Linguistic Matrices

Linguistic Matrices

W. B. Vasantha Kandasamy Ilanthenral K Florentin Smarandache

Editorial Global Knowledge 2022

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Peerreviewers:

Dr.S. A.Edalatpanah,DepartmentofAppliedMathematics,Ayandegan InstituteofHigherEducation,Tonekabon,Iran.

Dr.SaidBroumi,UniversityofHassanIIMohammedia, HayElBarakaBenM'sik,CasablancaB.P.7951.Morocco. Dr.HudaE.Khalid,DepartmentofMathematics, TelaferUniversity,Iraq.

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ISBN-13:978-1-59973-750-8 EAN:9781599737508 PrintedintheUnitedStatesofAmerica

2

CONTENTS

Preface 4

Chapter One LINGUISTIC VARIABLES, LINGUISTIC TERMS AND THEIR PROPERTIES 7

Chapter Two BASIC PROPERTIES OF LINGUISTIC MATRICES 48

Chapter Three LINGUISTIC MATRICES AND OPERATIONS ON THEM 102

3

FURTHER READING 182

INDEX 187

ABOUT THE AUTHORS 189

PREFACE

In this book, the authors introduce the linguistic set associated with a linguistic variable and the structure of matrices, which they define as linguistic matrices. The authors build linguistic matrices only for those linguistic variables whichyieldalinguisticcontinuumoranorderedlinguisticset.

This book is organised into three chapters. The first chapter is introductory, in which we introduce all the basic concepts of linguistic variables and the associated linguistic set to make this book self-contained. Chapter two introduces linguistic matrices and develops basic properties associated with them, like types of matrices, transpose of matrices and diagonal matrices. Most of the properties enjoyed by real or complex matrices are satisfied by these linguistic matrices. Chapter three deals with operationsonthelinguisticmatrices.

However, we see when we try to define operations on linguistic matrices that they behave in very different ways. As the meaning of +; addition, or product × are not defined or cannot be defined on linguistic words, we cannot have the notion of addition or product in the case of linguistic matrices. For instance, the linguistic terms good and very bad cannot be addedormultiplied.We see (good +verybad)hasnomeaning, so on (good × very bad) is not defined. The possible operations thatcanbe definedonlinguisticmatricesareonlythemaximum and minimum operations; the maximum is denoted by max, and theminimumoperationbymin.

Max and min operations can be defined both on the set of linguistic words.Nowforthe linguistic continuum[worst,best], takegoodandverybadfromthelinguisticinterval[worst, best], we see min{good, very bad} = very bad and max{ good, very bad} = good. Thus, these operators, min and max, are well definedonthelinguisticcontinuum[worst,best].Hence,wecan define min and max operators on linguistic matrices provided theysatisfycompatibilitywiththeseoperations.

Further, we have different identities for min and max operations on these matrices for the linguistic variable and its associated linguistic words. The empty matrix is one in which everyentryisphi(),theemptysymbol.

Results in this direction are developed and discussed. Finally, the notion of linguistic submatrices is discussed and developed. All these concepts are described with examples for theclearunderstandingofthereader.

Another essential feature of this book is that we suggest many problems at the end of each chapter. As matrix theory happens to be the basis for all mathematical and linguistic models for networks, in AI, in all fuzzy models, these concepts will be a boon to non-mathematicians, for they can quickly work with linguistic matrices in place of usual matrices. This bookisanintroductorybookonlinguisticmatrices.

We acknowledge Dr K.Kandasamy with gratitude for his continuoussupport.

W.B.VASANTHAKANDASAMY K.ILANTHENRAL FLORENTINSMARANDACHE

LINGUISTIC VARIABLES

, LINGUISTIC TERMS AND THEIR PROPERTIES

In this chapter we proceed on to define the notion of linguistic variables, linguistic terms or linguistic words or linguisticcontinuumassociatedwitheachlinguisticvariable.

We prove some of these linguistic words / terms form a totally order set, be a finite collection of linguistic terms or be an infinite collection of linguistic terms or be a linguistic continuum. We define on these totally ordered linguistic terms thetwobinaryoperationminandmax.

At this juncture it is mandatory to recall the notion of linguistic variables was defined in 1975 by Zadeh [25-7]. He definedlinguisticvariablesasavariablewhosevaluesarewords orsentencesinanartificialorinanaturallanguage.

However, we do not use the concept of sentences as we are trying to build a world of linguistic mathematics based on these linguistic terms / words / sets. So throughout this book by

w(L), we denote a linguistic set or linguistic words or linguistic termsassociatedwiththelinguisticvariableL.

As we have different linguistic variables the linguistic wordsetw(L)willvary.

For instance, if we want to talk about heights of people then the linguistic variable L in this case is the ‘height’ so the linguisticset/wordswillbe

{tall, tallest, just tall, short, very tall, very very tall, very short, short, just short, very very short, medium, just medium, shortest veryveryveryshort,andsoon}.

If on the other hand height is to be represented for all humans then this can also be represented as a linguistic continuumandisdenotedbytheintervalS=[shortest,tallest].

This interval [shortest, tallest] corresponds to [0.6 foot, 8 feet], for a human can be shortest at birth say 6 inches to and growin unusualcasesupto 8feet sonumerical [0.6feet,8 feet] = [6 inches, 8 feet] is equivalent to the linguistic interval [shortest,tallest].

We see there are infinitely many numbers between 6 and 10; and it is a continuum one for no person after birth suddenly just grows from 1 foot to 1 foot 2” or so on, as time goes the growth take place very slowly from very small fractions of an inchtofullinchandsoon.

So [0.6 foot, 8 feet] is a section of a continuum on the real scale that is number [0.6, 8] is a section or subinterval on therealline,(–, ).

Likewise [shortest, tallest] is a linguistic continuum and shortest may correspond to 1 2 feet or 6 inches and largest corresponds to the value 8 feet. Thus instead of giving height in the real scale we can also give the height of a person in a linguisticscalewhichcorrespondstothelinguisticcontinuum.

Now it is very natural for a passerby you see or from a group of people you move with; you and your friend discuss about their heights you cannot say he is 6 2 or 6 or 5 9 but instead you may say tall or just tall or short or very tall or medium height and so on. However in this case the linguistic term of assessing his height is more natural though approximate; than saying 62 or 6 or 59 for none can be true for instance he may exactly when measured be 6 1 but we cannotaskastrangerhisexactheightorsoon.

Thus whileassessingtheheightofapersonbyseeinghim it is more natural and appropriate to express it as a linguistic term. This sort of study when introduced in primary and secondary school children it will not only make them understand in a fearless way but they will be forced to appreciate such study which will develop them with strong basicsforlogicandcommonintelligence/sense.

Hence the study of linguistic terms and probable mathematics developed based on these happens to be more natural, appropriate and less artificial. So the need for linguistic terms and linguistic variables are a necessity to remove the grip ofmathematicsphobiainschoolchildren.

Another vital reason for making this study is that Artificial Intelligence functions basically on atomic words and atomic sentences so this linguistic language happens to be very useful. It is also pertinent to keep on record all the linguistic variablesneednotbedependentontime.

Next we consider the linguistic variable age. Age of any human varies linguistically from [youngest, oldest] in this linguistic continuum. The changein age of a person isalso only time dependent. Suppose we have set of 10 people and we are interested in finding their age immediately we has to assess that as we cannot ask them and get, it, then in that case we cannot say one person is 60 years 9 months, another one 40 years 2 months another one 4 years 2 months etc., by look we can just say the person is old, older, just old, very old, middle age, just young,youth,justyouth,veryyoungandyoungest.

These linguistic words or terms on the linguistic continuum can also be referred to as linguistic features or features associated with age. This form of assigning the age is much more natural than putting the age in number when we are notawareofit.

Suppose we want to assess the age we know it lies numerically in the continuum [0, 100] however in linguistic continuum it lies in the range [youngest, oldest]  {} where  corresponds to the empty linguistic term as that of 0 in case of [0, 100] in numerals. We say this as a linguistic continuum in parwiththerealcontinuum[0,100].

Now if it is the assessment of colour of a person it is not time variant. The colour of a person may be from dark to white,

very dark, darkest, just dark, brown, brownish, just brown so on andsoforth.

So if a person is given the task of assessing the colour of apersonthenhemaygiveanyofthetermsvaryingfromdarkest to whitehoweverthecoloursblue,red,orange,green,violetetc; are removed and no ordering is easily possible for the colour may be light brown or dark brown, or just brown or slight brown and so on. A linguistic continuum cannot be formed it is also time independent. Further we cannot give a comparative order in a very precise way. We can only say black, brown , palewhite,paleyellowandwhite.

So the colour red to yellow ordering is not possible or no such ordering can be made. Only sectional ordering can be made from [dark blue to light blue or say up to white] in a little vague way. However the assignment of colour and its linguistic set is not a continuum so it is not in general, possible to give an ordering.

Onecanordereachcolourfromdarktolightorfromlight to dark other form of ordering is not possible. Further colours cannot be represented numerically unless the fuzzy form of membership is made which is beyond the scope of this book as we not absolutely interested in such study. We proceed on to give an example of the students’ performance in the classroom toseethedifferenceintypesoflinguisticvariables.

The performance of 12 students in a class room is given by the following linguistic set / words / terms S associated with thelinguisticvariable;performanceofthestudents.

S = {very average, average, good, just average, bad, verybad,justbad,verygood,veryverygood}.

Further it is assumed 3 students performance in the class roomisgoodandtwostudentsperformanceisverybad.

Now certain observations about these linguistic set is vital.

First of all we cannot easily give marks because while assessing without any answer paper written, it is not easy to mark them say between 0 to 100, buthowever a classteacheror a teacher who teaches them can easily say the performance aspect of a student of her class students are like good, average performer, bad, fair (not that good), very good, or very bad and soon.

Now suppose we try to order them; will it be a totally ordered set or a partially ordered set. It is important in order to makethis book a self-contained one we just recall thedefinition of a totally ordered set and a partially ordered set and illustrate themwithsomeexamples.

Definition 1.1 Let S be a set of elements. We say S is totally ordered if for any pair of elements a, b S we have a b (or b a). Thus {S, } is a totally ordered set if the elements of S can be putintheform s1 s2 s3  sn , n can be finite or infinite depending on the number of elements in S.

S = { s1,…, sn }.

We call s1  s2 s3  sn ; to be an increasing chain or anincreasing order.

On the otherhandif we write S as sn  sn–1  sn–2  …  s3  s2  s1 ; then we call this as decreasing order or a decreasing chain.

In this book we only use the increasing order chain just likethelineofrealsorrationalsorintegers.Formorerefer[].

Example1.1.LetS={–28,2,4,–1,0,7,8,90,–42,+1,3}bea set of positive and negative integers, S is a totally order finite settheincreasingchainofSisasfollows. –42  –28  –1  0  1  2  3  4  7  8  90

Order of S is 11 and that of the chain of S is also of length o(S)=11.

We now proceed onto give another example of a finite totallyorderedset.

Example 1.2. Let S = {{a}, {}, {a, b, c}, {a, b}, {a, b, c, d}, {a, b, c, d, e}, {a, b, c, d, e, f}, {a, b, c, d, e, f, g}, {a, b, c, d, e, f,g,h}}beasetoforder9.

We show S is a totally ordered set the total ordering is done by the ‘inclusion’ relation  that is the set containment relation. We may call the inclusion relation as the containment relation.

{}  {a}  {a,b}  {a, b, c}  {a, b, c, d}  {a, b, c, d, e}  {a,b,c,d,e,f}  {a,b,c,d,e,f,g}  {a,b,c,d,e,f,g,h} ...I

Thisisachainoforder9andthisSisatotallyorderedset.

ConsideranysubsetPofSsay

P={{a},{a,b,c,d},{a,b,c,d,e},{a,b,c,d,e,f,g,h}}  S.

Clearly P is also a totally ordered set under containment relationfor {a}  {a,b,c,d}  {a,b,c,d,e}  {a,b,c,d,e,f,g,h} …II

II is called the subchain of I and the length of the subchain II is oforderfour.

We have all lengths of subchains of I from 2 to 8, for the lengthofthechainIis9.

Example1.3.LetRbethelineofrealsorrealcontinuum.

R is a totally ordered set for any two real numbers are comparable.

ThechainassociatedwithRisasfollows. – <…<–n<…–1<0<1<2<…<n<…<  …I

IisthechainofincreasingorderassociatedwithR.

In fact R is an infinite chain of real numbers as R the realsisofinfinitecardinalityanditisatotallyorderedset.

The chain I contains subchains of both finite and infinite order.

For take P1 = {–0.0001, –0.004, 81, –9, –0.01, 1, – 1, 0, 0.001,0.08,0.9,–8,19,8,9,98,18,1485}  R.

ThecardinalityofP1 is 18andP1 is atotally ordersetand thesubchainofIisasfollows.

–9  –8  –1  –0.01  4  –0.001  –0.0001  0  0.001

 0.08  0.9  8  9  18  19  81  98  1485 …II

IIis a subchain of I of finite order,thatis lengthofthe subchain IIis18.

Consider the subset of R given by P2 = {–, 0} the total orderingoftheinfinitesetP2 is – <…<–n<…<–5  –4  –3  –2  –1<0 …III isthesubchainofthechainI.

The subchain III is also increasing one but of infinite order.

Thus the infinite chain can have subchains of both finite andinfiniteorder.

Now having seen examples of totally ordered set now we proceed onto define partially ordered set and give examples of them.

Definition 1.2 Let S be a nonempty set, we say S is a partially ordered set if there exists at least two distinct elements a, b S suchthata b (or b a).

It is to be noted that unlike the totally ordered set in which every pair should be comparable in a partially ordered set where at least two distinct elements are comparable; S can be finitesetor aninfinite set.

Wegiveexamplesofpartiallyorderedsets.

Example1.4. LetP(S)bethepowersetofthesetS.

S = {a, b, c, d, e} and P(S) = {{}, {a}, {b}, {c}, {d}, {e}, {a, b},{a,c},{a,d},{a,e},{b,c},{b,d},{b,e},{c,d},{c,e},{d, e},{a,b,c},{a,b,d},{a,b,e},{a,c,d},{a,c,e},{a,e,d},{b, c,d},{b,c,e},{b,e,d},{c,e,d},{a,b,c,d},{a,b,c,e},{a,b, d,e},{a,c,d,e},{b,c,d,e},{a,b,c,d,e}=S} bethepowersetofS.

Clearly S is only a partially ordered set as we have subsetsinP(S)whicharenotcomparable.

For take {a, e} and {b, d} in P(S) we cannot relate them byacontainmentrelationofsubsetsofS.

Consider{a,b,c},{c,d,e}  P(S)weseetheycannotbe relatedusinganycontainmentrelation.

Consider{a,b,c}and{a,b,c,d}  P(S),wesee

{a,b,c}  {a,b,c,d}soP(S)isapartiallyorderedset.

HoweverP(S)is notatotallyordered set aswehavepairs ofdistinctelementswhichdonotsatisfycontainmentrelation.

We give some subsets of P(S) which forms a totally orderedset.

Consider B1 ={{},{a,b},{a,b,c},{a,b,c,d},{a,b,c,d,e}}  P(S).

ItiseasilyverifiedB1 isatotallyorderchain.

ThetotallyorderedchainofB1 isasfollows.

{}  {a,b}  {a,b,c}  {a,b,c,d}  {a,b,c,d,e} …I

ClearlythechainIoflength5andorderofB1 isalso5.

Nowconsider

B2 ={{},{a,b},{a,b,c},{b,c,d},{a,b,c,d},{a,b,c,d,e}} beasubsetofP(S).

Clearly B2 is not a totally ordered set. B2 is only a partially ordered set as {a, b} is not continued in {b, c, d} and {a, b, c} and {b, c, d} are not comparable we get the following partiallyorderedfigure1.1.

{}  {a,b}  {a,b,c}  {a,b,c,d}  {a,b,c,d,e}

{b,c,d}  {a,b,c,d}  {a,b,c,d,e}.

Thuswehavetwobranchesortwodistincttotallyordered chainsgivenbythefollowing.

{}  {a,b}  {a,b,c}  {a,b,c,d}  {a,b,c,d,e} …(A)

{}  {b,c,d}  {a,b,c,d}  {a,b,c,d,e} …(B)

WeseeAandBaretwodistinctchains

{}

{a,b} {b,c,d} {a,b,c} {a,b,c,d} {a,b,c,d,e}

 

Figure1.1

We can say the longest totally ordered chain in P(S) is only 6.Someofthetotallyorderedchainsin P(S)is giveninthe following.

