Intuitionistic Fuzzy 2-norm

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1 In. Journal of Mah. Analysis, Vol. 5, 2011, no. 14, Inuiionisic Fuzzy 2-norm B. Surender Reddy Deparmen of Mahemaics, PGCS, Saifabad, Osmania Universiy Hyderabad , A.P., India bsrmahou@yahoo.com Absrac In his paper, we redefine he noion of fuzzy 2-normed linear space using -norm and fuzzy ani-2-normed linear space using -conorm and inroduced he definiion of inuiionisic fuzzy 2-norm on a linear space. Mahemaics Subjec Classificaion: 46S40, 03E72 Keywords: -norm, -conorm, fuzzy -2-norm, fuzzy -ani-2-norm 1 Inroducion Fuzzy se heory is a useful ool o describe siuaions in which he daa are imprecise or vague. Fuzzy ses handle such siuaion by aribuing a degree o which a cerain objec belongs o a se. The idea of fuzzy norm was iniiaed by Kasaras in [8]. Felbin [5] defined a fuzzy norm on a linear space whose associaed fuzzy meric is of Kaleva and Seikkala ype [7]. Cheng and Mordeson [4] inroduced an idea of a fuzzy norm on a linear space whose associaed meric is Kramosil and Michalek ype [9]. Bag and Samana in [1] gave a definiion of a fuzzy norm in such a manner ha he corresponding fuzzy meric is of Kramosil and Michalek ype [9]. They also sudied some properies of he fuzzy norm in [2] and [3]. Bag and Samana discussed he noions of convergen sequence and Cauchy sequence in fuzzy normed linear space in [1]. They also made in [3] a comparaive sudy of he fuzzy norms defined by Kasaras [8], Felbin [5], and Bag and Samana [1]. In [6] Iqbal H. Jebril and Samana redefined he noion of fuzzy normed linear space using -norm and fuzzy ani-normed linear space using -conorm and inroduced he definiion of inuiionisic fuzzy norm on a linear space. In [12] Surender Reddy inroduced he noion fuzzy ani-2-norm on a linear space and some resuls are inroduced in fuzzy ani-2-norms on a linear space.

2 652 B. Surender Reddy In his paper, we redefine he noion of fuzzy 2-normed linear space using -norm and fuzzy ani-2-normed linear space using -conorm and inroduced he definiion of inuiionisic fuzzy 2-norm on a linear space. In [5], Felbin inroduced he concep of a fuzzy norm based on a Kaleva and Seikkala ype [7] of fuzzy meric using he noion of fuzzy number. Le X be a vecor space over R(se of real numbers). Le : X R (I) be a mapping and le he mappings L, U :[0, 1] [0, 1] [0, 1], be symmeric, non-decreasing in boh argumens and saisfying L(0, 0) = 0 and U(1, 1) = 1. Wrie [ x ] α =[ x 1 α R x 2 α ] for x X, 0<α 1 and suppose for all x X, x 0here exiss α 0 (0, 1] independen of x such ha for all α α 0, (A) x 2 α < (B) inf x 1 α > 0 The quadruple (X,,L,U) is called a Felbin-fuzzy normed linear space and is a Felbin-fuzzy norm if (i) x = 0 if and only if x =0(he null vecor), (ii) rx = r x, x X, r R, (iii) For all x, y X, (a) Whenever s x 1 1, y 1 1 and s+ x + y 1 1, x + y (s + ) L( x (s), y ()). (b)whenever s x 1 1, y 1 1 and s + x + y 1 1, x + y (s + ) U( x (s), y ()). Definiion 1.1 Le X be a vecor space over R(se of real numbers). Le, : X X R (I) be a mapping and le he mappings L, U :[0, 1] [0, 1] [0, 1], be symmeric, non-decreasing in boh argumens and saisfying L(0, 0) = 0 and U(1, 1) = 1. Wrie [ x, z ] α =[ x, z 1 α R x, z 2 α ] for x, z X, 0 <α 1and suppose for all x, z X, x 0, z 0, here exiss α 0 (0, 1] independen of x, z such ha for all α α 0, (A) x, z 2 α < (B) inf x, z 1 α > 0 The quadruple (X,,, L,U) is called a Felbin-fuzzy 2-normed linear space and, is a Felbin-fuzzy 2-norm if (i) x, z =0if and only if x, z are linearly dependen, (ii) rx, z = r x, z, x, z X, r R (iii) x, z is invarian under any permuaion of x, z, (iv) For all x, y, z X, (a) Whenever s x, z 1 1, y, z 1 1 and s + x + y, z 1 1, x + y, z (s + ) L ( x, z (s), y, z ()). (b) Whenever s x, z 1 1, y, z 1 1 and s + x + y, z 1 1, x + y, z (s + ) U ( x, z (s), y, z ()). 2 Preliminaries This secion conains a few basic definiions and preliminary resuls which will be needed in he sequel.

