Vague Cosets. (2) aa Aa A is vague invariant. b G then. X [0,1], where as a vague set A of set X is a pair. (1) aa ba Aa Ab if A is vague symmetric.

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1 World Journl of Rsrch nd Rviw (WJRR) ISSN: , Volum-3, Issu-2, ugust 206 Pgs 28-3 Vgu Costs Ch Mllik, N Rmkrishn, nndh Ro strct- In this ppr w study th vgu costs nd thir proprtisths concpts r usd in th dvlopmnt of som importnt rsults nd thorms out vgu groups nd vgu normlgroupslso som of thir importnt proprtis hv n invstigtd Indx Trms vgu st,vgu group, vgu norml group, vgu cost I INTRODUCTION Th concpt proposd y ZdhL$[0]$ dfining fuzzy sust of givn univrs X chrctrizing th mmrship of n lmnt x of X longing to y mns of mmrship function dfind from X into [0, ] hs rvolutionizd th thory of Mthmticl modling, Dcision mking tc, in hndling th imprcis rl lif situtions Mthmticlly Now, svrl rnchs of fuzzy mthmtics lik fuzzy lgr, Fuzzy topology, fuzzy control thory, fuzzy msur thory tc, hv mrgd But in th dcision mking, th fuzzy thory tks cr of mmrship of n lmnt x only, tht is th vidnc of x longing to It dos not tk cr of th vidnc of x not longing to It is flt y svrl dcision mkrs nd rsrchrs tht in propr dcision mking, th vidnc of x longing to nd vidnc of x not longing to r oth ncssry, nd how much x longs to or how much x dos not long to r ncssry Svrl gnrliztions of Zdh's fuzzy st thory hv n proposd, on of thm such s Vgu sts of uwl,nd BuhrrDJ [3] (or quivlntly Intuituinistic fuzzy sts of tnsssovkt[] This concpt ws pplid in svrl rs lik dcision mking, Fuzzy control, vgu crrir dcistion mking,lctro dmocrcy modl,knowldg discovry nd fult dignosis tc It is livd tht vgu sts will mor usful in dcision mking,nd othr rs of mthmticl modling fuzzy st t( x ) of st X is mpping from X [0,], whr s vgu st of st X is pir ( t, f ), whr t, f r functions from X [0,] with 0 t( x) f ( x) for ll x in X Hr t is clld th mmrship function nd f is clld non-mmrship function of fuzzy st t of X cn idntifid with th pir ( t, t ) Thus; th thory of vgu sts is gnrliztion of th thory of fuzzy sts Th Ch Mllik, Dprtmnt of Mthmtics, MrsVN Collg, Viskhptnm-53000,ndhrprdsh,Indi NRmkrishn, Dprtmnt of Mthmtics, MrsVN Collg, Viskhptnm-53000,ndhrprdsh,Indi nndh Ro,Dprtmnt of Mthmtics, ITM Univrsity Viskhptnm, ndhr Prdsh, Indi lgric spcts of vgu sts wr inititd y Rnjit Bisws [8] y studying th concpts of vgu groups, vgu norml groups tc, s gnrliztion of th thory of fuzzy groups tc Furthr Rmkrishn [5],[6],[7] continud th study of vgu norml groups, homologous vgu groups, vgu normliz, vgu cntrlizr, Vgu wights nd chrctriztions of cyclic groups in trms of vgu groups tc In this ppr w study th vgu costs, vgu symmtry, vgu invrint nd som of thir importnt proprtis lso w provd, if vgu group of group, hn for ll x,y () if is vgu symmtric (2) is vgu invrint (3) if is vgu norml, nd (4) Lt vgu group of group nd, thn ( i) c c, ( ii) c c II PRELIMINRIES W giv hr rviw of som dfinitions nd rsults which r in uwl nd Buhrr DJ[3],RmkrishnN[5],Rnjit Bisws [8] Dfinition 2: vgu st in th univrs of discours U is pir ( t, f ) whr t : U [0,], f : U [0,], r mppings such tht t ( u) f ( u),for ll u U Th functions t nd f r clld tru mmrship function nd fls mmrship function rspctivly Dfinition 22: Th intrvl [ t ( u), f ( u)] is clld th vgu vlu of u in, nd it is dnotd y V ( u ) i V ( ) u = [ t( u), f ( u)] Dfinition 23: Lt (, ) group vgu st of is clld vgu group of if,for ll x,y in, V ( xy) min{ V ( x), V ( y)} nd V ( x ) V ( x) i, t ( xy) min{ t ( x), t ( y)}, f ( xy) mx{ f ( x), f ( y)} nd ( ) ( ), t x t x f ( x ) f ( x) Hr th lmnt x y stnds for x y Dfinition 24: vgu st of univrs with tru-mmrship function function t, nd fls mmrship f For ò, [0,] with, th (, ) cut 28 wwwwjrrorg