{}  {a}  {a,b}  {a,b,c}  {a,b,c,d}  {a,b,c,d,e} …1

{}  {a}  {a,c}  {a,b,c}  {a,b,d,c}  {a,b,c,d,e}…2

{}  {a}  {a,d}  {a,d,c}  {a,b,c,d}  {a,b,c,d,e}…3

{}  {a}  {a,d}  {a,d,c}  {a,d,c,e}  {a,b,c,d,e}…4

{} {a}  {a,e}  {a,e,d}  {a,e,d,c}  {a,b,c,d,e}…5

{}  {a}  {a,e}  {a,e,c}  {a,e,d,c}  {a,b,c,d,e}…6 andsoon.

The task of finding all totally ordered chains of largest orderviz6isleftasanexerciseforthereader.

Wehaveseenexamplesofpartiallyorderedsets.

In fact, the following observations are given without proof which can be taken as a results by the readers for more refer[6,9].

Result 1.1. Every totally order set {S, } is also a partially orderedsetunder‘’.

Follows from the very definition of a partially ordered sets which demands that there which at least a pair of distinct elementsinSwhichareordered.Hencetheclaim.

Result 1.2. A partially ordered set (S, ) in general is not a totallyorderedset.

Proof. All totally ordered sets, (S, ) are partially ordered set (S, ) for we can compare any pair of elements in S. However, if(A, )isonlyapartiallyorderedsetthenthereexists atleast a pair of elements a, b  A such that a and b are not related. Hence the claim that a partially ordered set is not a totally orderedset.

The classical or very common example of this situation is {P(S), } the powerset of any non-empty set under the inclusion (or containment) operation,  as only a partially orderedsetandisnotatotallyorderedset.

Result 1.3 Let S be a non-empty set. P(S) the power set of S. {P(S), }isonlyapartiallyorderedset.

For P(S) has subsets P1, P2 such that P1  P2 and also has subset R1, R2 such that R1  R2 and R2  R1 for instance take

singleton set {a}, {b}, (a  b a, b  S) then {a} and {b} are not orderedinanywayso{P(S), }isonlyapartiallyorderedset.

Wecanshowseveralexamplesofthesame.

Result 1.4 Let o(S) = n and P(S) the power set of S. {P(S), } the partially ordered set P(S) has a totally ordered chain of maximallengthofordern+1.

Prooffollowsfromthesimplefactwehaveachain {}  {a1}  {a1,a2}  {a,a2,a3}  {a1,a2,a3,a4}  {a1,a2,a3,a4,a5}   {a1,a2,a3,a4,a5,…,an}=S …I

Clearly I is an increasing totally ordered chain and the lengthofIisn+1.

Result 1.5 Let {S, } be a totally ordered set. Every subset of S isatotallyorderedset.

Proof . If there is a subset say; P1 of S which is not a totally ordered set then we have some elements in P1 which are not comparable.

But as P1  S and every pair of elements in S is comparable hence the above assumption P1 is not totally orderableisnotpossible.

Hence every subset of the set S is again a totally ordered set.

Now we proceed onto obtain results relating with linguistic sets associated with a linguistic variable. If we are

studying time dependent linguistic variables certainly they are totallyorderable.

Concepts like colour etc, may not be totally orderable howeversomeofthemarepartiallyorderable.

However throughout this book we use only those linguistic variables which has a totally ordered linguistic set associatedwithit.

Only with totally ordered linguistic sets we can perform min and max operations. This assumption is very vital in this book.

We do not deny the existence of linguistic sets which cannot be totally ordered, those linguistic sets which are also time dependent still not totally orderable we avoid them for the present, we are not in a position to use them in this book for we construct linguistic matrix theory in par with classical matrix theory where total or partial order is mandatory for us to performoperationsonlinguisticmatricesonlyinthisbook.

Or we feel this is beyond the scope of this book, for we assume every linguistic variables considered in this book yield their respective linguistic set to be a totally ordered set or a linguisticcontinuum.

We now prove some examples of totally ordered linguisticsetsassociatedwithsomeofthelinguisticvariables.

Example 1.5. Let S = w(L) be the linguistic set associated with the linguistic variable L age of the person. This linguistic variable is obviously time dependent. In case of age of persons

we see the age increases from o to maximum age and each day theybecomeoldbyadayandsoon.

Let

W(L) = {old, just old, middle age, just young, very young, just middleage,youngandyoungest}

bethelinguisticsetassignedbyanexpertforsomepeopleunder investigation.

It is easily seen that w(L) is a totally ordered set. For the age of the person changes from very young or youngest to very oldoroldestastimegoeson.

youngest  very young  just young  young  just middle age  middleage  justold  old …I

This is the totally ordered set and is an increasing chain oflength8whichisthesameasthatoftheorderofS.

Suppose as discussed earlier, age of a person is discussed from his birth to his very old age then we can use the linguistic continuum

[youngest,veryold]  {}

the linguistic empty term is included as his the age is assessed used linguistic technique this term  denotes the empty linguistic term. This is a continuous one hence it is a totally orderedset.

Wegetthefollowingincreasingtotallyorderedchain;

  youngest  veryveryveryyoung  veryveryyoung  veryyoung  justyoung  young   justmiddleage

…  middleage  …  justold  old  veryold.

Thisis aninfinitetotally orderedlinguisticchainwhichis differentfromlinguisticvariable,ageofaperson.

Now we give yet another real world example of a linguisticvariableanditsassociatedlinguisticset.

Example 1.6. Let us consider the speed of a vehicle on a busy roadwhichisconsideredasthelinguisticvariable.

The expert wishes to work only with a finite number of linguistictermsassociatedwiththislinguisticvariable.

First of all in the signals the speed of any car will tend to droptozero,whichisdenotedbytheemptylinguisticterm ,so itsmandatoryto add  inourlinguisticset.Thestartingspeedat each signal will be very slow or slow and so on for any vehicle ontheroad.

LetS w(speedofavehicleintheroad)

= {, very very slow, very slow, slow, medium, just medium, fast,justfast,veryfast,veryveryfast,fastest}.

Now this is a totally ordered set and is totally increasing orderchainisasfollows.

  veryveryslow  veryslow  slow  justmedium  medium  justfast  fast  veryfast  veryveryfast  fastest …I

Here we wish to state the following facts about the linguistic variable age, is different from the linguistic variable speed of the car; for the linguistic variable age the linguistic terms goes on increasing there is no possibility of decreasing steadily but increases with time; however in case of the linguistic variable speed of the car the starting point is the empty linguistic term  then slow and goes on increasing but just as the time goes on and when the vehicle is near the signal the speed has to reduce and if signal falls the speed reduces to the empty linguistic term  and then starts to increase; further on the speed breakers the speed is again reduced though not to theemptylinguisticterm .

Thus one sees in case of speed theterms increases slowly reaching a maximum (that may occur) and slows down depending on the traffic in the road and signals the vehicle has tocross,howeverthissortofincreaseordecreaseisnotpossible in case of the linguistic variable age it is ever increasing and afterdeathstopsorterminatesthechain.

Thus it is important to note the difference between the twolinguisticvariables.

We give some more examples of changes of a linguistic variablerelatedtotime.

Example 1.7. Consider the height of a person from birth to death. The height of a person increases up to some level and thenremainsaconstant.

Thus for the linguistic variable ‘height’ of a person it changes from [shortest, tallest], the shortest being height of the

person at birth and the tallest the height of a person which remainsaconstantafteraage.

The growth of height for some may be up to 18 years for some upto the age of 20 yearsandfor someothersitmaybeup to theageof21andso onbut afterage 22they(humans)do not grow.

This linguistic variable height of a person is distinctly different from the other linguistic variables; age of a person which steadily increases till death of a person or the speed of a carwhichincreasesanddecreaseswithincreaseoftime.

A researcher must observe these linguistic variables with intensecarefortheyarenotthesameforalllinguisticvariables.

Nowweprovideyetanotherexample.

Example 1.8. Let us consider the weight of a person. The weight of a person is the linguistic variable. The weight of a person varies from the birth weight increasing with time and also decreases with some health problems after a stage may remain constant for some years increase in the middle age and afterthatmaydecreaseorreducewitholdageandsoon.

This linguistic variable is very different from the other 3 linguistic variables, weight, age and height of a person and speedofcar.

Howeverjustlikethespeedofthevehicletheweightmay nottouchtheemptylinguistictermortheleastterm.

The researcher or reader is requested to observe all the fourlinguisticvariables.

Now let us consider the linguistic variable “performance” ofaworkerworkinginafactory.

How does the performance of a person described as a linguistic set varies from other linguistic variables described so far.

Example 1.9. Let us consider the performance of a specific workerworkinginafactory.

He may be a very good worker at the time of joining as time goes he may be the best (let us assume) he may become a less performer if he has some problems in the family or with other co-workers or with the employer or if he has some health problems so his performance may fluctuate at this period and after that it may improve or it may deteriorate. Thus the performance of a person may also be changing from time to time. This linguistic variable is different from the other linguistic variable so far discussed. However this linguistic variableisalsoonlytimedependent.

Furtherthisvariable“performance”alsodrasticallyvaries from person to person also not like the linguistic variables age or height which increases steadily (in case of height up to some age).

Finally we will briefly discuss about the linguistic variabletemperatureofwaterfrom0o toboilingpoint.

Example 1.10. Let us consider the variable; temperature of waterfromsolidicestatetoitsboilingpoint.

The temperature of water is a linguistic variable which is under study, while trying to find the change in temperatures whenheated.

The temperature under uniform heat increases its temperature from 0oC to 100oC steadily increases and it can be easilymappedonthelinguisticscaleorintervalfrom [lowest,highest]  {}

where empty linguistic term word denotes the temperature of water at the 0oC and lowest just the process of increasing and highest at the 100oC. This is like the linguistic age variable with uniform temperature; increasing steadily with time, why it is heateduniformly,

Howeverifthesamelinguisticvariablewatertemperature in a pond or well or sea or lake is studied the condition or the linguistic set associated with it would be entirely different. The trend will be increasing to decreasing, constant for some time decreasingtoincreases.

Next we discuss the linguistic variable weather report of anyplaceforaweekoramonthoradayorayear.

We are only measuring the temperature not the humidity orrainfall.

Now for the present the linguistic variable of the weather report with specification to temperature is under study of a particularplaceinIndiasay,Vellore.

Now the temperature will vary in a dayfrom high or very high in the daytime to lowor very low inthenight orveryearly morning of 3 am to 4 am etc. So the linguistic variable of temperature of the day in a specific place will be very varying fromhightolow.

This is dependent not only in the place we live, it is also dependentontheseasonthetemperaturerecorded andsoonand so forth. This is also different from the otherlinguisticvariables under study. This linguistic variable is different from other linguistic variables though time dependent yet functions very differently.

Furtherthroughoutthisbookwetakeonlythoselinguistic variables which has its associated linguistic set / term / word to beatotallyorderedsetforotherwisewewillnotbeinaposition toworkminormaxorbothoperationsonthem.

We do not mention those linguistic variables which contribute to linguisticwords / terms which are not orderable or partiallyorderable.

We have given examples of them this never occurs in case of usual number systems or even the memberships which wegiveusingfuzzymembership.

Oncewegivenumbersthesituationisverydifferentforit becomesorderablestructure.

Sayifwehavesome9colours

{black,white,red,brown,blue,green,yellow,orange,pink}

we cannot order them however we can order the colour red if bright red to light red among the ‘red’ property but ordering red withwhiteorblueorgreenisnotpossible.

But some fuzzy membership property is used these membership values associated with them becomes ordered. However at this juncture we do not want to discuss or debate aboutitsvalidityinthisbook.

Repeatedly we keep on record we use in this book only those linguistic variables whose linguistic terms are totally orderable,thatiswhywecandefineoperationsonmatricesbuilt overthem.

Now we proceed onto describe operations on the linguisticsetsassociatedwithalinguisticvariable.

Example 1.11. Let L be the linguistic variable associated with thequalityofthedurabilityofproductsmadebyanindustry. Let w(L)={verygood,good,fair,bad,justfair,justgood,(notupto mark=notbad)notbad,verybad,veryverygood}.

Now w(L) is totally ordered set it is time independent for itdoesnotvarywithtime.

Now we have the totally ordered chain associated with w(L)inthefollowing:

very bad  bad  not bad  just fair  fair  just good  good  verygood  veryverygood.

Nowweknow

xif x  y min{x,y}= yif y  x

So min{fair,veryverygood}

=fairasfair  veryverygood min{bad,good}

=bad;fromthetotallyorderedchainasbad  good. (Recall in a totally ordered system every pair is ordered or‘comparable’).

Now we can also define max operator on S. For x, y  S (x  t)

xify  x max{x,y}= yifx  y

Further max{fair,veryverygood}

=veryverygoodasfair  veryverygood. max{bad,good}=goodasbad  good.

Clearlymax{x,y}  min{x,y}unlessx=y.

When x=ywehavemin{x,x}=x=max{x,x}. max{x,y}=min{x,y}ifandonlyifx=y.

Theorem 1.1. Let S be a linguistic set associated with a linguistic variable.S is a totally ordered set.

max {x, y} = min{x,y}if and onlyif x =y; x, y S.

Proof. To provemax {x,y}= min {x, y}if and only if x = y for x,y  S.

Letmax{x,y}=min{x,y}forx,y  Stoprovex=y.

max{x,y} = if x y  x …I min{x,y} = if x x  y …II

From Iand IIwe see x  yand y  x this is possible if and only ifx=y.Hencetheonewayclaim.

Suppose x=y(x  yandy  x)toprove min{x,y}=max{x,y}.

Consider min{x,x}=xand max{x,x)=x.

Somin{x,y}=max{x,y}asx=y.Hencetheclaim.

We have found min and max operators on a pair of elementsfromthetotallyorderedset.

Now we find the min and max operators on a finite sub setofthetotallyorderedsetS.

Considerthesubsets

A={verygood,good,notbad,verybad,bad,fair}and

B={justgood,justfair}

be two proper subsets of the linguistic set w(L) given in example related to the variable durability of products made by anindustry.

Wefindoutmin{A,B}min{A,B}

= min{{very good, good, not bad, very bad, bad, fair}, {justgood,justfair}}

= {min{very good, just good}, min{very good, just fair}, min{good, just good}, min {good, just fair}, min{not bad, just good}, min{not bad, just fair}, min{very bad, just good}, min{very bad, just fair}, min{bad, just good}, min{bad, just fair},min{fair,justgood},min{fair,jutfair}}

={justgood,justfair,notbad,verybad,bad,fair} …I

Thisisthewayminoperationisperformedonsets. NowweperformmaxoperationonthesubsetsAandBofS

max{A,B}=

max{{very good, good, not bad, very bad, bad, fair}, {just good,justfair}}

= {max {very good, just good}, max{very good, just fair}, max{good, just good}, max{good, just fair}, max{not bad, just good}, max{not bad, just fair}, max{bad, just good}, max{bad, just fair}, max{very bad, just good}, max{very bad, just air}, max{fair,justfair},max{fair,justgood}}

={verygood,good,justfair,justgood} …II

Clearly I and II are distinct and number of elements in max{A,B}is4whereasthatofmin{A,B}is6.

We just observe A  B = , that is there are no common elementsbetweenAandB.

Howevermin{A,B}  max{A,B} ={justgood,justfair,notbad,verybad,bad,fair}  {justfair, justgood}

= {just fair, just good} that is they have the very subset B to be containedinbothmin{A,B}andmax{A,B}.

ThusB  min{A,B}andB  max{A,B}and min{A,B}  max{A,B}=B,howeverA  B= .

Now we proceed onto give an example of two linguistic subsets of w(L) for the same linguistic variable “durability of theproduct”ofsomefactorygiveninexample1.10.

LetM={verygood,good,bad,fair,notbad}and N = {bad good, just fair, just good, very bad, very very good} betwosubsetsofw(L).