3 Inuiionisic fuzzy 2-norm 653 Definiion 2.1 Le X be a real linear space of dimension greaer han one and le, be a real valued funcion on X X saisfying he following condiions 2N 1 : x, y =0if and only if x and y are linearly dependen, 2N 2 : x, y = y, x, 2N 3 : αx, y α x, y, for every α R 2N 4 : x, y + z x, y + x, z hen he funcion, is called a 2-norm on X and he pair (X,, ) is called a 2-normed linear space. Definiion 2.2 [10] Le X be a linear space over a field F. A fuzzy subse N of X X R is such ha for all x, y, z X, c F (2 N1): For all R wih 0, N(x, y, ) =0, (2 N2): For all R wih >0, N(x, y, ) =1if and only if x, y are linearly dependen (2 N3): N(x, y, ) is invarian under any permuaion of x, y (2 N4): For all R wih >0, N(x, cy, ) =N(x, y, ) if c 0, (2 N5): For all s, R, N(x, y + z, s + ) min{n(x, y, s),n(x, z, )} (2 N6): N(x, y, ) is a non-decreasing funcion of R and lim N(x, y, ) = 1. Then N is said o be a fuzzy 2-norm on a linear space X and he pair (X, N) is called a fuzzy 2-normed linear space (briefly F-2-NLS) Example 2.3 Le (X,, ) be a 2-normed linear space. Define N(x, y, ) =, + x, y when > 0, R, x, y X = 0, when 0, R, x, y X. Then (X, N) is an F-2-NLS. Example 2.4.Le(X,, ) be a 2-normed linear space. Define N(x, y, ) = 0, when x, y, R, x, y X = 1, when > x, y, R, x, y X. Then (X, N) is an F-2-NLS. Definiion 2.5 Le U be a linear space over a real field F. A fuzzy subse N of U U R such ha for all x, y, u U, c F (a 2 N1): For all R wih 0, N (x, y, ) =1, (a 2 N2): For all R wih >0, N (x, y, ) =0if and only if x, y are linearly dependen (a 2 N3): N (x, y, ) is invarian under any permuaion of x, y

4 654 B. Surender Reddy (a 2 N4): For all R wih >0, N (x, cy, ) =N (x, y, ) if c 0, (a 2 N5): For all s, R, N (x, y+u, s+) max{n (x, y, s),n (x, u, )} (a 2 N6): N (x, y, ) is a non-increasing funcion of R and lim N (x, y, ) = 0. Then N is said o be a fuzzy ani-2-norm on a linear space U and he pair (U, N ) is called a fuzzy ani-2-normed linear space (briefly Fa-2-NLS). Example 2.6 Le (U,, ) be a 2-normed linear space. Define N x, y (x, y, ) =, when > 0, R, x, y U + x, y = 1, when 0, R, x, y U. Then (U, N ) is an Fa-2-NLS. Proof. Now we have o show ha N (x, y, ) is a fuzzy ani-2-norm in U. (a 2 N1): For all R wih 0, we have by definiion N (x, y, ) =1. (a 2 N2): For all R wih >0, N (x, y, ) =0 x,y =0 + x,y x, y =0 x, y are linearly dependen (a 2 N3): As x, y is invarian under any permuaion of x, y, i follows ha N (x, y, ) is invarian under any permuaion of x, y (a 2 N4): For all R wih >0 and c 0,c F, we ge N (x, cy, ) = x, cy + x, cy = x, y + x, y = x, y + x, y = N (x, y, ). (a 2 N5): For all s, R and x, y, u U. We have o show ha N (x, y +u, s+) max{n (x, y, s),n (x, u, )}. If (a) s+ <0 (b) s = =0 (c) s + >0; s>0, <0; s<0, >0, hen in hese cases he relaion is obvious. If (d) s>0, >0, s + >0. Then assume ha N (x, y, s) N (x, u, ) x, y s + x, y x, u + x, u x, y ( + x, u ) x, u (s + x, y ) x, y s x, u. Now x, y + u s + + x, y + u = x, u + x, u x, y s x, u (s + + x, y + x, u )( + x, u ) x, y + u s + + x, y + u x, y + x, u s + + x, y + x, u x, u + x, u 0. (Since x, y s x, u ) x, u + x, u. Similarly x, y + u s + + x, y + u x, y s + x, y.

