, 3169-3178 3. P.K. Sharma “ Intuitionistic fuzzy groups”, International Journal of Data Ware housing & Mining, Vol. 1 , Issue 1 , (2011) , 86-94 4. P.K. Sharma “ Translates of Intuitionistic fuzzy subgroups” , International Journal of Pure and Applied Mathematics , Vol. 72, no.4 , 2011, 555-564 5. P.K. Sharma “On the Intuitionistic fuzzy order of an element in a group”, Research Cell: International Journal of Engineering Science” Special Issue , Sept. 2011 , Vol. 4 , 355362. 6. P.K. Sharma “ On Intuitionistic Magnified Translation in Groups”, International Journal of Mathematical Sciences and Applications, Vol. 2, no.1, 2012, 139-144 7. P.K. Sharma and Vandana Bansal “ Anti Homomorphism of Intuitionistic Magnified Translation in Groups”, International Journal of Mathematical Sciences and Applications, Vol. 2, no.1 , 2012, 161-166 8. P.K. Sharma “ On Intuitionistic Magnified Translation in Rings” International Journal of Algebra, Vol.5 , 2011 , no. 30 , 1451-1458. 9. P.K. Sharma “(, )-Cut of Intuitionistic fuzzy modules”, International Journal of Mathematical Sciences and Applications, Vol. 1 , no.3 , Sept. 2011, 1489-1492 10. P.K. Sharma “ Translates of Intuitionistic fuzzy subrings”, International Review of Fuzzy Mathematics” Vol. 6 , No. 2 , 2011, pp. 77-84 11. P.K. Sharma “ Intuitionistic fuzzy ideal in near rings”, International Mathematics Forum, Vol. 7 , 2012 , no. 16, 769-776 12. P.K. Sharma and M. Yamin “Relative relation modules of SL( 2, p) and PSL(2, p) groups” Journal of Indian Mathematics Society (JIMS), Vol., 80 , no. 1-4 , (2013), 13. P.K. Sharma “ Common fixed point for two pair of compatible mapping” , Proceedings of International Conference on Mathematics and soft computing ( Application in Engineering),held at N.C. College of Engg. Israra , Panipat 4-5th Dec. , 2010 , page 107-113. 14. P.K. Sharma “ On the direct product of Intuitionistic fuzzy groups”, International Mathematical Forum , Vol. 7 , 2012 , no. 11, 523-530 15. P.K. Sharma “ On intuitionistic fuzzy abelian subgroups” Advances of Fuzzy Sets and Systems , Vol. 12(1), 2012 , pp. 1-16 16. P.K. Sharma “ On Translates of intuitionistic fuzzy abelian subgroups” (Communicated) 17. P.K.Sharma “ t- Intuitionistic fuzzy Quotient modules” International Journal of Fuzzy Mathematics & System (IJFMS) Vol., 2, no. 1, 2012, 37-45 18. P.K.Sharma and Ashu Bahl “ Translates of Intuitionistic fuzzy submodules” International Journal of Mathematicals Sciences, Vol. 11, no. 3-4 , 2012, pp. 277-287 19. P.K.Sharma “ Intuitionistic fuzzy module over Intuitionistic fuzzy ring”, International Journal of Fuzzy Mathematics & System (IJFMS), Vol. 2 , no., 2, 2012, 133-140 20. P.K. Sharma “ Anti Fuzzy Submodule of a Module”, Advances of Fuzzy Sets and Systems , Vol. 12(1), 2012 , pp. 49-57 21. P.K.Sharma “ t-Intuitionitic Fuzzy Quotient Group”, Advances in Fuzzy Mathematics, Vol. 7 , no. 1 , 2012 , pp. 1-9 22. P.K.Sharma “ t- Intuitionistic Fuzzy Quotient Ring”, International Journal of Fuzzy Mathematics & Systems, Vol. 2, No. 3 , (2012), pp. 207-216 23. P.K.Sharma and Vandana Bansal, “ On Intuitionistic anti-fuzzy ideals in rings” International Journal of Mathematicals Sciences, Vol. 11, no. 3-4 , 2012, pp. 237-243 24. P.K.Sharma, “ On Intuitionistic Anti-fuzzy submodule of a module”, International Journal of Fuzzy Mathematics & System (IJFMS) Vol. 2, No 2 (2012), 127-132 25. P.K.Sharma, “ Intuitionistic fuzzy submodules with thresholds ( , ) of modules”, International Journal of Mathematics Sciences and Engineering Applications, Vol. 6, No. III, May , 2012 , 21-30 26. P.K.Sharma, “ Intuitionistic Fuzzy Relation on Modules”, International Review of Fuzzy Mathematics (Accepted) 27. P.K.Sharma, “ t- intuitionistic fuzzy subgroups”, International Journal of Fuzzy Mathematics and Systems (IJFMS) , Vol. 2 , no. 3 , 2012 , pp. 233-243 28. P.K.Sharma, “ t- intuitionistic fuzzy subrings”, International Journal of Mathematicals Sciences , Vol. 11 , no. 3-4 , 2012, pp. 265-275 29. P.K.Sharma, “ Lattices of Intuitionistic fuzzy Submodules”, The Journal of Mathematics and Computer Sciences ( Communicated ) 30. P.K.Sharma, “ Lattices of Anti fuzzy Submodules”, ( Communicated ) 31. P.K.Sharma, “ Q-Intuitionistic L-Fuzzy Submodules”, International Journal of Fuzzy Mathematics and Systems (IJFMS ) , Vol. 2 , No. 3, 2012 , pp. 245-252 32. P.K.Sharma, “Structure of Q - Intuitionistic L-Fuzzy Subgroups” (Under preparation) 33. P.K.Sharma, “ On Intuitionistic fuzzy characteristic subgroup”, (Under preparation) 34. P.K.Sharma, “(, )-Cut of Intuitionistic fuzzy modules- II” (Communicated) 35. P.K.Sharma, “Intuitionistic Fuzzy Prime Ideals (Under Preparation)