ClearlyM  N={bad,good}  

Nowwefindout

min{M, N} = min{{very good, good, bad, fair, not bad}, {bad, good,justfair,justgood,verybad,veryverygood}}

= {min{very good, bad}, min{very good, good}, min{very good, just fair}, min{very good, just good}, min{very good, very bad}, min{very good, very very good}, min{good, bad}, min{good, good}, min{good, just fair}, min{good, just good}, min {good, very bad}, min{good, very very good}, min{bad, bad},min{bad,good},min{bad,justfair},min{bad,justgood}, min{bad, very bad}, min{bad, very very good}, min{fair, bad}, min{fair, good}, min{fair, just fair}, min{fair, just good}, min{fair, very bad}, min{fair, very very good}, min{not bad, bad},min{not bad,good},min{not bad,justfair},min{notbad, just good}, min{not bad, very bad}, min{not bad, very very good}}

={bad,good,justfair,justgood,verybad,verygood,notbad} …I

Nowwefind

max{A,B}=max{{verygood,good,bad,fair,notbad}},{bad, good,justfair,justgood,verybad,veryverygood}}

= {max{very good, bad}, max{very good, good}, max{very good, just fair}, max{very good, just good}, max{very good, very bad}, max{very good, very very good}, max{good, bad}, max{good, good}, max{good, just fair}, max{good, just good}, max{good, very bad}, max{good, very very good}, max{bad, bad}, max{bad, good}, max{bad, just fair}, max{bad, just good}, max{bad, very bad}, max{bad, very very good}, max{fair, bad}, max{fair, good}, max{fair, just fair}, max{fair, just good}, max{fair, very bad}, max[fair, very very good}, max{not bad, bad}, max{not bad, good}, max {not bad, just

fair}, max{not bad, just good}, max {not bad, very bad}, max{notbad,veryverygood}}

= {very good, very very good, good, bad, just fair, just good, fair,notbad} …II

IandIIaredistinctlydifferenti.e.max{M,N}  min{M,N}.

FurtherM  N={good,bad}and max{M, N}  min{M, N} = {bad, good, very good, not bad, fair,justfair,justgood}.

ClearlyM  min{M,N}  min{M,N}and N  max{M,N}  min{M,N}aswellas max{M,N}  min{M,N}  N.

Having seen examples of max and min operator on S and subsets of S we now proceed onto give examples of partially orderedlinguisticsets.

Example 1.12. Let S = {best, fair, good, very good, bad}} be the linguistic set associated with the linguistic variable performanceofastudentinaclassroom.

The power set of S is given by P(S) = { {best}, {fair}, {good}, {very good}, {bad}, {best, fair}, {best, good}, {best, very good}, {best, bad}, {fair, good}, {fair, very good}, {fair, bad}, {good, very good}, {good, bad}, {very good, bad}, {best, fair,good},{bestfair,verygood},{best,fair,bad},{best,good, very good}, {best good, bad}, {fair, good, very good}, {fair, good, bad}, {fair, very good, bad}, {good, very good, bad},

{best,bad,verygood},{best,fair,good,verygood},{best,fair, good, bad}, {best, bad, good, very good}, {fair, good, very good,bad},{best,fair,verygood,bad}}.

WeseeP(S)thelinguisticpowersetofthelinguisticsetS isonlyapartiallyorderedset.

In case of partially ordered set also we define two types oftwooperations ,  andotherisminandmax.

In case {P(S), min} is not the same as {P(S), }. Similarly{P(S),max}isnotsameasthatof{P(S), }. Weprovethisbytheexample.

Consider A={fair,good,verygood,bad}and B={best,fair,good,bad}betwolinguisticsubsetsofP(S).

WeseeA  B={good,bad},min{A,B}

=min{{fair,good,bad,verygood},{best,fair,good,bad}}

= {min {fair, best}, min{fair, fair} min{fair, good}, min{fair, bad}, min{good, fair}, min{good, fair}, min{good, good}, min{good, bad}, min{bad, best}, min{bad, fair}, min{bad, good}, min{bad, bad}, min{very good, best}, min{very good, fair},min{verygood,good},min{verygood,bad}}

={fair,bad,good,verygood}.

Clearlymin{A,B}  A  Bfor

{fair,bad,good,verygood}  {good,bad}.

InfactA  B  min{A,B}.

NextwefindA  Bandmax{A,B}.

A  B={fair,good,verygood,bad,best} …II

max{A, B} = max{{fair, good, very good, bad}, {best, fair, good,bad}}

= {max {fair best}, max{fair, fair}, max{fair, good}, max{fair, bad}, max{good, best}, max {good, fair}, max{good, good}, max{good, bad}, max{very good, best}, max{very good, fair}, max{very good, good}, max{very good, bad}, max{bad, best}, max{bad,fair},max{bad,good},max{bad,bad}}

={best,fair,good,verygood,bad} …III

A  B={fair,good,verygood,bad,best}

={best,fair,good,verygood,bad}=max{A,B}.

InthissetbothA  Bandmax{A,B}areequal.

Wenowfindforthesets

M={verygood,fair}and N={bad,best,good}

betwolinguisticsubsetsofthelinguisticpowersetS.

WeseeM  N={verygood,fair}  {bad,best,good}

={verygood,fair,bad,best,good} …(a)

Nowwefindmax{M,N}

=max{{fair,verygood},{bad,best,good}}

= {max{fair, bad}, max{fair, best}, max{fair, good}, max{very good,bad},max{verygood,best},max{verygood,good}}

={verygood,fair,best,good} …(b)

Clearly(a)  (b)thatisA  B  max{A,B}.

ThusweseeA  B=max{A,B}and

A  B = min{A, B} in case of the operations on linguistic subsets of a linguistic set or a linguistic power set P(S) are not ingeneraltrue.

So we see on linguistic subset the two operations  and  do not coincide so they will not be the same on the linguistic powersetP(S)ofalinguisticsetS.Thesamefor  andmax.

However on linguistic continuum the two operations coincidewhenwetakesingletonsetinpairs.

That a  b = min{a, b}, a, b  S(S a totally ordered linguistic setoralinguisticcontinuum).

Similarly a  b = max{a, b} on totally ordered linguistic setoratotallyorderedlinguisticcontinuuma,b  S.

Letusgiveonemoreexampleofthissituation.

Example 1.13. Let S = {high, low, lowest, medium, very low, very high, highest, just low, just high, just medium, very very low,veryveryhigh}

be a linguistic set associated with the linguistic variable temperatureofweatherinaday.

LetM={high,low,lowest,medium,verylow,veryhigh, highest,justlow}and

N = {just high, just medium, very very low, very very high} be twolinguisticsubsetofthelinguisticsetS.

ClearlyM  N={} …I and M  N=S …II

Nowwefind

min{M, N} = min{{high, low, lowest, medium, very low, very high, highest, just low}, {just high, just medium, very verylow,veryveryhigh}}

= {min{high, just high}, min{high, just medium}, min{high, very very low}, min{high, very very high}, min{low, just high}, min{low, just medium}, min{low, very very low}, min{low, very very high}, min{lowest, just high} min{lowest, just medium), min{lowest, very very low}, min{lowest, very very high}, min{medium, just high}, min{medium, just medium}, min{medium, very very low}, min{medium, very very high}, min{very low, just high}, min{very low, just medium}, min{very low, very very high}, min{very low, very very low}, min{very high, just high}, min{very high, just medium}, min{very high, very very low}, min{very high, very veryhigh},min{highest,justmedium},min{highest,justhigh}, min{highest, very very low}, min{highest, very very high}

min{just low, just high}, min{just low, just medium}, min{just low,veryverylow},min{justlow,veryveryhigh}}

= {just high, just medium, very very low, high, low, lowest, medium,veryverylow,verylow} …III

ClearlyM  N  min{M,N}.

Nowfindmax{M,N}

= max{high, low,lowest, medium, very low,veryhigh, highest, just low}, {just high, just medium, very very low, very very high}}

= {high, very very high, just high, just medium, medium, very verylow,verylow,veryhigh,highest,justlow} …IV

ClearlyIIandIVaredistinctforS  max{M,N}.

Now we find for the same set max{min, M, N}} and min{max{M,N}}.

EquationIIIgivesmin{M,N}andIVgivesmax{M,N}.

Nowmax{min{M,N}}usingIIIis

max{just high, just medium, very very low, high low, lowest, medium,veryverylow,verylow}=high …V

WefindequationIVmin{max{M,N}}

= min{high, very very high, just high, just medium, medium, veryverylow,verylow,veryhigh,highest,justlow} =veryverylow.

Thusmax{min{M,N}}  min{max{M,N}}.

Clearly min-max or max min operation can be performed onlyonpropersubsetsofthesetS.

Now we suggest a few problems for the reader to familiarizethenotionoflinguisticsetsassociatedwithlinguistic variables and min and max operations on linguistic sets. The starredproblemsaredifficulttobesolved.

SUGGESTEDPROBLEMS

1. Give at least four linguistic variables which cannot be ordered (different from colour eyes or ears or skin of humans).

2. Give four examples of linguistic variables which are time dependentandorderable.

3. Does there exist a linguistic variable which is time dependentyetnotorderable?

4. Suppose there are 5 persons belonging to five-different partsoftheworld,theircoloursare {white,yellow,darkbrown,black,lightbrown}=C

i) CanCbetotallyordered?

ii) CanCbepartiallyordered?Justifyyourclaim.

5. Heights of 7 persons are given by the linguistic set S; S = {verytall,tall,short, veryshort,justtall, very veryshort,mediumheight}.

i) Isheightatimedependentvariable?(ingeneral).

ii) CanSbeatotallyorderedset?(ifsoorderS).

iii) IsSapartiallyorderedset?

iv) Can we say S is the subset of the linguistic continuum [shortest, tallest] associated with heightofaperson?

6. Let S = {0.7, 9, 2, 4, –3, 0.09, 4, 1} be the set of real numbers.

Prove S is a totally ordered set and total order S by ‘’ lessthanor  greaterthanrelation.

7. LetP(S)={collectionofall subsetsofthesetS={a,b, c, d}} ={{a},{b},{c},{d},{a,b},{b,c},{c,d},{a,c},{a,d}, {b,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d},{},{a,b, c,d}=S}.

i) ProveSisonlyapartiallyorderedset.

ii) Give at least there are 5 subsets (elements) in P(S)whichistotallyordered.

iii) Prove S has no collection of sets whose cardinality is greater than 5 can be totally ordered.

iv) GivesomeexamplesofsubsetsofSwhichcannot beordered.

8. Let A={{},{1,2}, {1,2,4},{1,2,4,7},{1,2,4,7,9}, {1, 2, 8, 4, 7, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9}} be a set of subsets.

i) IsAatotallyorderedset?

ii) WhatisthelengthofthechainofA?

iii) Is Aundertheoperationof  closed?Justifyyour claim!

iv) IsAclosedundertheoperation ?Justify.

9. Can we say if a set X is a totally ordered set then can we say X is closed under the operation  and not closed undertheoperation ? Justifyyourclaimandproveitinageneralcase!

10. Let S be the linguistic continuum related with the linguisticvariableweightofaperson[lowest,highest}

i) Find linguistic subsets of finite order and prove theyaretotallyorderedsets.

ii) Let M = {low, very low, just low, very very low, medium,justmedium,high,veryhigh,justhigh},

N = {lowest, very very high, very medium, very very very low, higher, just higher, just lower, lower} be two linguistic subsets of the linguistic continuumS=[lowest,highest].

a) Calculatethefollowingvalues.

i) min{M,N}, ii) max{M,N}, iii) M  N, iv) M  N, v) min{max{M,N}}and vi) max{min{M,N}}.

Compare the 6 values. How many of them are equalwitheachother?

b) Find two finite subchains constructed using thechainofSoflength16and18.

c) Findthe linguistic totally orderedsubchains of the linguistic subsets M and N and find theirlengths.

11. Let S = {brown, light brown, very light brown, just brown, very very light brown, whitish brown, very very dark brown, very dark brown} be the linguistic set on the linguisticvariableshadesofthebrowncolour.

i) Find the linguistic totally ordered increasing chainCofS.

ii) WhatisthelengthofthelinguisticchainCofS?

iii) Is every linguistic subset of the linguistic set S a totallyorderedsubchainofC?Justifyyourclaim.

iv) Let A = {brown, light brown, very light brown, just brown, dark brown, just light brown} and B = {just brown, dark brown, very very light brown, whitish brown, very very dark brown} be twolinguisticsubsetsofthelinguisticsetS.

Find (a) A  B, (b) A  B, (c) min{A,B}, (d) max{A,B}andcomparethem.

v) min{max{A,B}}and

vi) max{min{A,B}}andcomparethem.

12. Let B = {brown, black, white, red, dark red, light red, orange, pink, dark blue, light blue, very light blue, green, light green} be a linguistic set associated with the linguisticvariablecolour.

i) IsBatotallyorderedlinguisticset? ii) IsBapartiallyorderedlinguisticset? iii) FortwolinguisticsubsetsXandYof(B). Findthevaluesof i) X  Y, ii) X  Y, iii) min{max{X,Y}},

iv) max{min{X,Y}},

v) min{X,Y}

vi) max{X,Y}andcomparethem.

iv) What is the length of the maximal linguistic chainsinP(B)?

v) How many maximal linguistic chains does P(B) contribute?

vi) Find any other special feature enjoyed by the linguisticpowersetP(B).

13. Give anexample ofa linguistic variable other than colour forwhichthelinguisticsetassociatedwithitis not totally orderablebutpartiallyorderable.

14. Let S = {dense, very dense, just dense, scattered, very scattered, just scattered, slightly dense, light barren, slightly scattered, barren} be the linguistic set associated vegetationofaplaceinIndia.

i) IsSatotallyorderedset?

ii) WhatisthelengthofthelinguisticthechainofS?

15. Let S be a linguistic set associated with the linguistic variable interpersonal relations in a family (between husband and wife); S = {hatred, loving, understanding, fights over frivolous matters, hates each other, neutral,no understanding,…}.

Clearly S cannot be ordered. However it has partial order andP(S)thelinguisticpowersetofScanbeordered.

FindallpossiblepropertiesassociatedwithSandP(S).

16*. LetSbethelinguisticset“indeterminate”

a) Find W(S) the set of linguistic words associated withS.

b) Associate a real world problem which is a linguistic variable L associated with S the indeterminatevariable.

BASIC PROPERTIES OF LINGUISTIC MATRICES

In this chapter we define the notion of linguistic matrices and discuss some the basic properties with examples. The basic notions and notations of linguistic variable L, and its associated linguistic word set w(L) or the linguistic terms w(L) is introducedinchapterI.

Further we have to define on any linguistic term / set we have an ordering as that of the ordering found in the line of integersorrationalsorreals.

The linguistic set can be finite or infinite or can also be a continuum. However for any linguistic set we also have with it an associated linguistic variable; the ordering of the linguistic set is dependent on the ling variable. It may be totally orderable orattimesunorderable.

Further unlike line of reals or rationals or integers the continuum given by a linguistic variable can be many which solelydependsonthelinguisticvariableL.

All these has been explained earlier but however we recall again and again mainly for the readers to remember them as these concepts of distinctly different from the usual number system.

The matrix is defined as a rectangular array of numbers forinstanceAis

A=

9324 10.557 0701 1102 4568 1010

         

Thisisa6  4rectangularmatrix.

LetBbeamatrixofnumbergiveninthefollowing

B=

       

60124 12006 1370.50 01020.7 10.1606

Bisa5  5rectangularmatrix.

ConsiderCthematrixofnumbers; C=(7,0,8,0.5,4,7,8,1)

Cisa1  8matrix.

Cwillbeknownasa1  8rowmatrix.

LetDbeamatrixofnumberofgivenbythefollowing D=

0.8 0.7 0.8 5 4 2 1 7 9 0.6 1.2

                

Disknownasa11  1columnmatrix.

Thus we have seen 4 types of matrices. We will proceed onto describe zero matrix of all the four types. Wefirst provide anexampleofa1  9zeromatrixdenotedby (0)19 =(0,0,0,0,0,0,0,0,0)

Weproveandexampleofa7  4zeromatrix

(0)74 =

(0)66=

          

0000 0000 0000 0000 0000 0000 0000

         

000000 000000 000000 000000 000000 000000

isa6  6zerosquarematrix, (0)81 = 81

            

0 0 0 0 0 0 0 0 

isazero-columnmatrix.

Now we define the diagonal matrix of a 7  7 square matrix

M=

          

4000000 0900000 00100000 0001000 0000700 0000060 0000009

When all the entries of the diagonal are only one and the rest of the elements are zero then we call the square matrix as theidentitymatrix.

Weprovideexamplesofthem.

I44 =

1000 0100 0010 0001

     

, I77 =

1000000 0100000 0010000 0001000 0000100 0000010 0000001

          

,

I22 = 10 01    andI99 =

100000000 010000000 001000000 000100000 000010000 000001000 000000100 000000010 000000001

             

are identity matrices of orders 4  4, 7  7, 2  2 and 9  9 respectively.

Inn =

       

  isan  nidentitymatrix.

1000 0100 0010 0001

Now in case of linguistic matrices we proceed onto give examplesofthem.

Definition 2.1. Let S be a linguistic set (or (I) a linguistic continuum)and A be a rectangular array of linguisticterms.

We define A to be linguistic matrix with entries from S (or fromthelinguistic continuum I).