5 Inuiionisic fuzzy 2-norm 655 Hence N (x, y + u, s + ) max{n (x, y, s),n (x, u, )}. (a 2 N6): If 1 < 2 0, hen we have N (x, y, 1 )=N (x, y, 2 ) = 1. If 0 < 1 < 2 hen x, y 1 + x, y x, y 2 + x, y = x, y ( 2 1 ) ( 1 + x, y )( 2 + x, y ) > 0 N (x, y, 1 ) N (x, y, 2 ). Thus N (x, y, ) is a non-increasing funcion of R. Again for all x, y U lim N x, y (x, y, ) = lim + x, y =0. Hence (U, N ) is an F a 2 NLS. Example 2.7 Le (U,, ) be a 2-normed linear space. Define N : U U R [0, 1] by N (x, y, ) = 0, when > x, y, R, x, y U = 1, when x, y, R, x, y U. Then (U, N ) is an Fa-2-NLS. 3 Inuiionisic Fuzzy 2-norm In his secion we redefine he noion of fuzzy 2-normed linear space using -norm and fuzzy ani 2-normed linear space using -conorm hen we inroduce he definiion of inuiionisic fuzzy 2-norm on a linear space. Definiion 3.1 [11] A binary operaion :[0, 1] [0, 1] [0, 1] is coninuous -norm if saisfies he following condiions (a) is commuaive and associaive (b) is coninuous (c) a 1=a for all a [0, 1] (d) a b c d, whenever a c and b d, and a, b, c, d [0, 1]. Example 3.2 (i) a b = ab (ii) a b = min{a, b} (iii) a b = max{a + b 10} are coninuous -norms. Definiion 3.3 Le X be a linear space over a real field F. A fuzzy subse N of X X R(se of real numbers) is such ha for all x, y, z X, c F (F 2 N1): For all R wih 0, N(x, y, ) =0, (F 2 N2): For all R wih >0, N(x, y, ) =1if and only if x, y are linearly dependen (F 2 N3): N(x, y, ) is invarian under any permuaion of x, y (F 2 N4): For all R wih >0, N(x, cy, ) =N(x, y, ) if c 0, (F 2 N5): For all s, R, N(x, y + z, s + ) N(x, y, s) N(x, z, ) (F 2 N6): N(x, y, ) is a non-decreasing funcion of R and lim N(x, y, ) = 1. Then N is said o be a fuzzy -2-norm on a linear space X and he pair (X, N) is called a fuzzy -2-normed linear space (briefly F- -2-NLS).

6 656 B. Surender Reddy Definiion 3.4 [11] A binary operaion :[0, 1] [0, 1] [0, 1] is coninuous -conorm if saisfies he following condiions (a) is commuaive and associaive (b) is coninuous (c) a 0 =a for all a [0, 1] (d) a b c d, whenever a c and b d, and a, b, c, d [0, 1]. Example 3.5 (i) a b = min{a+b, 1} (ii) a b = max{a, b} (iii) a b = a + b ab are coninuous -conorms. Definiion 3.6 Le X be a linear space over a real field F. A fuzzy subse M of X X R(se of real numbers) is such ha for all x, y, z X, c F (Fa 2 N1): For all R wih 0, M(x, y, ) =1, (Fa 2 N2): For all R wih >0, M(x, y, ) =0if and only if x, y are linearly dependen (Fa 2 N3): M(x, y, ) is invarian under any permuaion of x, y (Fa 2 N4): For all R wih >0, M(x, cy, ) =M(x, y, ) if c 0, (Fa 2 N5): For all s, R, M(x, y + z, s + ) M(x, y, s) M(x, z, ) (Fa 2 N6): M(x, y, ) is a non-increasing funcion of R and lim M(x, y, ) = 0. Then M is said o be a fuzzy -ani-2-norm on a linear space X and he pair (X, M) is called a fuzzy -ani-2-normed linear space (briefly F- -a-2-nls). Definiion 3.7 [13] Le E be any se. An inuiionisic fuzzy se A of E is an objec of he form A = {(x, μ A (x),ν A (x)) : x E}, where μ A : E [0, 1] and ν A : E [0, 1] denoes he degree of membership and he nonmembership of he elemen x E respecively and for every x E, 0 μ A (x)+ν A (x) 1. Definiion 3.8 Le E be any se. An inuiionisic fuzzy 2-se A of E is an objec of he form A = {((x, y),μ A (x, y),ν A (x, y)) : (x, y) E E}, where μ A : E E [0, 1] and ν A : E E [0, 1] denoes he degree of membership and he non-membership of he elemen (x, y) E E respecively and for every (x, y) E E, 0 μ A (x, y)+ν A (x, y) 1. Definiion 3.9 Le be coninuous -norm, be a coninuous -conorm and X be a linear space over he field F, (where F = R or C). An inuiionisic fuzzy 2-norm on X is an objec of he form A = {((x, y, ),N(x, y, ),M(x, y, )) : (x, y, ) X X R + }, where N,M are fuzzy ses on X X R +, N denoes he degree of membership and M denoes he degree of non-membership of he elemen (x, y, ) X X R + saisfying he following condiions (i) N is a fuzzy -2-norm on a linear space X (ii) M is a fuzzy -ani-2-norm on a linear space X (iii) N(x, y, )+M(x, y, ) 1, for all (x, y, ) X X R +.