INTRODUCTION Th concpt proposd y ZdhL$[0]$ dfining fuzzy sust of givn univrs X chrctrizing th mmrship of n lmnt x of X longing to y mns of mmrship function dfind from X into [0, ] hs rvolutionizd th

2 Vgu Costs or vgu cut of vgu st is th crisp sust of is = { x : xò, V ( x) [, ]} givn y (, ) i, (, ) = { x xò, t ( x),nd f ( x) } Dfinition 25: Th -cut, of th vgu st is th (, ) cut of,nd hnc givn y { x xò, t ( x) } Dfinition 26: Lt vgu group of group Thn is clld vgu norml group if for ll x,y, V ( xy) V ( yx) ltrntivly, w cn sy tht, vgu group is sid to vgu norml group of if V ( x) V ( yxy ) for ll x,y Nottion 27: Lt I[0,] dnot th fmily of ll closd su intrvls of [0,] If I [, ] nd I2 [ 2, 2 ] two lmnts of I[0,] W cll I I2 if 2 nd 2 similrly w undrstnd th rltions I I2 nd I I2 Clrly th rltion I I2 dos not ncssrily imply tht I I2 nd convrsly lso for ny two unqul intrvls I nd I 2, thr is no ncssity tht I I2, or I I will tru Th trm ` imx' mns th mximum 2 of two intrvls s imx{ I, I2} [ mx{, 2}, mx{, 2}], similrly dfind imin Th concpt of ìmx' nd ìmin' could xtndd to dfin ìsup' nd ìinf' of infinit numr of lmnts of I[0,] III VUE COSETS W hv th following Dfinition 3: Lt vgu group of group ( ), For ny, vgu lft cost of is dnotd y nd dfind y V ( x) V ( x) t () t(( ) ) t( ) t( ) 07 f () f (( ) ) f ( ) f ( ) 02 2 ti( i) t(( i) i) t( i i) t( i ) 07 2 fi() f (( i) i) f ( i i) f ( i ) f ( ) 02 2 ti( i) t(( i) i) t( i i) t( i ) t() 07 2 fi( i) f (( i) i) f ( i i) f ( i ) f () 02 Thrfor {(,07,02),(,07,02),( i,07,02),( i,07,02)} is vgu lft cost of Dfinition 35: vgu group of group is sid to () vgu symmtric if V ( x ) V ( x) for ll x I t ( x ) t ( x) nd it( x) t( x) nd f( x) f ( x) ( is vgu symmtric ) Dfinition 32: Lt vgu group of group ( ) For i ny, vgu right cost of is dnotd y nd dfind y (2) Suppos is vgu invrint V ( x) V( x ) V( x) V( ( x)) V( x ) it( x) t( x ) nd f ( x) f ( x ) V ( x) V ( x) for ll x Rmrk 33: Clrly vgu lft cost is vgu st I t( x) f( x) t( x) f ( x) (3) Suppos is vgu norml nd Similrly in cs of vgu right cost y () y(2) Exmpl 34: Considr th group {,, i, i} with Thorm 37: Lt vgu group of group nd inry oprtion s complx multipliction, clrly th st, thn {(,08,0),(,07,02),( i,06,02),( i,05,02)} () c c is vgu st of group lso (2) c c f ( x ) f ( x) for ll x (2) vgu invrint if V( xy) V( yx) for ll x, y i t( xy) t( yx) nd f ( xy) f ( yx) for ll x, y (3) Vgu norml if is oth vgu symmtry nd vgu invrint Thorm 36: Lt vgu group of group Thn for ll x,y () if is vgu symmtric (2) is vgu invrint (3) if is vgu norml Proof: () Suppos is vgu symmtric nd = V ( x) V ( x) ( ) ( ) [( )) ] [( ) ] ( ( ) ) ( ( ) ) V ( x ) V ( x ) V x V x V x V x V x V x V ( x ) V ( x ) 29 wwwwjrrorg

x,y Nottion 27: Lt I[0,] dnot th fmily of ll closd su intrvls of [0,] If I [, ] nd I2 [ 2, 2 ] two lmnts of I[0,] W cll I I2 if 2 nd 2 similrly w undrstnd th rltions I I2 nd I I2 Clrly th rltion I I2