A =

      

aa…a aa…a aa…a

11121n 21222n m1m2mn

where aij S (or I); 1 i m and 1 j n. We say order of the linguistic matrix Ais m n.

Wewillprovidesomeexamplesofthem.

Example2.1. Let S={verytall,tall,medium height,short,very short, just tall, very very short, just medium height, shortest} be a linguistic finite set associated with the ling variable height ofaperson.

A = (very short, tall, shortest, short, just tall, tall, short, verytall,justmedium)

bea1  9linguisticrowmatrix. OrderofAis1  9.

                

shortest short tall verytall mediumheight tall veryshort short tall shortest justmediumheight 

B= 111

be a 11  1 linguistic column matrix. B is a column linguistic matrixoforder11  1.

C=

tallshortshortesttall tallshortesttallveryshort shortmediumshortshort veryshorttalltallmedium talljusttallshorttall shortmediumveryshortshort shortestjusttalltalltall justtallshortshortshort mediumtallshortesttall

             

Cis9  4linguisticmatrix.

OrderofCis9  4.

Now we provide an example of a 5  5 square linguistic matrixwithentriesfromS.

D =

veryshorttallshortmediumjusttall tallshorttallveryshorttall shortmediumshorttalltall mediumtalljusttallshorttall veryshorttallshortshortestshort

       

Disalinguisticsquarematrixoforder5  5.

LetE= very shorttalltallshorttallshort short talltalltalltallshortestshortshort justveryvery shortmediumshorttall tallshortshort veryveryverytalltalltalltall shortshortshort

          

Eisalinguisticmatrixoforder4  7.

Now we proceed onto work with the linguistic matrices withentriesfromalinguisticcontinuum.

Example 2.2. Let LC = [best, worst] be a linguistic continuum associatedwiththeperformanceofstudentsinaclassroom.

ClearlythecardinalityofLC is infiniteaswehaveinfinite numberoflinguistictermsinLC

Now we give examples of linguistic matrices which are row linguistic matrices, column linguistic matrices, rectangular linguisticmatricesandsquarelinguisticmatrices.

B = (worst, just good, good, very fair, fair, bad, very bad, best,better,justgood)

bearowlinguisticmatrixoforder1  10.

Just as in case ofmatrices with real numbers we can have entries of a linguistic matrix also to repeat here the term just goodrepeatstwiceinthelinguisticrowmatrixB.

It is to be noted that in case there are infinite number of 1  10 linguistic matrices if the linguistic set used is of infinite orderorisalinguisticcontinuum.

If on the other hand the linguistic set is finite the number oflinguisticmatricesoforder1  10isonlyfiniteinnumber. Infacttheexactnumberwillbe|S|10 = 10times |S||S||S| 

Now using LC the linguistic continuum of the linguistic variable; performance of students, we give examples of linguisticcolumnmatricesoforder9  1.

             

best bad good fair verygood bad justfair good bad 

D= 91

isalinguisticcolumnmatrixoforder9  1.

Herethelinguisticterms goodandbadhasrepeatedtwice andthricerespectively.

We see there are infinite number of such 9  1 linguistic columnmatricesusingLC

IfweuseafinitelinguisticsetSthereare |S|9 =|S|  |S|   |S|

number of 1  9 linguistic column matrices where |S| denotes thenumberofelementsinS,thelinguisticsetS.

LetM=

goodbadfairbest goodfairbadworst fairbadverygoodfair fairverybadworstbest

     

be a linguistic square matrix of order 4  4 with entries from LC.

We see the linguistic term bad has repeated twice, the linguistic terms good, best and worst has repeated twice and the linguisticterm,fairhasrepeatedfourtimes.

Number of linguistic square matrices of order 4  4 using thelinguisticcontinuumisinfinite.

If however instead of LC we use a finite linguistic set S thenthenumberof4  4linguisticsquarematrixis |S|16= 16times |S||S| 

Now let N be a 16  3 rectangular linguistic matrix with entriesfromLC.

NowletN=

                        

goodbadbest badbestfair fairbestbad goodgoodgood badbadbad bestbadgood worstverygoodfair veryfairbadbad goodgoodgoosd goodgoodgood bestbadbad badbadbad goodbestbest veryfairverygoodverybad justfairgoodbad goodbadgoo 163 d 

We see there are infinite number of 16  3 rectangular linguisticmatrices.

If we replace the linguistic continuum LC by a finite linguisticsetSthenwehave

We cannot provide examples of linguistic diagonal matrices unless we adjoin with the linguistic set S (or with the linguisticcontinuumLC)theemptylinguisticterm 

This linguistic empty term  of a linguistic set has been introducedinthechapter1ofthisbook.

Example 2.3. S be a finite linguistic set associated with the linguisticvariableageofaperson

S = {old, older, oldest, very young, just old, very old, younger, youngest},

weadjointtheemptylinguisticterm  withS.

LetS =S  {}.

The linguistic column matrix C with entries from S of order7  1isasfollows.

C=

             

old young old veryold

The linguistic empty column matrix of order 9  1 is as follows.

()91 =

                      

Now the linguistic row matrix D of order 1  9 with entriesfromS isasfollows.

D=(,young, , ,old,veryyoung, ,youngest,oldest). The1  9linguisticrowemptymatrix ()19 =(, , , , , , , , ).

Let E be a 4  9 linguistic row matrix with entries from S =S  {}giveninthefollowing.

E =

         

youngoldyoungest oldestold oldyoungveryold justoldjustyoung

oldoldoldest youngoldoldest youngestyoung oldestoldest

         

Nowthe4  9linguisticemptysetisasfollows. ()49 =

         

.

.

Finallyconsider alinguistic squarematrix N of order6  6 with entriesfromS =S {}isgiveninthefollowing: N=

               

oldyoungold veryyoungoldold youngoldestoldyoung justoldoldest justoldoldolderyoung justoldyoungestoldold

Wehaveonly|S|36 = 36times |S||S|    

number linguistic square matrices of order 6  6 with entries fromS

Nowthelinguisticemptysquarematrixoforder6  6is asfollows.

()66=

               

Now we give a few examples of diagonal linguistic squarematriceswithentriesfromS =S{}.

 

D

youngest oldest justold

D44, D33 and D66 are linguistic diagonal matrices of order 4  4,3  3and6  6respectively.

However we cannot define the linguistic identity matrix in auniquewayasincaseofusualmatrices. Theidentitymatrix isdependentontwofactors

i) The linguistic identity depends on the operation which is defined on that linguistic matrices in contrast with the classical product  operation in caseofusualmatrices.

ii) The linguistic identity depends on the linguistic setusedtoconstructthelinguisticmatrices.

However it is pertinent to recall we have defined two operationsonthelinguisticsetSvizminandmaxwhere min{x,y} =x  y and max{x,y}=x  y inchapter1.

We have not defined these operations on linguistic matrices. We will be defining the notion of min and max on linguistic matrices which can have compatible under min or max.

We canalso asincaseof usualmatricesdefine thenotion of transpose, symmetry etc. in case of linguistic matrices also. Wewillfirstillustratesituationbysomeexamples.

Example 2.4. Let S= {heavy, drizzle, veryheavy, heaviest, just heavy,justdrizzle,verylightdrizzle, }

be the linguistic terms associated with the linguistic variable ‘rain’.Here  denotesnorain.

Let R = (just drizzle, , heavy, just heavy, , heaviest, , verylightdrizzle)

bethe1  9linguisticrowmatrixwiththeentriesfromS. Nowasincaseofusualmatrices

Rt = (just drizzle, , heavy, just heavy, , , heaviest, , very lightdrizzle)t =

justdrizzle heavy justheavy heaviest verylightdrizzle

                 

the transpose of a linguistic row matrix of R is a linguistic columnmatrix.

ThusifRisalinguisticrowmatrixoforder1  9thenthe transposeofR; Rt isalinguisticcolumnmatrixoforder9  1.

Nowwefindout

(Rt)t =

justdrizzle heavy justheavy heaviest verylightdrizzle

                 

=(justdrizzle, ,heavy,justheavy, , ,heaviest, ,verylight) =R.

whichisthelinguisticrowmatrixoforder1  9westartedwith. Now on the other hand if C is a linguistic column matrix of order7  1withentriesfromSgivenby

C=

heavy justheavy veryheavy justdrizzle

             

.NowCt =

             

t heavy justheavy veryheavy justdrizzle

=(heavy, ,justheavy,veryheavy, ,justdrizzle, ).

Thus transpose of the linguistic column matrix Cof order 7  1isalinguisticrowmatrixoforder1  7.

NowwefindthetransposeoftransposeofC.

(Ct)t =

                        

t heavy justheavy veryheavy justdrizzle

=(heavy, ,justheavy,veryheavy, ,justdrizzle, )t =

heavy justheavy veryheavy justdrizzle

             

=C.

Thus as in case of usual matrices we see in of linguistic matricesalsotransposeofthetransposeoflinguisticmatrixMis alinguisticmatrixM.

Nowwefindthelinguistictransposeofasquarematrixof Morder5

5withentriesfromS.

                M.

veryheavyveryheavy drizzle drizzleheavyheavy

We will be characterizing the condition under which a transpose of a linguistic square matrix is equal to linguistic matrixM,i.e.M=Mt

Before we proceed do this about square linguistic matrices we will find the transpose of a linguistic matrix N of order8  3withentriesfromS.

N=

                  

heavyjustdrizzle drizzle drizzleverylightdrizzle heavyheavyheavy justheavydrizzlejustheavy verylightdrizzleheaviest heaviest

NowweproceedontofindthetransposeofN.

.

Nt =

t heavyjustdrizzle drizzle drizzleverylightdrizzle heavyheavyheavy justheavydrizzlejustheavy verylightdrizzleheaviest heaviest

                  

= heavydrizzle drizzle justdrizzleverylightdrizzle

       

heavyjustheavyverylightdrizzleheaviest heavydrizzle heavyjustheavyheaviest

      

We see N is a linguistic matrix of order 8  3 whereas Nt isalinguisticmatrixoforder3  8.

Itisleftforthereadertoprove(Nt)t =N.

We just define symmetric linguistic matrix. It is pertinent to keep on record that the notion of symmetry in case of linguistic square matrices it the same as that of usual square matrices.

Definition 2.2 Let S be linguistic set (or a linguistic continuum). Let M =

      

aa…a aaa aa…a

11121n 21222n n1n2nn

be a n n square linguistic matrix with aij S(orIC) 1 i,jn.

We say M is a symmetric linguistic matrixif and onlyif

aij = aji for alli j1i,j n. That is M =

aa…a aaa aa…a

      

11121n 21222n 1n2nnn

or equivalently we can say M is a symmetric linguistic matrix if and only ifM =Mt

Allsquarelinguisticmatricesarenotsymmetric.

We give examples of symmetric linguistic matrices and non symmetric linguistic square matrices; however we cannot define antisymmetric (or equivalently skew symmetric (or equivalently skew symmetric linguistic matrices as we do not havetheconceptofnegativelinguisticterms).

However later on when linguistic logic is developed and when higher level linguistic theory books are defined we will proceed onto developthe negation ofthe linguistic words which willserveasthenegativenumber.

At this juncture we only talk about linguistic symmetric matricesandnotlinguisticskew(oranti)symmetricmatrices.

Nowweprovideexamplesofthem.

Example 2.5. S = {low, lowest, just low, high, medium, , just high,veryhigh,verylow}

be a linguistic set associated with the linguistic variable temperatureofwater.

LetA= lowlowesthigh verylowhigh veryhighjustlow

      

bea3  3linguisticsquarematrixwithentriesfromS.

ClearlyAisnotasymmetriclinguisticmatrixasA  At

ConsiderH=

NowHt =

         

verylowlowhigh lowveryhighhighest highestlowest highlowestmedium

         

verylowlowhigh lowveryhighhighest highestlowest highlowestmedium

ThusHisasymmetriclinguisticmatrixasH=Ht . K=

=H.

lowhighlow lowlowlowlowhigh highhighestlow highlowhigh lowhighlow

           

bea5  5squarelinguisticmatrixwithentriesfromS.

ConsiderKt; Kt =

lowlowhighlow highlowhighesthigh lowlowlowhigh lowlowlow highhigh

            

.

Clearly K  Kt so K is not a symmetric linguistic square matrix.

Thus in general all square linguistic matrices are not symmetric.

Consider 6  6 linguistic diagonal matrix W with entries fromS. W=

low high lowest highest

                NowwefindWt;

Wt =

low high lowest highest

               

ClearlyW=Wt

Sowecansayalwaysadiagonallinguisticmatrixissymmetric.

Theorem 2.1 Let S be a linguistic finite set (or IC a linguistic continuum). LetDbe a n n square linguistic diagonal matrix.

Dis a symmetriclinguistic matrix.

Proof. To prove D is a symmetric linguistic matrix it is enough ifweshowD=Dt foranydiagonallinguisticmatrixD. LetD=

           

a a a

  

11 22 nn

bean  nlinguisticdiagonalmatrix.

Clearly Dt = D for every aij = aji =  if i  j; hence the linguisticdiagonalmatrixissymmetric.

Next we proceed onto prove that transpose of a transpose ofalinguisticmatrixMisM.

That(Mt)t =MandMt  M.

Theorem 2.2 Let S be a linguistic finite set (or IC the linguistic continuum). Let M be a s  n (s  n) linguistic rectangular matrix.

i) Mt M (that is transpose of M is notequal toM)

ii) (Mt)t = M (transpose of the transpose of a linguistic matrix Mis M).

Proof.GivenMisas  nrectangularlinguisticmatrix(s  n). M=

       

mmm mmm mmm 

11121n 21222n s1s2sn sn

wheremij  S(orIC);1  i  sand1  j  n.

NowwefindMt =

mmm mmm mmm

t 11121n 21222n s1s2sn

     

 

=

     



mmm mmm mmm 

1121s1 1222s2 1n2nsn sn

Clearly Mt is a s  n linguistic matrix (s  n). So M  Mt asbothhavedifferentorders.Hence(1)isproved.

Nowwefind(Mt)t . (Mt)t =

            

mmm mmm mmm 

t t 11121n 21222n s1s2sn sn

=



 =

     

mmm mmm mmm 

     

mmm mmm mmm 

t 1121s1 1222s2 1n2nsn ns

11121n 21222n s1s2sn sn

Thus(Mt)t =M,hence(ii)ofthetheoremisproved.

Now we proceed onto discuss about the notion of submatrixofalinguisticmatrix.

Wewillfirstdefinethisconcept.

Definition 2.3. Let S be a finite linguistic set (or a linguistic continuumIC).

SupposeA =

      

aa…a aaa aa…a

11121n 21222n m1m2mn

be a m  n linguistic matrix with aij  S (or IC); 1 i m and 1 j n.

We call M to be linguistic submatrix of A if we remove some columns or some rows or both some columns and some rows from A.

Wewillillustratethissituationsomeexamples.

Example 2.6. Let S= {good, very good,average,bad,verybad, lazy,worst,justgood,justaverage, ,best}

be linguistic set of order 11 related with the linguistic variable, performanceaspectsofaworkerinanindustry.

                 

goodaveragebestworst lazyworst badverybadaverage badgoodbad badgoodbad goodbad worstbadgood 

LetM= 75

bea7  5linguisticwithentriesfromS.

Suppose the first row of the linguistic matrix M is removed we get the linguistic submatrix S1 of M which is as follows.

               

lazyworst badverybadaverage badgoodbad badgoodbad goodbad worstbadgood 

S1 = 65

Now the 3rd column of M is removed we get the correspond linguistic submatrix S2 of M of order 6  5 which is asfollows.

                 

goodbestworst lazyworst badaverage badbad badgoodbad good worstbadgood 

S2 = 74

S2 isalinguisticsubmatrixoforder7  4.

Let S3 be the linguistic submatrix obtained from M by removingthe4th rowandfourthcolumn.

Thisisgiveninthefollowing.

               

goodaverageworst lazy badverybadaverage badbad goodbad worstgood 

S3 = 64

Theorderofthislinguisticsubmatrixis6  4.

Let S4 be linguistic submatrix obtained from M by the first3rowsand2nd and3rd columngiveninthefollowing.

       

bad badgoodbad worstbadgood 

S4 = 43

TheorderofthelinguisticsubmatrixS4 is4  3.

Let S5 be the linguistic submatrix of M obtained by removing the last 6 rows and last column. The linguistic submatrixS5 isasfollows.

S5 =[good  average best]14

Weseethelinguisticsubmatrixisarowmatrixoforder1  4.

Let S6 be the linguistic submatrix of M obtained by movingthelastrowandfirstfourcolumns.