7 Inuiionisic fuzzy 2-norm 657 Example 3.10 Le (X = R,, ) be a 2-normed linear space. a b = min{a, b} and a b = max{a, b} for all a, b [0, 1] Also define Define N(x, y, ) =, when > 0, R, x, y X, k > 0 + k x, y = 0, when 0, R, x, y X. k x, y and M(x, y, ) =, when > 0, R, x, y X, k > 0 + k x, y = 1, when 0, R, x, y X. Now consider A = {((x, y, ),N(x, y, ),M(x, y, )) : (x, y, ) X X R + } hen A is an inuiionisic fuzzy 2-norm on X. Proof.(i) To prove ha N is a fuzzy -2-norm on a linear space X (F 2 N1): For all R wih 0, we have by definiion N(x, y, ) =0. (F 2 N2): For all R wih >0, N(x, y, ) =1 =1 +k x,y x, y =0 x, y are linearly dependen (F 2 N3): As x, y is invarian under any permuaion of x, y, i follows ha N(x, y, ) is invarian under any permuaion of x, y (F 2 N4): For all R wih >0 and c 0,c F, we ge N(x, cy, ) = + k x, cy = + k x, y = + k x, y = N(x, y, ). (F 2 N5): For all s, R and x, y, u X. We have o show ha N(x, y + u, s + ) min{n(x, y, s),n(x, u, )} = N(x, y, s) N(x, y, ). If (a) s + <0 (b) s = = 0 (c) s + >0; s>0, <0; s<0, >0, hen in hese cases he relaion is obvious. If (d) s>0, >0, s + >0. Then assume ha N(x, y, s) N(x, u, ) s s + k x, y + k x, u (+k x, u ) ( (s + k x, y ) s Now s++k x, y+u s + + k x, y + k x, u =(s+k x, y )+(+k x, u ) (s + k x, y )+ (s + k x, y ) ) s =( s+ s (s + k x, y ). s + + k x, y + u s + s + k x, y s s + s + + k x, y + u s s + k x, y. Hence N(x, y + u, s + ) min{n(x, y, s),n(x, u, )} = N(x, y, s) N(x, u, ). (F 2 N6): If 1 < 2 0, hen we have N(x, y, 1 )=N(x, y, 2 ) = 0. If 0 < 1 < 2 hen k x, y k x, y = k x, y ( 1 2 ) ( 1 + k x, y )( 2 + k x, y ) < 0

