Rough neutrosophic set in a lattice

Transcription

1 nternational Journal of ppld esearch 6; 5: 43-5 SSN Print: SSN Online: mpact actor: 5 J 6; 5: 43-5 wwwallresearchjournalcom eceived: -3-6 ccepted: -4-6 rockiarani Nirmala College for Women Coimbatore amilnadu ndia C nton Crispin Sweet Nirmala College for Women Coimbatore amilnadu ndia ough neutrosophic set in a lattice rockiarani C nton Crispin Sweet bstract n this paper we eamine the relationship between rough fuzz neutrosophic sets lattice theor We introduce the notion of ough fuzz neutrosophic set ough fuzz neutrosophic lattice resp ough fuzz neutrosophic ideals urther we discuss about fuzz neutrosophic rough set corresponding to a rough set define the terms conditions for fuzz neutrosophic rough lattice We also prove that a fuzz neutrosophic rough set in is a fuzz neutrosophic rough lattice iff it s level rough sets is a rough sub lattice of Kewords: ough set rough fuzz neutrosophic set fuzz Neutrosophic rough sets ntroduction n 98 Pawlak 6 introduced the concept of rough set as a formal tool for modeling processing incomplete information in information sstems his concept is fundamental to the eamination of granularit in knowledge he basic idea of rough set is based upon the approimation of sets b a pair of sets known as the lower approimation the upper approimation of a set Here the lower upper approimation operators are based on equivalence relation fter Pawlak there have been man models built upon different aspect ie universe relations object operators b man scholars Various notions that combine rough sets fuzz sets vague set intuitionistic fuzz sets are introduced such as rough fuzz sets fuzz rough sets generalized fuzz rough sets rough vague sets he theor of rough sets is based upon the classification mechanism from which the classification can be vwed as an equivalence relation knowledge blocks induced b it be a partition on universe One of the interesting generalizations of the theor of fuzz sets intuitionistic fuzz sets is the theor of neutrosophic sets introduced b Smarache 8 Neutrosophic sets described b three functions: ruth function indeterminac function false function that are independentl related he theors of neutrosophic set have achved great success in various areas such as medical diagnosis database topolog image processing decision making problem While the neutrosophic set is a powerful tool to deal with indeterminate inconsistent data the theor of rough sets is a powerful mathematical tool to deal with incompleteness ecentl man researchers appld the notion of fuzz neutrosophic sets to relations group theor ring theor lattice theor etc n this paper we studd relationship between rough sets fuzz neutrosophic sets Here we give the rough approimation of fuzz neutrosophic set introduced rough fuzz neutrosophic sub lattices ideals etc lso we defined fuzz neutrosophic rough sets fuzz neutrosophic rough sub lattices ideals studd their properts Preliminars: Definition Neutrosophic set on the universe of discourse X is defined as Correspondence rockiarani Nirmala College for Women Coimbatore amilnadu ndia = Where : ~ 43 ~

2 nternational Journal of ppld esearch Definition 3: uzz Neutrosophic set on the universe of discourse X is defined as = where : 3 Definition : neutrosophic set is contained in another neutrosophic set ie X Definition 4: he complement of a neutrosophic set denoted b c is defined as c = c Where c c c Definition 5: Let be two neutrosophic sets over the common universe U is said to be neutrosophic subset of if E U Definition 6: wo neutrosophic sets G over the common universe U are said to be equal if G G We denote it b = G Definition 7: Let X be a non empt set are fuzz neutrosophic sets hen ~ ma ma ~ min min min ma Definition : 3 Let U be an non-empt set Suppose is an equivalence relation over U or an non-null subset X of U the sets X = {: X} X = {: X } are called lower approimation upper approimation respectivel of X the pair S= U is called approimation space he equivalence relation is called indiscernibilit relation he pair X = X X is called the rough set of X in S Here denotes the equivalence class of containing 3 ough uzz Neutrosophic Sets n Lattice n this section we define rough fuzz neutrosophic set some of their operations urther we introduce ough fuzz neutrosophic lattices L ideals stud certain properts of them Definition 3: Let U be a non-null set be an equivalence relation on U Let be a neutrosophic set in U with the truth value indeterminate value ~ 44 ~ false value he lower the upper approimations of in the approimation U denoted b are respectivel defined as follows: { / U} = / } { U where: = = = = = So 3 3 : Where mean ma min operators respectivel are the truth indeterminac false values of with respect to t is eas to see that are two neutrosophic sets in U : are respectivel referred to as the lower upper rough NS approimation operators the pair is called the rough neutrosophic set in U rom the above definition we can see that have constant membership on the equivalence classes of U Eample 3: Let U= {S S S 3 S 4 S 5} be the universe of discourse Let be an equivalence relation where its partition of U is given b U/= {{S S } {S 3} {S 4 S 5}} ={ S 345 S 43 S 3567} be a neutrosophic set of U he lower upper approimations are obtained as = { S 343 S 343 S 3567} ={ S 45 S 445 S 3567} nother neutrosophic set can be defined as = { S 34 S 4354 S 546} he lower upper approimations are obtained as = {S 354 S 434 S 46 S 546} ={ S 34 S 34}