ThelinguisticsubmatrixS6 ofMisasfollows.

            

worst average bad 

S6 = 61

TheorderofS6 is6  1acolumnmatrix.

Let S7 be the linguistic submatrix of M got by removing thefirstthreecolumnsandthefirstandseventhrowsofM.

The resulting linguistic submatrix S7 of M is given in the following.

           

worst average bad goodbad 

S7 = 52

isalinguisticsubmatrixof5  2.

Let S8 be the linguistic submatrix of M got by removing the first column and the last four rows. The linguistic submatrix S8 ofMisasfollows.

       

averagebestworst lazyworst verybadaverage 

S8 = 34

S8 islinguisticmatrixoforder3  4.

Let S9 be the linguistic submatrix of M obtained by movingthelast3columnsandlast5rows.

ThelinguisticsubmatrixS9 isasfollows.

    

good lazy 

S9 = 22

ThelinguisticsubmatrixS9 isoforder2  2.

Let S10 be the linguistic submatrix of M obtained by removingthefirstfourcolumnsandfirst6rowsofM.

S10 isgiveninthefollowing.

S10 =[good]

Fromthiswemakethefollowingobservations.

Every single linguistic term / element in the linguistic matrixMisalinguisticsubmatrixofM.

We have linguistic matrices are all order bound by 7  5 tobetrueoflinguisticsubmatrices.

But all things cannot contribute to linguistic submatrices. For instance [bad worst] or worst lazy    are not linguistic submatrices of M; for they cannot be got by removing some stipulatednumberofrowsandcolumnsofM.

Now finding the number of linguistic submatrices is also adifficultproblem. Thiswehaveforalinguisticmatrixoforder 3  4 so that interested students / researchers can find the total number of linguistic submatrices of any m  n linguistic matrix M.

Example 2.7. Let S be the linguistic set given in example 2.6. Nowlet.

     

goodlazyverygood badbestverybadaverage justgoodbadbestgood 

B= 34

bealinguisticmatrixoforder3  4withentriesfromS.

If we removethe last threecolumnsandlasttwo rowsthe resultinglinguisticsubmatrix

S1 =[good]11 ofBisatrivial1  1lingusiticsubmatrix.

Similarly if the first, third and fourth columns and last tworowsareremovedtheresulting1  1linguisticsubmatrixis

S2 =[]11 ofB.

On similar lines if the first, second and fourth columns and the first and 3rd row from B are removed the resulting linguistic submatrix of B is [very bad] is 1  1 trivial linguistic submatrixofB.

Thus we have 12 trivial 1  1 linguistic submatrices of B isgiveninthefollowing

{[good, [], [lazy], [very good], [bad], [best], [very bad], [average],[justgood],[bad],[best]and[good]}

Thereare12trivial1  1linguisticsubmatricesofB.

Suppose we remove the last 2 columns and last two rows of B. Then we get S3 to be linguistic submatrix of B given by S3 =[good, ];S2 isalinguisticrowmatrixoforder1  2.

On the other hand if we remove the first two rows and lasttwocolumnswegetalinguisticsubmatrixS4 ofBgivenby S4 =[justgood,bad]1  2 whichisagainalinguisticrowmatrixoforder1  2.

Likewiseweget thefollowinglinguisticsubmatricesofB oforder1  2giveninthefollowing.

{[good, ], [good, lazy], [good, very good], [, lazy], [, very good], [lazy, very good], [bad, best], [bad, very bad], [bad, average], [best, very bad], [best, average], [very bad, average], [just good, bad], [just good, best], [just good, good], [bad, best], [bad,good],[best,good]}.

Thusthereare18linguisticsubmatricesofBoforder1  2.

Now suppose we remove the last two rows and the first column,thenwegetalinguisticsubmatrixS5 ofBgiveby S5 =[,lazy,verygood]13 alinguisticrowsubmatrixoforder1  3.

We enumerate all linguistic submatrices of B of order 1  3inthefollowing.

{[good, , lazy], [good, lazy, very good], [good, , very good], [, lazy, very good], [bad, best, very bad], [bad, best, average], [bad, very bad, average], [best, very bad, average], [just good, bad, best], [just good, bad, good], [just good, best, good], [bad,best,good]}.

Weseethere12 1  3linguisticrowsubmatricesofB,.

Suppose in the matrix B we remove the first two row the resultinglinguisticsubmatrixS6 ofBisgivenby

S6 =[justgood,bad,best,good]14

TheorderS6 is1  4.

We enumerate all the linguistic submatrices of B of order 1  4inthefollowing.

{[good  lazyverygood],[bad,best,verybad,average], [justgood,bad,best,good]}.

Thereare3linguisticsubmatricesofBoforder1  3.

Now let S7 be a linguistic submatrix of B got by removingthelastrowandthelastthreecolumnsofB.

S7 = good bad   

isalinguisticsubmatrixofBoforder2  1.

We now enumerate all linguistic submatrices of B of order2  1isthefollowing

{ good bad    , good justgood    , bad justgood    , best     , bad     , best bad    ,

lazy verybad    , lazy best    , verybad best    ,

verygood average    , verygood good    , average good    }

There12linguisticcolumnsubmatricesoforder2  1.

Let S8 be a linguistic submatrix of B got by removing the first3colunofB.Now S8 = 31

    

verygood average good 

S8 isalinguisticcolumnsubmatrixofBoforder3  1.

We now enumerate all linguistic submatrices of B of order3  1inthefollowing.

{ good bad justgood

     , best bad

      ,

     ,

lazy verybad best

     }

verygood average good

therearefour(4)linguisticsubmatricesofBoforder3  1.

Let S9 be a linguistic submatrix of B got by removing the firstrowandlasttwocolumnsofB.

  

badbest justgoodbad 

Weget S9 = 22

S9 isalinguisticsubmatrixofBoforder2  2.

We now enumerate all the linguistic submatrices of B of order2  2inthefollowing.

{ good badbest     , good justgoodbad     ,

badbest justgoodbad    ,

goodlazy badverybad    , goodlazy justgoodbest    , badverybad justgoodbest    ,

goodverygood badaverage    , goodverygood justgoodgood    ,

badaverage justgoodgood    , lazy bestverybad     , lazy badbest     ,

bestverybad badbest    , verygood verybadaverage     , verygood badgood     ,

bestaverage badgood    , lazyverygood verybadaverage    , lazyverygood bestgood    ,

verybadaverage bestgood    }.

Weseethereare18linguisticsubmatricesoforder2  2.

Let S10 be a linguistic submatrix of B of order 3  2 by deletingthelasttworowsofB.

     

good badbest justgoodbad 

S10 = 32

isalinguisticsubmatrixofBoforder3  2.

We now enumerate of linguistic submatrices of B of order3  2inthefollowing.

{ good badbest justgoodbad

      ,

     ,

goodlazy badverybad justgoodbest

goodverygood badaverage justgoodgood

     ,

verygood bestaverage badgood

      ,

      ,

lazy bestverybad badbest

     }

lazyverygood verybadaverage bestgood

Thereare6linguisticsubmatricesofBoforder3  2.

Now let S11 be a linguistic submatrix of B of order 2  3 gotbyremovingthelastrowandlastcolumnofB.

   

goodlazy badbestverybad 

S11 = 23

S11 is of order 2  3, we enumerate all linguistic submatricesofBoforder2  3inthefollowing.

{ goodlazy badbestverybad     , goodlazy justgoodbadbest     ,

badbestverybad justgoodbadbest    , lazyverygood bestverybadaverage     , lazyverygood badbestgood     , bestverybadaverage badbestgood    ,

goodlazyverygood badverybadaverage    ,

goodlazyverygood justgoodbestgood    ,

badverybadaverage justgoodbestgood    ,

goodverygood badbestaverage     ,

goodverygood badverybadaverage     , badbestaverage justgoodbestgood    }.

Thereare12linguisticsubmatricesofBoforder2  3.

Now let S12 be a linguistic submatrix of B got by removingthelastcolumn.

     

goodlazy badbestverybad justgoodbadbest 

S12 = 33

S12 isalinguisticsubmatrixofBoforder3  3.

Now we enumerate all linguistic submatrices of B of order3  3inthefollowing.

{ goodlazy badbestverybad justgoodbadbest

      ,

      ,

goodverygood badbestaverage justgoodbadgood

goodlazyverygood badverybadaverage justgoodbestgood

     , lazyverygood bestverybadaverage badbestgood

      }

Now we see there are 4 linguistic submatrices of B of order3  3. Thuswehave 12+18+12+3+12+4+18+6+12+4+1=92

such trivial and non trivial linguistic submatrices of B where orderofBis3  4.

We leave it as an exercise for the interested researcher to find a suitable program to find the number of linguistic submatricesofalinguisticmatrix.

Next we proceed onto describe the notion of upper triangular and low triangular linguistic matrices. This can be done only in case of square matrices and also the linguistic set (linguistic continuum) under consideration should contain the emptylinguisticterm .

Wewillfirstdefinethisconcept. Definition 2.4 Let S be a linguistic set of finite order containing the empty linguistic term  (or the linguistic continuumIC which contains thelinguistic term ).

A linguistic square matrix is said to be a upper triangular linguistic matrix if all the entries below the main diagonal are emptylinguistic term 

A low triangular linguistic matrix is one in which all the entriesabovethe main diagonal are emptylinguistic terms  That is U =

            

aaa aa a a a

11121n 222n 3n n1n nn

    where aij S (or IC) 1 i, j n is the upper triangularlinguistic matrix oforder n n.



L=

a aa aa aa aaa

    

         

11 2122 3132 n11n12 n1n2nn

  

with aij S (or IC); 1 i, j n is the lower triangular linguistic matrix oforder n n.

Nowwewillillustratethissituationbysomeexamples.

Example 2.8. Let S = {, fastest, fast, very fast, just fast, slow, veryslow,mediumspeed,justslow,slowest}. be the linguistic set associated with the linguistic variable, “speed of the car”.  the empty linguistic term / word occurs when the car is in the signal (i.e. waiting for the signal so its speedis ).

All other linguistic terms of  are self-explanatory. Consider L=

fast slowfastest justslowfastslow slowfastfastestslow fastslowfastslow veryfastfastslowslowestjustslowslow

              

Lisalinguisticlowertriangularmatrixoforder6  6.

Now we give the linguistic upper triangular, matrix of order4  4inthefollowing

U=

slowfastfastestslow veryslowslowfast veryfastfast justslow

        

NowwefindtransposeofU,

UT =

slowfastfastestslow veryslowslowfast veryfastfast justslow

         = 44

        

slow fastestveryslow fastestslowveryfast slowfastfastjustfast 

ClearlyUT,thetransposeofthelinguisticuppertriangular matrixisalinguisticlowertriangularmatrix.

NowwefindLT ofthelinguisticlowtriangularmatrixL.

Lt =

fast slowfastest justslowfastslow slowfastfastestslow fastslowfastslow veryfastfastslowslowestjustslowslow

               =

              

fastslowjustslowslowfastveryfast fastestfastfastfast slowfastestslowslow slowfastslowest slowjustslow slow

Clearly the transpose of a lower triangular linguistic matrixisauppertriangularlinguisticmatrix.

Now we will show when are two linguistic matrices M andNareequal.

FirstofalltodefineortwolinguisticmatricesMandNto equalweneedorderofMandNmustbethesamesays  t.

bealinguisticmatrixoforders  tand N=

     

nnn nnn nnn

11121t 21222t s1s2st

  

be another s  t linguistic matrix N = M if and only if mij = nij for1  i  sand1  j  t,thenwesayMandNareequal.

Now we proceed onto suggest some problems to the reader.

By solving these problem one can become more familiar with the concept of linguistic matrices and some of their basic properties.

SUGGESTEDPROBLEMS

1. Give a 4  5 linguistic matrix using the linguistic set S = {, good, bad, fair, very bad, best, just fair, just good}

i) How many linguistic matrices of order 4  5 canbegotusingS?

ii) How any linguistic matrices of order 5  4 can beobtainedusingS?

iii) How many linguistic matrices of order 1  3 canbegotusingS?

iv) Find the number linguistic matrices of order 4  4usingS.

                   

badverybadbest fairjustfairbad justgoodgoodbest verybadbadbad fairgood goodbad badfairfairbest justgoodfair 

2. GivenM= 84

withentriesfromS(Sgivenproblem1).

i) FindMt andproveM  Mt

ii) Find(Mt)t andprove(Mt)t =M.

iii) FindalllinguisticsubmatricesofM.

iv) How manylinguistic submatrices of order 3  3 existinM?

v) Find all linguistic submatrics of M of order 3  2.

vi) Find all linguistic submatrices of M of order 2  3.

vii) Compare the number of linguistic submatrices oforder2  3and3  2.

viii) Find all linguistic submatrices of order 4  4 in M.

ix) How many 6  6 upper triangular linguistic matrices using the linguistic set S given problem1?

3. Find 3 linguistic matrices which are symmetric using the linguistic set S given in problem 1 of orders 10  10, 12  12and9  9.

4. Let IC = [worst, best] be the linguistic continuum related with the linguistic variable performance of a teacher in theclassroom.

i) Let C =

ii) FindCt

worst bad verybad fair medium best justgood good verygood justgood best

                

bea11  1linguisticmatrix.

iii) FindalllinguisticsubmatricesofC.

iv) How many such linguistic submatrices of the linguisticmatrixCt exist?

5. Let IC = [slowest, fastest]  {} be the linguistic continuum associatedwiththelinguisticvariablespeedof thecarontheroad.

LetC=

slowslowestfast fastslow veryslowslow slowslowslowslowest fastfasterfast fastjustfast justfastjustslow fasterfastest

                     

bethelinguisticmatrixoforder9  4

i) FindalllinguisticsubmatricesofC. ii) FindthetransposeofC. iii) Find(Ct)t .

6. Obtainanyofthestrikingfeaturesoflinguisticmatrices.

7. Give six distinct linguistic continuum associated with six distinctlinguisticvariables.

8. Compare the real continuum with the linguistic continuums and show there are infinitely many linguistic continuums.

9. ForanygivenlinguisticmatrixMofordern  m(n  m)

i) FindthenumberoflinguisticsubmatricesofM.

ii) Write a programme to find the number of linguisticsubmatricesofM.

iii) What change takes place if M is a square linguisticmatrix(i.e,n=m)?

10. For the linguistic matrix C given in problem 4 find at least 4 distinct matrices which are not linguistic submatricesofC.

11. Is it possible to find all linguistic matrices of C in problem4whicharenotlinguisticsubmatricesofC.

12. What collection is larger the linguistic submatrices of C or those linguistic matrices built using C which are not linguisticsubmatricesofC?

13. Canthelinguisticrowmatrix

D = (good, bad, , best, fair, , worst, just good, very good) have linguistic matrices which are not linguistic submatricesofD?Justifyyourclaim.

14. Let F =

best bad worst good verygood justgood fair justfair

                   

bealinguisticcolumnmatrixoforder11  1.

Can F have linguistic matrices which are not linguistic submatrices?Justifyyourclaim!

15. Obtainanyotherspecial feature associated with linguistic submatricesofalinguisticmatrixingeneral.

16. Suppose N = (a1, …, an) is a linguistic row matrix of order1  n.

Prove or disprove N has only 2n – 1number of linguistic submatrices and every linguistic matrix of N is a linguisticsubmatrixofN.

17. Suppose M be a linguistic column matrix of order n  1. Prove Mhas only 2n – 1 number oflinguistic submatrices and no linguistic matrices of M and every linguistic matrixofMisalinguisticsubmatrixofM.

18. Let D = 11121p 21222p 31323p

    

aaa aaa aaa

where aij  S (S a linguistic set) 1  i  3 and 1 j  p be alinguisticmatrixoforder3  p.

i) Prove D has linguistic matrix which are not linguisticsubmatricesofD.

ii) FindthenumberoflinguisticsubmatricesofD.

19. Let E =

        

aaa aaa aaa aaa

111213 212223 313233 r1r2r3

bealinguisticmatrixoforderr  3.

i) Prove E has linguistic matrices which are not linguisticsubmatrices.

ii) Find the total number of linguistic submatrices ofE.

iii) AreS1 = 11 32 r2

     andS2 =

a a a

     

aa aa aa aa

1123 3243 6371 8293

linguisticsubmatricesofE?Justifyyourclaim.

LINGUISTIC MATRICES AND OPERATIONS ON THEM

In this chapterwe for the first time introduce all the basic properties of linguistic matrices in a methodical way. This is mainly carried out with a view to benefit non mathematicians like socio scientist, economists, engineers, medical experts, computer scientist and others can approach this book and construct linguistic mathematical models to analyze any mathematicalproblems/research.