3 Proof: Suppos is vgu norml nd lt V [ c x] V [ c x] for c x ( ) ( ) (( ) ) (( ) ) V c x V c x V c x V c x Vc ( x) Vc ( x) c c (2)Suppos z z y() ( z) ( z) c c Now w hv th following Dfinition 38: Lt vgu group of group, thn () { x : x } (2) { x : x } (3) { x : x } (4) { xy : x nd y } Rmrk 39: If is invrint thn w hv World Journl of Rsrch nd Rviw (WJRR) ISSN: , Volum-3, Issu-2, ugust 206 Pgs 28-3 for ll Thorm 30: If vgu group of group is vgu norml thn () For ny,, (2) is crisp norml sugroup of (3) for ll (4) for ll, proof: () is vgu norml nd suppos (2), nd is crisp su group of Lt x, x x xx xx xx xx thus is norml sugroup of (3) Tk { x : x } { x : x } { x : x } { x : x } { y : y } whr y x (4) Lt x, y x x x nd y x nd is vgu norml tht is x x similrly y xy x implis xy xy Thus~ ( i) On th othr hnd,lt x x x x x i x x For ny x, x x x whr, x x x x Thus ( ii) From (i) nd (ii) Dfinition 3: Lt group nd x, y w sy tht x is conjugt to y if thr xists such tht y x It is known tht th conjugcy is n quivlnc rltion on Th quivlnc clss in undr th rltion of conjugcy is clld conjugt clss Thorm 32: Lt vgu group of group Thn is vgu norml iff is constnt on th conjugt clsss of proof: Suppos is vgu norml group of nd, thn V ( ) V ( ) V ( ) V ( ) hnc is constnt on th conjugt clsss of on th othr hnd is constnt on th conjugt clsss of nd Lt, thn V ( ) V ( ) V = V ( ) ( ( ) ) Hnc is vgu norml group of Thorm 33: Lt vgu group of group thn th following r quivlnt () forch (2) for ch proof: W hv is vgu group of group nd Suppos On th othr hnd, suppos 30 wwwwjrrorg

28-3 for ll Thorm 30: If vgu group of group is vgu norml thn () For ny,, (2) is crisp norml sugroup of (3) for ll (4) for ll, proof: () is vgu norml nd suppos (2), nd is crisp su group of Lt x, x x

4 Vgu Costs cknowldgmnts CKNOWLEDEMENTS Th uthors r grtful to ProfKLNSwmy for his vlul suggstions nd discussions on this work REFERENCES [] tnssovkt,mor on Intuitionistic [2] fuzzy sts,fuzzy sts nd systms,33(989),37-45 [3] Dmirici,M,vgu groups jnmthnl ppl,230(999),42-56 [4] huwlbuhrrdj,vgu sts IEEE [5] Trnsctions on systms,mn nd cyrntics [6] vol23(993),60-64 [7] Rosnfld,,Fuzzy groups, JnMthsnl [8] ppli,35 (97),52-57 [9] Rmkrishn,N,vgu Norml roups,int journl of [0] computtionl cognition,6(2)(2008),0-3 [] Rmkrishn,N, chrctririztion of cyclic groups [2] in trms of vgu groups,intjournl of computtionl [3] Congnition,6(2) (jun 2008),7-9 [4] Rnjit Bisws,vgu groups,int journl of computtionl Congnition, 4(2) (2006),20-23 [5] Rnjit Bisws, Vgu roups,intjournl of computtionl cognition,vol4 No2,Jun 2006 [6] Zdh,L, Fuzzy Sts, Infor nd Control,volum 8 (965) First uthor ChMllik, Rsrch Scholr,Dprtmnt of Mthmtics,ITM Univrsity Scond uthor CptDrNRmkrishn, ssocit Profssor of MthmticsMrsVN Collg,Viskhptnm 24 rsrch pprs pulishd in Intrntionl rputd Journl H guidd 3 MPhil nd PhD scholrs H is lif mmr in PSMS nd ssocition of Mthmtics Tchr of Indi, sso fllow of kdimy of Scincs, P H ws wrdd ntionl nd stt wrds Third uthor Profnndh Ro, Profssor of Mthmtics, itm Univrsity, Viskhptnm rsrch pprs wr pulishd in ntionl nd Intrntionl rputd journls H guidd 3MPhils nd 5 Ph D scholrs H rcivd ITM Univrsity Bst Tchr wrd H is lif mmr in ISTM nd ISTE 3 wwwwjrrorg

[8] ppli,35 (97),52-57 [9] Rmkrishn,N,vgu Norml roups,int journl of [0] computtionl cognition,6(2)(2008),0-3 [] Rmkrishn,N, chrctririztion of cyclic groups [2] in trms of vgu groups,intjournl of

Higher. Exponentials and Logarithms 160

Higher. Exponentials and Logarithms 160 hsn uknt Highr Mthmtics UNIT UTCME Eponntils nd Logrithms Contnts Eponntils nd Logrithms 6 Eponntils 6 Logrithms 6 Lws of Logrithms 6 Eponntils nd Logrithms to th Bs 65 5 Eponntil nd Logrithmic Equtions