3 nternational Journal of ppld esearch Definition 33: f is a rough fuzz neutrosophic set in U the rough fuzz neutrosophic complement of C neutrosophic set denoted b c c c c c where are defined as c = / } { U c = { / U} Definition34: f are two rough fuzz neutrosophic set of the neutrosophic sets X X respectivel in then we define the following: iff = = iff Definition 35: Let L be a lattice { / L} be a fuzz neutrosophic set then is called fuzz neutrosophic sublattice of L if the following conditions are satisfd i } } } ma{ ma{ } ii } iii } he set of all NLs of L is denoted b NL Definition 36: NS of L is called a fuzz neutrosophic ideal of L if he following conditions are satisfd i } ma{ } ma{ } ma{ } ii } iii } he set of all Ns of L is denoted as N Definition 37: Hence is NL of L n of L is called a fuzz neutrosophic prime Proof for N is similar ideal if ma{ } Proposition 39: Let L be a lattice is an L of hen are also NL s N s of ~ 45 ~ ma{ } } L heorem 38: f are two NLs Ns of a lattice L then is a NLN of Proof: Let { / X} { / X} are two NS of hen { X} Where } } ma{ } so that } } }} } }} } as are NLs of L } L Similarl we get } L } as are NLs of L } } } } L }} }} Siilarl we get } L lso ma{ } ma{ma{ } }} ma{ma{ } }} ma{ } as are NLs of L ma{ } L Similarl we get ma{ } L

4 ~ 46 ~ nternational Journal of ppld esearch Proof We will prove the case of N Proof for N is similar We have } } } NL of since is } } } } NL of since is } } } min } since is NL of } } } } NL of since is } } ma{ } ma{ } ma{ NL of since is } ma{ } ma{ } ma{ } ma{ NL of since is } ma{ Hence is a NL of } } } NL of since is } } similarl } } } NL of since is } } similarl } ma{ } ma{ } ma{ NL of since is } ma{ } ma{ similarl Hence is a NL of Definition 3: rough fuzz neutrosophic set of L is called a rough fuzz neutrosophic lattice L rough neutrosophic fuzz ideal if both are NL s N s of heorem 3: f is an NL N of L then is a L of Proof ollow from Proposition 34 heorem 3: f are NL s s then is also a LN N Proof We have Since are NL s N s we have are NL s N shen are NL s N s b heorem 5 So is a NL N b Def 35 emark 33: he union of two N s need not be a N Consider the lattice L= { } of divisors of Let ={364}be the equivalence class We define } / { L b

5 nternational Journal of ppld esearch { } { / L} { } Here are s of Now where { } is { } { } lso where { } is { } { } clearl are N s { } } 7 Hence is not an emark 34: Ever is a ut the converse is not true Consider the lattice the equivalence relation given in the esult 38Let { } { } { } t can be easil verifd that is NL but NL because ma{ uzz Neutrosophic ough Set NS n this section we introduce uzz Neutrosophic rough sublattices ideals define certain characterization of uzz Neutrosophic rough sublattice ideal in terms of level rough set Definition 4 Let X be a rough set is a NS in X hen we can define an interval valued fuzz neutrosophic rough set Where { } if X if X if X if X if X if X Where X X X we denote ~ 47 ~

6 nternational Journal of ppld esearch Definition 4: Let X be a rough lattice is a NS in X hen is called a fuzz neutrosophic rough sublattice N if for ever X the following hold i } ii } iii } iv } v ma{ } v ma{ } L f conditions replaced b ma{ } ma{ } } hen is called a fuzz neutrosophic rough ideal N Definition 43: Let X be a rough lattice a NS in X hen we define = X / } { X / } then is called L-NS { heorem 44: Let X be a rough lattice is a NS in X hen is a NL iff is a rough sublattice of X Proof: irst assume that is a rough sublattice in X We have to prove that is a NL of X Set } } ma{ } hen } } } } } ma{ } ma{ hen Hence so ~ 48 ~ Let X either or X f or X then So that f X then so } } ma{ } since is a sublattice So Hence } } } }

7 nternational Journal of ppld esearch ma{ } ma{ } So is a NL Conversel assume that is a NL of X We have to prove that are sublattices of Let then So } } } } ma{ } ma{ } Hence } } } } ma{ } ma{ } rom these inequalits we get Since X We have = = = Hence hus is a sublattice Similarl if then Hence Hence is a sublattice so hus 5 eferences nton Crispin Sweet C rockiarani ough sets in uzz Neutrosophic approimation space" accepted nton Crispin Sweet C rockiarani uzz neutrosophic rough sets JGM 4; 3: Dubios D Prade H ough fuzz sets fuzz rough sets nt J Gen Sst 99; 7:9-8 4 Gong Z Sun Chen D ough set theor for the interval-valued fuzz information sstems nf Sci 8; 78: Mi J-S Leung Y Zhao H-Y eng Generalized uzz ough Sets determined b a triangular norm nf Sci 8; 78: Pawlak Z ough Sets nt J Comput nform Sci 98; : Smarache Unifing ld in Logics Neutrosoph: Neutrosophic Probabilit 8 Set Logic ehoboth: merican esearch Press Smarache Neutrosoph Neutrosophic Logic irst nternational Conference on Neutrosoph Neutrosophic Logic Set Probabilit Statistics Universit of New Meico Gallup NM 873 US Sun Gong Z Chen D uzz rough set theor for the interval-valued fuzz information sstems nf Sci 8; 78: Vasantha Kadasam lorentin Smarache ~ Neutrosophic Lattices Neutrosophic Lattices Neutrosophic Sets Sstems 3; :4-5 Wu WZ Mi J-S Zhang W-X Generalized uzz ough Sets nf Sci 3; 5: Zhang Z On interval tpe- rough fuzz sets Knowledge ased Sst ; 35:-3 ~ 49 ~