The general concept of linguistic theory or the basic introductionoflinguistictheorycanbehadfrom[25].

However, a brief introduction of linguistic theory and their properties are recalled in chapter I of this book to make thisbookaself-containedone.

Throughout this book S will denote a linguistic set / a continuum and it is always assumed to be kept in an increasing order.

So without loss of generality when we say S is a linguistic set it is assumed always to be ordered and more specificallyitisorderedintheincreasingorderonly.

ThusifS={a1, a2,…,an}thenwehave a1<a2<…an whereeachai isalinguisticterm.

S can be a collection of linguistic terms associated with height or performance of workers in an industry or teachers’ performance in any educational institute or students’ performanceinstudiesetc.

Weshallfirstdefinethefollowingmatrices.

Example3.1.LetS={a1,a2,…,a8,a9}

= {very poor, just poor, poor, very medium, just medium, medium,justgood,good,verygood}

be the evaluation of an expert in the linguistic expression about theperformanceofaworker,workinginanindustry.

ClearlySisatotallyorderedlinguisticsetas

very poor < just poor < poor < very medium < just medium<medium<justgood<good<verygood.

Weseea=(poor,good,verygood)

is a 1  3 linguistic row matrix with entries from S or we say orderofais1  3.

Thatisa=(a3,a8,a9)=(poor,good,verygood).

By varying the linguistic terms from S in the 1  3 linguistic row matrix, we get 9  9  9 = 93 number of such 1  3 linguisticmatrices.

LetM=

a a a a a

       

9 1 2 3 6

bealinguistic5  1columnmatrixwherea9,a1,a2,a3,a6  Sor orderofMis5  1.

LetN={

       

a a a a a

i j k t s

/ai,aj,ak,at as  S;1  i,j,k,t,s  9}

be the collection of all 5  1 linguistic column matrices (of order5  1).

Clearly|N|=orderofN=9  9  9  9  9=95 .

LetC=

     

bbbb bbbb bbbb bbbb

1234 5678 9101112 13141516

bea4  4linguisticsquarematrixwithbi  S;1  i  16.

IfT={

bbbb bbbb bbbb bbbb

     

1234 5678 9101112 13141516

/bi  S;1  i  16}

be the collection of all 4  4 linguistic square matrices then o(T)=916

Letd=

     

bbbbbb bbbbbb bbbbbb bbbbbb

123456 789101112 131415161718 192021222324

bea4  6linguisticmatrixwithbi  S;1  i  24.

Suppose W={

bbbbbb bbbbbb bbbbbb bbbbbb

     

123456 789101112 131415161718 192021222324

/bi  S;1  i  24}

be the collection of all 4  6 linguistic matrices with entries fromS.

Clearly o(W) = 924, we have 924 such 4  6 linguistic matriceswithentriesfromS.

In viewofthese we present nowthe abstract definitionof thelinguisticm  nmatriceswithentriesfrom

S = {a1, a2, …, as}, where 2  s < , a linguistic totally ordered set.

Definition 3.1 Let S = {a1, a2, …, an} be a linguistic set associated with some problem / concept which is a totally orderedsetintheincreasingorder

A=

aa aa aa

      

1m m p1pm

11 212

be rectangular array of linguistic terms with entries from S and aij S;1 i pand1 j m.

We define A as the p  m linguistic matrix having p numberof rowsandmnumberof columns.Wecallp masthe orderofthelinguisticmatrixA.

Ifp=1thenwedefineasa1 mlinguisticrowmatrix.If m=1thendefineAasap 1linguisticcolumnmatrix.

If p = m then we define A as a p  p linguistic square matrix. If p  m we define A as a p  m linguistic rectangular matrix. Wewillillustratethissituationbysomeexamples.

Example3.2. Let S = {lowest, very low, just low, low, moderate, very moderate, justhigh,high,highest}

be the linguistic terms associated with a industry during a span of5years. A=(lowest,low,moderate,high,highest) isa1  5linguistic1  5rowmatrixwithentriesfromS.

Let B =

            

verylow low low high high highest moderate low

bea8  1linguisticcolumnmatrixwithentriesfromS.

LetC=

highlowmoderatejustlowlowest highlowestmoderatejustlowhighest highlowverylowjusthighmoderate highhighmoderateverylowverylow lowhighlowestverylowverylow

       

bea linguistic5  5squarematrixwithentriesfromS.

Let D=

lowestveryghighhighest lowhighverylow highlowestjusthigh justhighlowlow verylowlowhigh justhighhighlowest losesthighlow lowlowhigh moderatejustlowlow highjusthighhighest

               

bea10  3rectangularlinguisticmatrixwithentriesfromS.

Wehaveseenallfourtypesoflinguisticmatrices.

Example 3.3. Let S = {worst, very very bad, very bad,bad, just fair,fair,veryfair,good,best}

be a linguistic set which pertain to the performance of a student inaschool. Let A=(worst,bad,fair,good,best) bea1  5linguisticmatrix.

LetB=

            

bad bad worst good good fair fair bad

bea8  1columnlinguisticmatrix. Forthe6  6linguarematrixwehavethefollowing. C=

      



     bethelinguisticsquarematrixoforder6  6.

Thefollowinggivestherectangularlinguisticmatrix

D=

     

fairfairverybadgood goodbadveryfairbad badbestjustfairbest bestgoodverybadfair

worstverybadgoodfairbest veryverybadworstbadbadgood veryfairbadfairgoodbest bestgoodfairworstbad

     

bea4  9rectangularlinguisticmatrixwithentriesfromS.

Now having seen examples of linguistic matrix we now proceedontodescribeafewofitsproperties.

Now we first describe the notion of transpose of any linguistic matrix. It isimportanttorecordthatthetranspose of a linguistic matrix is also the same as that of a classical matrix. However to make this book a complete one on linguistic matriceswefistexhibitthembysomeexamples.

Example 3.4. Let A = (worst, bad, fair, good, best) be the 1  5 linguistic row matrix described in example 3.3. We denote the transposeofalinguisticmatrixAasAt .

At =

worst bad fair good best

       

isthelinguistictransposeofthe1  5linguisticrowmatrix.

ClearlyAt isa5  1linguisticcolumnmatrix.

Consider the 8  1 linguistic column matrix B given in example3.3.WefindthetransposeofB;

Bt = (bad, bad, worst, good, good, fair, fair, bad) is the 1  8 linguisticrowmatrix.

So we see as in the case of classical matrices the transpose of a linguistic row matrix is a linguistic column matrix and the transpose of a column linguistic matrix is a linguisticrowmatrix.

Thus(At)t =Aand(Bt)t =B.

Fornow (At)t = (worst, bad, fair, good, best) = A which is the linguistic 1  5rowmatrix.

Now(Bt)t =

            

bad bad worst good good fair fair bad

=Bislinguisticcolumn8  1matrix.

Now we find the transpose of the 6  6 square linguistic matrixgiveninexample3.3.

Ct = fairbadgoodgoodbadfair verybadveryfairjustfairverybadverybadgood goodbadgoodbadgoodverybad fairjustfairbadbadgoodverybad bestbestbadgoodbestfair very verybadjustfairveryfairgoodbest verybad

isagaina6  6squarelinguisticmatrix,butCt isdifferentfrom C.

Now we find the transpose of D, the 4  9 linguistic matrixfromexample3.3.

   

          

Dt =

             

fairgoodbadbest fairbadveryfairbad verybadveryfairjustfairverybad goodbadbestfair worstveryverybadveryfairbest verybadworstbadgood goodbadfairfair fairbadgoodworst bestgoodbestbad

isthe9  4linguisticmatrix. ClearlyD  Dt .

It is interesting to note that one can have only in case of linguisticsquarematricesunderspecialconditions

wemayhavethematrixanditstransposeareidentical.

We will first illustrate this situation by an example and thengiveadefinition.

Example 3.5.Letsbeasanexample3.3.Consideralinguistic5  5squarematrixAgivenby

A=

bestgoodfairbadworst goodfairbadworstgood fairbadveryfairgoodbest badworstgoodgoodbad worstgoodbestbadbest

       

nowthefindtheAt ofA.

At =

bestgoodfairbadworst goodfairbadworstgood fairbadveryfairgoodbest badworstgoodgoodbad worstgoodbestbadbest

       

. WeseeAt =A;thisisthecasewhenbothareequal.

Wemakethefollowingobservations.

Only in case of square linguistic matrices (as in classical case)wecangetthemaindiagonalconcept.

In case of A the main diagonal elements are denoted by (best,fair,veryfair,good,best).

Forthusweneedtohavethediagonallinguisticmatrix.

Now in our linguistic sets we do not have the concept of zero. So we in order to define a non-mathematical term we denote in any linguistic matrix A by ‘’ the linguistic term is empty. Thencomes the problem will Scontain . Tothis endto adjoinwithStheemptysymbolS.

So S = {, S} and  will be leading least element in any increasingorderset(thatisanorderedset).

WecandefineonlyincaseoftakingtermsfromS forma linguisticdiagonalmatrixwhichwillalwaysbeasquarematrix.

(thisS istakenfromExample3.3).

WecallAthediagonal4  4linguisticmatrix.

We see our linguistic matrices have  as an element; it is not mandatory that  must be present in every linguistic matrix. Only in case of diagonal linguistic matrices we have every element in it is  the diagonal terms can be empty or may not beempty.

Wehavetheemptyrowlinguisticmatrix ()=(, , , , , , , ) whichisa1  8rowemptylinguisticmatrix.

Let()=

                      

bea9  1linguisticemptycolumnmatrix. ()=

               

isa6  6linguisticemptymatrix. ()=

          isa4  9emptylinguisticmatrix. Wewillbeusingtheseconceptsinthenextsectionofthis chapter.

Thus using S = S  {} where S is taken from the example3.3.Wegivesomelinguisticmatriceswithentriesfrom S .

LetA=(good, ,bad,justfair,fair, ,best)

isa1 7rowlinguistic matrixwithentriesfromS

Consider the 9  1 linguistic column matrix with entries fromS

B=

bad good fair best best bad

                

isa9  1linguisticcolumnmatrixwithentriesfromS .

LetC=

         

goodbadfairveryfair badbestfairbadbad veryfairbadbestbest goodfairfairbestbad bestfairbadveryfairbest

is5  5linguisticsquarematrixwithentriesfromS

LikewiseD=

badbestgood goodbadfair veryfairbadbest badfair justfairbad badbestbad bestbad

                 

isa8  4linguisticrectangularmatrixwithentriesfromS

We now proceed onto define the notion of symmetric linguisticmatrix.

Definition 3.2. Let S= {S } be the linguistic set with the emptylinguisticterm.

Let A be any n  n linguistic square matrix with entries from S.Wesay Ais asymmetric linguisticsquare matrix if and onlyifA=At .

Thatisifandonlyifaij =aji fori j,1 i,j n(aij S)

whereA=(aij)=

     

aaa aaa aaa

n n nnnn

 

11121 21222 12

Clearlythediagonallinguisticmatrixissymmetric. Wehavealreadygivenexamplesofthem. Wewillnowprovidemoreexamplesofthem.

Example 3.6 Let S = {S  } where S is taken from example3.3.

A=

            

goodbadfairbest bestbad badbadveryfair fairbadjustfairbad bestveryfairbad

be a 5  5 linguistic square matrix. Clearly A = At so A is a symmetriclinguistic5  5 matrix.

Weseeaij =aji foralli  j;1  i,j  5.

ConsiderB= bad best justfair

       

Bisalinguisticdiagonalmatrixoforder3  3.

TriviallyBisasymmetriclinguisticmatrixas

aij =aji = .i  j;1  i,j  3.

It is pertinent to keep on record that we will not be in a position to define skewsymmetric matrix as we do not have the notionofnegativetermsinlinguisticsets.

Next we proceed onto define some operations on these linguisticmatrices.

 will betheleastelementweknowS is atotallyordered set (order is defined as an increasing order) with  as the least element.

We first define max operation on these linguistic matrices.

To have a better understanding of this concept we first define max operation on the set S non abstractly by taking an example.

Example 3.7. Let S = {, lowest, lower, very low, just low, low, moderate, very moderate, very high, high, higher, highest} be the linguistic set; say associated with the linguistic variable weatherreport.

S isorderedintheincreasingorderasfollows:  < lowest < lower < very low < just low < low < moderate < very moderate < high < very high < higher < highest.

IfwedefinemaxoperatoronS wesee max{low,veryhigh}=veryhighaslow<veryhigh. Clearlymax{low,veryhigh} =max{veryhigh,low}=veryhigh.

SowecandefinemaxoperatoronS as max{x,y}=yifandonlyifx  y(x  y).

Using this definition of max operator we can define max{A,B}whereAandBaretwo1  6linguisticrowmatrices givenas

A=(,fair,bad,good, ,best)and

B = (fair, , , bad, good, bad) the elements of them are takenfromexample3.7,whichare1  6rowlinguisticmatrices

max(A,B)

=max{(,fair,bad,good, ,best),(fair, , ,bad,goodbad)}

=(max{,fair},max{fair, },max{bad, },max{good,bad}, max{,good},max{best,bad}) =(fair,far,bad,good,good,best)

isagainalinguistic1  6rowmatrixwithentriesfromS

We show how max operation is performed on 8  1 columnlinguisticmatricesgivenbyMandNwhere

M=

               

bad good justfair best veryfair

andN=

              

best bad good best fair justfair

respectively.

max{M,W}=max{

=

bad good justfair best veryfair

               

,

              

best bad good best fair justfair

}

                

max{,best} max{bad,bad} max{good,} max{justfair,good} max{best,best} max{,} max{veryfair,fair} max{,justfair}

=

isagaina8  1linguisticcolumnmatrices.

best bad good good best veryfair justfair

             

Let A and B two 4  4 square linguistic matrices with entriesfromS =S {0}ofproblem3.3giveninthefollowing. A=

goodbad farigood veryfairbest badgood

         

and

B=

goodbadbestfair fairjustfairgoodbad badgood

       

. Wefind max{A,B}=max{

goodbad farigood veryfairbest badgood

         

, goodbadbestfair fairjustfairgoodbad badgood

       

} =

          max{bad,best}max{,fair} max{,}max{good,} max{,good}max{best,bad} max{good,}max{,good}

max{,good}max{good,bad} max{fair,}max{,} max{,fair}max{veryfair,justfair} max{bad,}max{,bad}

         

=

goodgoodbestfair fairgood fairveryfairgoodbest badbadgoodgood

      

isagaina4  4linguisticsquarematrix.

Finally we give the max operation on two rectangular 4  6 linguistic matrices X and Y using entries from S of problem3.3.

LetX=

fairgoodbadjustfair fairbadbestbad badgoodbadfair bestgoodbad

          and Y=

bestfairgood badbadbest justfairbadbad goodfairbadbad

         

be the 4  6 linguistic rectangular matrices with entries from S ={S  }whereSistakenfrom3.3.

NowweperformthemaxoperationonXandY;

max{X, Y} = max{

fairgoodbadjustfair fairbadbestbad badgoodbadfair bestgoodbad

          ,

         

} =

   



max{,best}max{fair,}max{good,fair} max{fair,}max{bad,bad}max{,} max{bad,justfair}max{good,}max{bad,bad} max{best,}max{,good}max{,fair} 

        =

bestfairgoodgoodjustfair fairbadbestbest justfairgoodbadfair bestgoodfairgoodbad

         

; clearly max{X, Y} is again a 4  6linguistic rectangular matrix withelementsfromS =S{},SgiveninExample3.3.

Now we make the formal definition of the max operator onlinguisticmatricesofsameorder.

Definition 3.3. Let Sbe alinguisticsetwith emptyset.Suppose

A and B are two linguistic matrices of same order p q, where A = (aij) and B = (bij); aij, bij  Swith 1 i p and 1 j q. We define maximum of A and B as max{A,B} = max{(aij), (bij)} foralli,jwith1 i pand1 j q.

We have already given several examples of this. In the first place we could define the operation on these linguistic matrices as only when they are of same order; otherwise it would not be possible to define max operation on them. However it is pertinent to keep on record we can define max operation on two linguistic matrices A and B of different orders provided we have the number of columns in A and the number ofrowsinBarethesame.

We make a little deviation for this, which we will first illustratebysomeexamples.

Example 3.8. Let S = S{} where S is taken as in example 3.3. We define instead of max define max{max} operatoron

AandBwhereA=(,badfair,good,justfair}and B=

best bad fair good

        

be two linguistic matrices with entries from S under the new composite operation max{max}; we see order of A is 1  5 and

order of B is 5  1 so we have compatibility of the max{max} ofAandB.

max{max{A,B}}

=max{max(,bad,fair,good,justfair}

best bad fair good

        

}

= max{{max{, best}, max{bad, bad}, max{fair, }, max{good,fair},max{justfair,good}}}

=max{best,bad,fair,good,good}

={best}isa1  1triviallinguisticmatrix.