8 658 B. Surender Reddy N(x, y, 1 ) N(x, y, 2 ). Thus N(x, y, ) is a non-decreasing funcion of R and for all x, y X lim N(x, y, ) = lim = 1. Hence N is a +k x,y fuzzy -2-norm on a linear space X. (ii)to prove M(x, y, ) is a fuzzy -ani-2-norm in X. (Fa 2 N1): For all R wih 0, we have by definiion M(x, y, ) =1. (Fa 2 N2): For all R wih >0, M(x, y, ) =0 k x,y =0 +k x,y x, y =0 x, y are linearly dependen (Fa 2 N3): As x, y is invarian under any permuaion of x, y, i follows ha M(x, y, ) is invarian under any permuaion of x, y (Fa 2 N4): For all R wih >0 and c 0,c F, we ge k x, cy M(x, cy, ) = + k x, cy = k x, y + k x, y = k x, y + k x, y = M(x, y, ). (Fa 2 N5): For all s, R and x, y, u X. We have o show ha M(x, y + u, s + ) max{m(x, y, s),m(x, u, )} = M(x, y, s) M(x, u, ). If (a) s + <0 (b) s = = 0 (c) s + >0; s>0, <0; s<0, >0, hen in hese cases he relaion is obvious. If (d) s>0, >0, s + >0. Then assume ha M(x, y, s) M(x, u, ) k x,y k x,u k x, y ( + k x, u ) s+k x,y +k x,u k x, u (s + k x, y ) x, y s x, u. Now = k x, y + u s + + k x, y + u k x, u + k x, u k x, y ks x, u (s + + k x, y + k x, u )( + k x, u ) Since x, y s x, u. Similarly k x, y + k x, u s + + k x, y + k x, u k x, y + u s + + k x, y + u k x, y + u s + + k x, y + u k x, y s + k x, y. k x, u + k x, u k x, u + k x, u. Hence M(x, y+u, s+) max{m(x, y, s),m(x, u, )} = M(x, y, s) M(x, u, ). (Fa 2 N6): If 1 < 2 0, hen we have M(x, y, 1 )=M(x, y, 2 ) = 1. If 0 < 1 < 2 hen k x, y 1 + k x, y k x, y 2 + k x, y = k x, y ( 2 1 ) ( 1 + k x, y )( 2 + k x, y ) > 0 M(x, y, 1 ) M(x, y, 2 ). Thus M(x, y, ) is a non-increasing funcion of k x,y R and for all x, y X lim M(x, y, ) = lim = 0. Hence M is a +k x,y fuzzy -ani-2-norm on a linear space X. (iii) Also N(x, y, )+M(x, y, ) 1, for all (x, y, ) X X R +. Hence A is an inuiionisic fuzzy 2-norm on X. Definiion 3.11 [10] If A is an inuiionisic fuzzy norm on X (where X is a linear space over he field F, where F = R or C) hen (X, A) is called an inuiionisic fuzzy normed linear space (briefly IF-NLS).

9 Inuiionisic fuzzy 2-norm 659 Definiion 3.12 If A is an inuiionisic fuzzy 2-norm on X (where X is a linear space over he field F, where F = R or C) hen (X, A) is called an inuiionisic fuzzy 2-normed linear space (briefly IF-2-NLS). References [1] T. Bag and T. K. Samana, Finie dimensional fuzzy normed linear spaces, The Journal of Fuzzy Mahemaics, 11(3) (2003), [2] T. Bag and T. K. Samana, Fuzzy bounded linear operaors, Fuzzy Ses and Sysems, 151 (2005), [3] T. Bag and T. K. Samana, A comparaive sudy of fuzzy norms on a linear space, Fuzzy Ses and Sysems, 159 (2008), [4] S. C. Cheng and J. N. Mordesen, Fuzzy linear operaors and fuzzy normed linear spaces, Bull. Cal. Mah. Soc., 86 (1994), [5] C. Felbin, Finie dimensional fuzzy normed linear spaces, Fuzzy Ses and Sysems, 48(1992) [6] Iqbal H. Jebril and T.K. Samana, Fuzzy Ani-Normed space, Journal of Mahemaics and Technology, February (2010), [7] O. Kaleva and S. Seikkala, On fuzzy meric spaces, Fuzzy Ses and Sysems, 12 (1984), [8] A.K. Kasaras, Fuzzy opological vecor spaces, Fuzzy Ses and Sysems, 12 (1984), [9] I. Kramosil and J. Michalek, Fuzzy meric and saisical meric space, Kyberneica, 11 (1975), [10] T.K Samana and Iqbal H. Jebril, Finie dimensional Inuiionisic fuzzy n-normed linear space, In. J. Open Prolbems Comp. Mah., 2(4) (2009). [11] B. Schweizer and A. Sklar, Probabilisic meric space, New York, Amserdam, Oxford, Norh Holland, [12] B. Surender Reddy, Fuzzy ani-2-normed linear space, Journal of Mahemaics Research, 3(2) May (2011). [13] S. Vijayabalaji, N. Thillaigovindan and Y. Bae Jun, Inuiionisic fuzzy n-normed linear space, Korean Bull. Mah. Soc., 44 (2007), Received: December, 2010