Now is the max{max{B, A}} defined}, it is defined as numberofcolumnsinBis1andthenumberofrowsinAis1.

Nowwefind

max{

best bad fair good

        

,(,bad,fair,good,justfair)}

=

  



 =

bestbestbestbestbest badbadfairgoodjustfair badfairgoodgood fairfairfairgoodgood goodgoodgoodgoodgood

        

.

Itistoberecordedthisisonewayusingonlymaxoperator.

We will give some more examples of using max operator onlinguisticmatrices.

LetA=

         

fairgood badfair bestfair worstfairgood

and

B=

        

fairgood bad best fair

beanytwolinguisticmatrices.

Clearly numberofcolumnsof Ais4 andthe numberof rows of Bisfoursomaxmax{A,B}iswelldefined.

Wefindmax{max(A,B)}

=max{max{

         

fairgood badfair bestfair worstfairgood

,

fairgood bad best fair

        

}

 

  ,}, max{good,fair} 

    ood,}}

max{max{fair,good,},max{,},max{good,best,max{,}} max{max{good}max{bad,},max{,best},max{fair,} max{max{best,good},max{,},max{{fair,best}max{,}, max{max{worst,good},max{fair,},max{,best} max{g

        

=

max{fair,bad,good,fair}max{good,,best,} max{fair,bad,,fair}max{good,bad,best,fair} max{best,fair,bad,fair}max{best,,best,} max{fair,fair,,good}max{good,fair,best,good}

                 =

goodbest fairbest bestbest goodbest

     

isadifferentlinguisticmatrixwhichisneitherAnorB. We give yet another example of the max{max{P, Q}} wherePandQ,2  3and3  2linguisticmatricesrespectively.

P= fairgood badbest      andQ= good fairbest bad

       with entries from the linguistic set S = S  {} where S is the linguisticsetdefinedinexample3.3.

Wefinemaxmax{Q,P};max{max{Q,P}}

=max{max good fairbest bad

       , fairgood badbest      )} = max{max{good,fair},max{,bad}} max{mad{fair,fair},max{best,bad}} max{max{,fair},max{bad,bad}

      

max{max{good,},max{,best}, max{max{good,good},max{,} max{max{fair,},max{best,best}, max{max{fair,good}max{best,} max{max{,},max{bad,best} max{max{,good}max{bad,}}

                = max{good,bad},max{good,best},max{good,} max{fair,best},max{fair,best},max{good,best} max{fair,bad},max{,best}max[good,bad}

       = goodbestgood bestbestbest fairbestgood

     …I

Clearly max{max{Q,P}} is a 3  3 linguistic matrix given by I. Considermax{max{P,Q}} =max{max fairgood badbest      ,

good fairbest bad

       }} = max{max{fair,good},max{,fair},max{good,} max{max{bad,good}max{best,fair},max{,}}     

max{max{fair,},max{,best},max{good,bad}} max{max{bad,},max{best,best},max{,bad}}     

= max{good,fair,good}max{fair,best,good} max{good,bestm,}max{bad,best,bad}    

= goodbest bestbest    …II

max{max{P,Q}}givesa2  2linguisticmatrix.

Wemakethefollowingobservations:

i) Both max{max{P,Q}} and max{max{Q, P}} are defined.

ii) max{max{P,Q}}  {max{max{Q,P}}.

It important to note if P and Q are of different order max{max{P,Q}}  max{max{Q,P}}.

iii) In this case both max{max{P, Q}} and max{max {Q,P}} are square linguistic matrices ofdifferentorder.

This occurs only when both max{max{P,Q}} and max{max{Q,P}}aredefined.

If one of them is defined then it will only be a linguistic squarematrix.

Let MandNbetwolinguisticmatricesoforder5  2and 2  3respectivelywhere

M=

good fairbad bestbad good

           and N= goodbad bestbad     

where M and N take their entries from S = S  {} where S is definedasinexample3.3.

Only max{max{M, N}} is defined, however max{max{N,M}}isundefined.

Wefindmax{max{M,N}};max{max{M,N}

max{max{good,good},max{,best}} max{max{fair,good},max{bad,best}} max{max{,good},max{,best}} max{max{best,good},max{bad,best}} max{max{,good},max{good,best}}

max{max{good,},max{,bad}} max{max{good,bad},max{,}} max{max{fair,},max{bad,bad}} max{max{fair,bad},max{bad,}} max{max{,},max{,bad}} max{max{,bad},max{,} max{max{best,},max{bad,bad}} max{max{bad,

      

best}max{bad,} maxmax{,},max{good,bad}} max{max{,bad},max{good,}}

                   =

  

  

max{good,best},max{good,bad}max[good,} max{good,best},max{fair,bad},max{fair,bad} max{good,best},max{,bad},max{bad,} max{good,best},max{best,bad}max{bad,bad} max{good,best},max{,good},max{bad,good}

       

=

bestgoodgood bestfairfair bestbadbad bestbestbad bestgoodgood

       

is a 5  3 linguistic matrix which is neither the order of M nor theorderofN.

Next we proceed onto define max{max {A, At}} and max{max{At,A}}whereAisalinguisticmatrix.

Wewillillustratethisbyanexample.

Example 3.9. Let S = S  {} where S is taken as in example 3.3.

LetA=

good bad goodfair bestgood fair best badbest

              

bea7  2linguisticmatrixwithentriesfromS ,

NowwegetthetransposeofA; At = goodgoodbestfairbad badfairgoodbestbest      ;

clearlyAt isa2  7linguisticmatrix.

Wefindmax{max{A,At}}=max{max{

              

good bad goodfair bestgood fair best badbest

, goodgoodbestfairbad badfairgoodbestbest      }}

=

    

goodgoodgoodbestgoodbestbest goodbadgoodbestfairbestbest goodgoodgoodbestgoodbestbest bestbestbestbestbestbestbest goodfairgoodbestfairbestbest bestbestbestbestbestbestbest bestbestbestbestbestbestbest

           isa7  7linguisticsquarematrixwhichissymmetric. Nowconsidertheoperationmax(max{At,A})ofAt with A; max(max{At,A})= max{max{ goodgoodbestfairbad badfairgoodbestbest      , good bad goodfair bestgood fair best badbest

              

         

}= max{max{good,good},max{,},max{good,good}, max{best,best},max{fair,fair},max{,},max{bad,bad}} max{max{,good},max{bad,},max{fair,good}, max{good,best},max{,fair},max{best,},max{best,bad}}

max{max{good,},max{,bad},max{good,fair} max{best,good},max{fair,},max{,best}max{bad,best}} max{max{,},max{bad,bad},max{fair,fair}, max{good,good},max{,},max{best,best}, max{best,best}}

            = max{good,,good,best,fair,,bad} max{good,bad,good,best,fair,best,best}    

max{good,bad,good,best,fair,best,best} max{,bad,fair,good,,best,best}     = bestbest bestbest   

isa2  2linguisticmatrix.

Wemakethefollowingobservation.

The order of the linguistic rectangular matrix A is 7  2 and its transpose At is of order 2  7. The max{max{A, At}} is a 7  7 square linguistic symmetric matrix whereas max{max{At, A}} is a 2  2 square linguistic matrix which is alsosymmetric.

Next for the same set S = S  {}; S given in example 3.3.

We find the max{max{A, At}} and max)max{At , A}} where A is a 5  1 column linguistic matrix given in the following:

A=

        

good bad fair best

isa5  1linguisticcolumnmatrixanditstranspose

At =(good,bad, ,fair,best)isa1  5rowlinguisticmatrix.

Wefind

max{max{A,At}}

=max{max{

good bad fair best

        

,(good,bad, ,fair,best)} =

goodgoodgoodgoodbest goodbadbadfairbest goodbadfairbest goodfairfairfairbest bestbestbestbestbest

        

isalinguistic5  5squarematrixwhichissymmetric. Nowwefindmax{max{At,A}}

=max{max{(good,bad, ,fair,best),

good bad fair best

        

=max{good,bad, ,fair,best} =(best)

isa1  1linguistictrivialmatrix.

Inviewofthisexamplewehavethefollowing.

}}

i) Since the order of A is 5  1 we have order of At to be 1  5 so max {max{A, At}} is a 5  5 a squarelinguisticsymmetricmatrix.

ii) Aisoforder5  1andthatofAt is1  5.

iii) max{max{At, A}} is a 1  1 trivial linguistic matrix which is nothing but the maximum of all thetermsusinginthecolumnlinguisticmatrixA.

Now consider A to be a 6  6 square linguistic matrix withentries fromS =S {}whereSisthelinguisticsettaken fromexample3.3.

A=

               

badfairbest badbestbad goodfairbad badgoodfairgood goodbadfair bestfairbad

ThetransposeofAis

At =

               

badbadbest badgoodgood fairbestgoodfair badfairbad bestfairfairbad badgood

At isalsoa6  6linguisticmatrix.

Our jobtofindmax{max{A,At}} andmax{max{At,A}} andcomparethem.

Considermax{max{

               

badfairbest badbestbad goodfairbad badgoodfairgood goodbadfair bestfairbad

,

badbadbest badgoodgood fairbestgoodfair badfairbad bestfairfairbad badgood

               

}} =

bestbestbestbestbestbest bestbestbestbestbestbest bestbestgoodgoodgoodbest bestbestgoodgoodgoodbest bestbestgoodgoodgoodbest bestbestbestbestbestbest

         

We seemax{max{A,At}} is a 6  6linguistic symmetric squarematrix. Nowwefindoutmax{max{At,A}}andcompareitwith max{max{A,At}}. max{max{At,A}} =max{max{

badbadbest badgoodgood fairbestgoodfair badfairbad bestfairfairbad badgood

               

,

badfairbest badbestbad goodfairbad badgoodfairgood goodbadfair bestfairbad

               

}}=

bestbestbestbestbestbest bestgoodbestgoodbestgood bestbestbestbestbestbest bestgoodbestfairbestbest bestbestbestbestbestbest bestgoodbestgoodbestgood

         

.

Clearly max{max{At, A}}  max{max{A, At}}.

However both max{max{A, At}} and max{max{At, A}} are 6  6 linguistic symmetric square matrices with entries from S =S{}whereSisasgiveninexample3.3.

Inviewofthisweprovethefollowingtheorem.

Theorem 3.1. Let Sbe a linguistic set together with the empty linguisticterm 

LetA=(aij)beam nlinguisticmatrixwithentriesfrom SthatisaijS;1 i mand1 j n.

max {max{A, At}} and max{max{At, A}} are square linguisticmatricesoforderm mandn nrespectively.

Incasem=nweseemax{maxAt,A}}and

max{max A, At}} are n  n square symmetric linguistic matriceswhichareingeneraldistinct.

If m = 1 (or n = 1) we get max{max{A, At}} be a trivial linguisticmatrix.

If n = 1 then max{max, {At, A}} is a trivial linguistic matrix.

Proof. If A is a m  m linguistic matrix At is a n  m matrix then it follows from the simple fact when we take the max max operatorthenwegetm  mandn  nsquarematricesonly.

Further as the column and rows are exchanged the max operator is performed and as the max operator is commutative we get max{aij, aji} in each case leading to a symmetry of the linguisticsquarematrix.

Finallyweseeincasem  n, max{max{At,A}}  max{max,{A,At}} withveryordertobedifferent.

Howeverincaseofm=nthatisAasquarethat max{max{A,At}}andmax{max{At,A}} are square linguistic symmetric matrices of same order however ingeneraltheyaredistinct.

Now we can say if A itself is symmetric linguistic matrix then we will have max{max{At, A}} = max{max{A, At}} as At =A.

Further if a researcher is interested in constructing such symmetric linguistic matrices one can easily do so by considering a linguistic matrix A of desired order and finding max{max{A,At}}ormax{max{At,A}}.

In the following section we study min{min{A, Bt}}, min{A, B} and min{min{B, A}} for any linguistic matrices A andB.

In the following we assume S is a linguistic set with the linguisticemptyword  includedinit.

That if S is a totally ordered set order and  the empty linguistictermistheleastlinguisticterm.

We by order of the linguistic matrix A mean the number of rows and number of columns A has and is denoted by m  n iftherearemrowsandncolumns.

SoifS isalinguisticsetorderedintheincreasingordersay

 <a1 < a2 …<an whereai  S;1  i  n,then

=ai ifai <aj

min(ai,aj)

=aj ifaj <ai

=ai ifai =aj

forall1 i,j  nandmin{,ai}=  forallai  S1  i  n.

With this in mind we first provide some examples of min operationonlinguisticmatricesofsameorder.

Example3.10.Let

A=

goodfair badbad bestbest fairfair good

             and B=

fairfairbad goodverybad fairgood badgoodbad goodbad

          

be two linguistic 5  3 matrices with entries by S = S {} where S is a linguistic set provided in example 3.3 of this chapter. Wefindmin{A,B};min{A,B} =min{

goodfair badbad bestbest fairfair good

            

,

          

fairfairbad goodverybad fairgood badgoodbad goodbad

min{,good}min{bad,verybad}min{bad,} min{best,fair}min{,}min{best,good} min{fair,bad}min{fair,good}min{,bad} min{,good}min{,}min{good,bad}

=

fairfair verybad fairgood badfair bad

            

.

is again a 5  3 linguistic matrix with entries from S = S {} whereSisdefinedasinexample3.3.

Consider M=

good bad fair best justfair verybad

              

andN=

best fair bad bad bad fair bad

             

beanytwo8  1linguisticmatriceswithentriesfrom S =S  {}whereSisasinexample3.3.

Wenowfind

min{M,N}=min{

              

good bad fair best justfair verybad

,

             

best fair bad bad bad fair bad

}

=

min{good,best} min{fair,bad} min{,} min{fair,bad} min{best,bad} min{bad,} min{fair,justfair} min{bad,verybad}

              

=

            

good bad bad justfair verybad

isagaina8  1linguisticmatrixwithentriesfromS .

Thefollowingobservationsismandatory;

min{A, B+ = min{B, A} that is the min operation is commutativewhichoperationonlinguisticmatrices.

LetP=

goodbadfairbest fairgood bestbadverybad justfairgood badgoodbadbad

             andQ=

fairbadverybad badbadbad goodbestbadfair fairgoodbad goodgoodbestverybad

            

be any two 5  5 square linguistic matrices with entries from S =S  {}Sgiveninexample3.3.

Wefindmin{P,Q};

            

goodbadfairbest fairgood bestbadverybad justfairgood badgoodbadbad

            

} =[

   



min{good,} min{,bad} min{,best} min{,fair} min{bad,good} 

 

    

min{,}min{fair,bad} min{good,bad}min{,} min{,best}min{verybad,bad} min(justfair,good}min{,} min{,}min{bad,best}

min{best,verybad} min{,bad} min{,fair} min{good,bad} min{bad,verybad}

          =

badbadverybad bad badverybad justfairbad badgoodbadverybad

            

is clearly a 5  5 linguistic square matrices with entries from S =S  {}.

LetA= goodbestjustfair goodverybad badfairworst

        and B= badgoodbad bestbad worstbadgood

       

be any two 3  4 linguistic rectangular matrices with entries from S = S {} where S is a linguistic set given in example 3.3. min{ goodbestjustfair goodverybad badfairworst

        , badgoodbad bestbad worstbadgood

        } = min{good,}min{,bad}min{best,good} min{,}min{good,best}min{,} min{bad,worst}miin{fair,bad}min{,}

       

min{justfair,bad} min{verybad,bad} min[worst,good}

    

= goodbad goodverybad worstbadworst

            

is a 3  4 linguistic rectangular matrix with entries from S =S {}whereSisgivenasinexample3.3.

Now having seen examples of min operator on linguistic matrices of same order we now proceed onto define the min operatorontwolinguisticmatricesofsameorder.

Definition 3.4. Let A and B be any two linguistic matrices of order m n with entriesfrom the set S= S {} where Sis a totallyorderedsetwithincreasingorder.

If A(aij) andB(bij) withaij,bij S; 1 i mand1 j  n; then min{A, B} = min{(aij), (bij)} = (cij) 1 i m; 1j n is againalinguisticmatrixoforderm n.

Here =aij =cij ifaij bij min{aij,bij} =bij =cij ifbij aij 1 i mand1 j n

Thenmin{A,B}=(cij).

Now having seen the min operator we proceed onto describe min min operator on linguistic matrices by some examples.

Example 3.11. Let S = S  {} be the linguistic set with linguisticemptyterm 

We take two linguistic matrices A and B of order 1  6 and6  3respectivelydescribedbelow.

A=(good, ,fair,bad,justfair,best) and B=

goodfair badfair badgood justfairbad fairgoodbad bestfair

              

belinguisticmatricesoforder1  6and6  3respectively.

TheentriesofAandBaretakenfromS =S {}where Sisthelinguisticsetgiveninexample3.3.

min{min{A,B}}

=min{min{(good, ,fair,bad,justfair,best), goodfair badfair badgood justfairbad fairgoodbad bestfair

              

}}

=

min{min{good,good}min{,bad},min{fair,} min{bad,justfair},min{justfair,fair}min{best,best}} min{min{goo,d},min{,fair},min{fair,bad} min{bad,bad},min{justfair,good}min{best,fair}} min{min{good,fair},min

  {,},min(fair,good},min{bad,} min{justfair,bad},min{best,}}

                     

= [(min good, , , bad, just fair, best), min{, , bad, bad, just fair,fair),min{fair, ,fair, ,bad, }]

= (, , ) is a 1  3 linguistic empty row matrix. Only min{min{A,B}} is defined however min{min{B, A}} is not defined.

Consider A= badbadfair goodfairjustfair bestbad

        and B=

fair fairbadgood verybadbadbad badfair

       

two linguistic matrices with entries from S = S  {} where S isgiveninexample3.3.

Wefindmin{min{A,B}}andmin{min{B,A}}.

min{min{A,B}}=min{min badbadfair goodfairjustfair bestbad

       

        , fair fairbadgood verybadbadbad badfair





}} =

  

min{min{bad,},min{,good},min{bad,bad} min{fair,}}

min{min}{good,},min{fair,good},min{,bad},

min{min{,},min{best,good},min{,bad},

               

=

       

min{,,bad,verybad}min{bad,,bad,fair} min{,fair,,bad}min{justfair,bad,,justfair} min[,fair,,bad}min{,bad,,bad}

min{,,bad,} min{,good,,} min{,good,,}

        =        

=()isa3  3emptylinguistictermmatrix. Triviallysymmetric. Nowwefindmin{min{B,A}}. Onsimilarlinesofcalculationwecanget min{min{B,A}}. min(min{B,A}}= bad verybad

         

whichisa4  4linguisticmatrix.

Now we show min{min{A, At}} and min{min{At, A}} aredistinctforanylinguisticmatrixAandbotharedefined.

In case A is a square linguistic matrix we see A both min{min{A, At}} and min{min{At, A}} are in general two distinctsymmetricsquarelinguisticmatrices.

min{min{A,At}}=min{min badbadfair goodfairjustfair bestbad

        

        , badgood fairbest bad fairjustfairbad

} =         =()

isa3  3linguistictriviallysymmetricmatrix.

Wenowfindmin{max{At,A}

=min{min

        

badgood fairbest bad fairjustfairbad

, badbadfair goodfairjustfair bestbad

        }}

= bad

       

isclearly4  4linguisticsquarematrixwhichissymmetric.

Furthermin{min{A,At}}  min{min{At,A}}.

NowconsiderthelinguisticmatrixAt =

whereAisa1  10linguisticmatrix.

min{min{At,A}}=min{min{

               

fair good bad justfair best good bad fair verybad good

,

               

fair good bad justfair best good bad fair verybad good

[fair, good, bad, just fair, best, good, bad, fair, very bad, good]}}

=(verybad);isa1  1triviallinguisticmatrix

min{min{A,At)=min{min{(fair,good,bad,justfair,best, good,bad,fair,verybad,good),

fair good bad justfair best good bad fair verybad good

               

}} = 1010

         

verybadverybadverybad verybadverybadverybad verybadverybadverybad verybadverybadverybad verybadverybadverybad 

  

isa10  10linguisticmatrix.

Itistriviallysymmetric.

Finally we find min{min{A, At}} and min{min{At, A}} where A is a 4  4 linguistic square matrix from S = S{} wherethelinguisticsetSistakenfromtheexample3.3.

A=

At=

bestgoodfairbad badbadbadbest fairbestbadbest goodbestgoodfair

     

and

     

bestbadfairgood goodbadbestbest fairbadbadgood badbestbestfair

be the given linguistic square matrix and its transpose respectively. min{min{A,At}} =min{min

bestgoodfairbad badbadbadbest fairbestbadbest goodbestgoodfair

     

,

bestbadfairgood goodbadbestbest fairbadbadgood badbestbestfair

     

}} =

     

badbadbadbad badbadbadbad badbadbadbad badbadbadbad

Clearly min{min{A, At}} is a 4  4 linguistic symmetric matrix.

Nowwefind

min{min{At,A}}=min{min

     

bestbadfairgood goodbadbestbest fairbadbadgood badbestbestfair

, bestgoodfairbad badbadbadbest fairbestbadbest goodbestgoodfair

     

}} =

badbadbadbad badbadbadbad badbadbadbad badbadbadbad

     

min{min{At,A}}isa4  4linguisticsymmetricmatrix. Infacttriviallysymmetric.

Further min{min{A, At}}  min{min{At, A}} in general.

However both yield a symmetric linguistic matrix of sameorderasthatofA.

Inviewofallthesewehavethefollowingresult.

Theorem 3.2 Let Sbe a linguistic set together with linguistic emptytermassociatedwithalinguisticvariable.

Suppose A = (aij) be a mn linguistic matrix with entries aij  S (1  i  m and 1  j  n). Then min(min(A, At)) and min(min(At, A)) are linguistic m  m and n  n square symmetricmatricesrespectively.

i) If m = 1 or n = 1 then it yield a trivial single elementlinguisticmatrix.

ii) If m  n then we get two symmetric linguistic matricesofdifferentorders.

iii) If m = n  1 we get two distinct symmetric linguistic matrices of same order; in fact as that ofA;thatism m.

Proof. Given A is a linguistic matrix of order m  m. If m = 1 or n = 1 then A is a 1  n row linguistic matrix or A is a m  1 columnlinguisticmatrixrespectively.

In both case min{min{A, At}} when m = 1 and min{min{A, At}} when n = 1 are trivial one element linguistic matrices.Thusproof(i)iscomplete.

Proofof (ii) If m  nwe see min{min{A, At}} is a m  n square linguistic symmetric matrix; proof as in case of max{max{A,At}}refertheorem3.1.

Similarly min{min{At, A}} is a n  n square linguistic symmetricmatrix(proofasincaseofmaxmaxoperator).

Proof of (iii) If m = n  1 then A is a square linguistic matrixandbothmin{min{A,At}}andmin{A,At}}areoforder m  m they are symmetric in nature as proved in theorem 3.1. Formax.max.

This method can be used to get many or desired order symmetriclinguisticmatrices.

Next we proceed onto describe the two operations max{min{A, B}} and min{max {A, B}} for compatible linguisticmatricesAandB.

In this section we proceed onto describe min max and maxminoperatorsonthelinguisticmatricesA andBwhichare such that the number of columns of A is equal to the number of rowsofBorthenumberofcolumnsofBequaltothenumberof rowofA.

We will first illustrate the operation min max by some examples.

Example 3.12. Let S = S {} where S is the linguistic set definedinexample3.3.

LetA=(verybad,fair,good,best,bad)and B=

bad good best bad fair

       

betwolinguisticmatricesoforder1  5and5  1respectively.

Clearly both min max{A, B} and min{max{B, A}} are defined.

Wefirstfindmin{max{A,B}}

=min{max{(verybad,fair,good,best,bad),

bad good best bad fair

       

}}

= min{max{very bad, bad}, max{fair, good}, max{good, best}, max{best,bad},max{bad,fair}}

=min{bad,good,best,best,fair}

=(bad)isasingletonlinguisticsetoforder1  1.

Nowwefindmax{min{A,B}} =max{min{(verybad,fair,good,best,bad)

bad good best bad fair

       

}}

= max{min {very bad, bad}, min{fair, good}, min{good, best}, min{best,bad},min{bad,fair}}

=max{verybad,fair,good,bad,bad}

={good}. Thisisa1  1matrix.

Clearlymax[min{A,B}}  min{max{A,B}}.

Now we proceed onto work with upper triangular linguisticmatricesandlowertriangularlinguisticmatricesinthe following.

Example 3.13. Let S = {very old, old, oldest, just old, young, very young, , just young, youngest} be a linguistic set associatedwiththelinguisticvariableageofaperson.

Let A=

              

old young veryyoungoldyoung oldyoungoldyoung oldoldyoung veryoldoldestyoungyoungoldold

bethelowertriangularlinguisticmatrix.

min{max{A,A}}=

min{max{=

               

old young very oldyoung young oldyoungoldyoung oldoldyoung veryoldoldestyoungyoungoldold

,

old young veryyoungoldyoung oldyoungoldyoung oldoldyoung veryoldoldestyoungyoungoldold

              

}} =

               

.

We see the resultant is not a lower triangular linguistic matrixundermin{max{A,B}}.

Now we find in this case of A min{max{A, A}} = {()} theemptylinguistictermmatrix.

Suppose we consider two linguistic matrices A and B where A is a upper triangular linguistic matrix of order 4  4 andthatofBisalinguisticorder4  4giveninthefollowing: A=

oldjustoldyoungold justyoungoldold youngold youngest

         and

B=

young oldyoung justoldyoungestjustold oldyoungyoungold

        

. Wefindmin{max{A,B}}

=min{max{

        

oldjustoldyoungold justyoungoldold youngold youngest

        

, young oldyoung }} justoldyoungestjustold oldyoungyoungold

oldyoungjustoldyoung young young young

         

.

This is not a upper triangular linguistic matrix or a low triangularlinguisticmatrix.

Nextwefindmax{min{A,B}

max{min{=

oldjustoldyoungold justyoungoldold youngold youngest

        

,

young oldyoung justoldyoungestyoung oldyoungyoungold

        

}} =

justoldyoungyoungold oldyoungyoungold oldyoungyoungold youngestyoungestyoungestyoungest

     

max{min{A,B}}  min{max{A,B}}.

It is important to note that none of them are upper triangularorlowertriangularlinguisticmatrices.

Wenowdeterminemin{max{B,A}} =min{max{

young oldyoung justoldyoungestjustold oldyoungyoungold

        

        

, oldjustoldyoungold justyoungoldold youngold youngest

}} =

youngest youngest youngest youngyoungyoungold

        

Nowwefindoutmax{min{B,A}}

=max{min{

        

young oldyoung justoldyoungestjustold oldyoungyoungold

,

oldjustoldyoungold justyoungoldold youngold youngest

        

}} =

youngyoungyoungyoung oldjustoldyoungold justoldjustoldyoungjustold oldjustoldyoungold

     

.

Bothmin{max{B,A}}  max{min{B,A}}.

Nowwefindmin{max{A,At}}

=min{max{

        

oldjustoldyoungold justyoungoldold youngold youngest

        

, old justoldjustyoung youngoldyoung oldoldoldyoungest

}}

=

youngjustyoungyoungyoung justold young old

        

Wefindmin{max{At,A}} =

youngest youngest youngest youngestyoungestyoungestyoungest

        

Weseemin{max{At,A}  min{max{A,At}.

Now we proceed onto propose some problems so that by solving them one becomes familiar with the new concept linguisticmatricesandtheirproperties.

SUGESSTEDPROBLEMS

1. Let S be a linguistic set associated with the linguistic variable speed of a car running in the road given by S={,fastest,fast,justfast,slow,veryslow,veryfast, justslow,slowest}.

Let B = (slow, ,fast, fastest, just slow, fast, fastest) be a 1  7rowlinguisticmatrix.

i) Show the number of linguistic row matrices of order 1  7 with entries from S is finite a number andfindthenumberofthem.

ii) FindBt

iii) Determinemin{B,Bt}andmax{B,Bt}.

iv) Findmax{Bt,B}andmin{Bt,B}.

v) Is the min-max and max-min operation defined on{B,Bt}and{Bt,B}?Justifyyourclaim.

vi) Whatistheidentitylinguisticrowmatrixoforder 1  7usingSunderminoperation.

vii) Is the identitymatrix ofthis linguistic rowmatrix of order 1  7 has the same identity linguistic matrixundermaxoperationalso?

2. Let S = [lowest, highest] be a linguistic continuum associated with the linguistic variable temperature of water(from0to100o whenstartstoboil).

          

low verylow lowest high veryhigh justlow highest 

i) LetA= 71

be a linguistic column matrix of order 7  1. Find min(max{A,At}}.

ii) LetB=

     

lowhighhighestlow highestjustlowlowlowest hgihhighhighlow highesthighlowlowest

Find a)min{max{A,A}}, b)max{min{A,A}}, c)max{min{A,At}}, d)min{max{At,A}}, e)max{min{At,A}}and f)min{max{At,A}}.

Areallthe6resultdistinct?Whichofthemareequal?

iv) LetM= 84

            

lowhighlowlowest justhighlowverylowhigh veryhighhighjusthighverylow lowesthighhighlow lowlowlowesthigh highhighhighlow lowverylowjustlowlow highveryhighjusthighlow 

bea linguisticmatrixof8  4order. a) Findmin{max{M,Mt}},

b) Findmax{min(M,Mt}},

c) Findminmax{{Mt,M}}and

d) minmax{{Mt,M}}. Proveallthefouraredistinct. Dotheycontributetosymmetriclinguistic matrices?

v) LetL=

     

veryhighlowhighhigest lowhighlowestlow highlowhighlow highesthighlowverylow

bealinguisticmatrix.

Find i) max{L,Lt}, ii) min{L,Lt}, iii) max{Lt,L}, iv) min{Lt,L}, v) max{min{L,Lt}}, vi) min{max{L,Lt}}, vii) max{min{Lt,L}}and viii) min{max{Lt,L}}.

v) Find the identity linguistic matrix of order 4  1 underminoperationandmaxoperation.

Are they different or the same? Justify your claim.

3. Let S be linguistic finite set given in problem 1, associatedwiththelinguisticvariablespeedofthecar.

                     

slowfast justfast slowestveryfast veryslow slowfast fastfastslow justfast justslow fastest 

LetM= 93

bealinguisticmatrixoforder9  3.

a) FindalllinguisticsubmatricesofM.

b) How many of these linguistic submatrices are squarematrices?

c) FindMt ofM.

d) Prove(Mt)t =M.

e) Findmin{max{M,Mt}}andmin{max{Mt,M}}.

f) Findmax{min{M,Mt}}andmax{min{Mt,M}}.

4. LetSbeasinproblem1,thelinguisticsetassociatedwith thelinguisticvariablespeedofthecar.

LetD=

        

fastslowjustfastveryfast fastslowestfastest slowfast justslow

betheuppertriangularlinguisticmatrixoforder4  4

a) FindDt,prove(Dt)t =D.

b) Suppose X = (fast  slow, slowest) is a row linguisticmatrixoforder1  4.

c) Findmax{X,D}.

d) Findmin{X,D}.

e) Calculatemax{X,Dt}andmin{X,Dt}.

f) Findmin{D,Xt}andmax{D,Xt}

5. LetA= 38

            

just fastfastveryfastslow slow very slowestfastslow slow just slowslowfastslow fast 

be a linguistic matrix of order 3  8 with entries from the linguistic set S given in problem 1 related with the linguistic variablespeedofthecar.

i) FindAt,prove(At)t =A.

ii) If X = (slow, fast, )13 a 1  3 linguistic row matrixfindmax{X,A}andmin{X,A}.

iii) Findmin{At,Xt}andmax{At,Xt}.

iv) Let Y = (fast, slow, just slow, very slow, very fast,justfast, ,slowest)18

bealinguisticrowmatrixoforder1  8

a) Findmax{Y,At}.

b) Findmin{Y,At}.

c) Calculatemin{A,Yt}andmax{A,Yt}.

6. Obtainanyotherspecial feature associatedwithlinguistic squarematrices.

7. UsingthelinguisticsetSgiveninproblem1.

a) Find linguistic diagonal matrices of order 4  4, 7  7,8  8and9  9.

b) Find linguistic symmetric matrices of order 3  3,6  6,10  10and8  8.

8. What are the advantages or drawbacks of using min-min and max-max operators instead of min-max or max-min operators?

9. Let B = {good, bad, , worst, very bad, fair, just good, verygood,veryfair,justbad,veryverybad,veryvery verygood,best}

be a linguistic set associated with the performance of a studentinaclassroom.

i) LetM= 94

                   

goodgoodbad fairverygoodbad verybadfair justgoodverybadjustbadbest worst goodbadbestfair fairbest goodbestbadbest verybadveryverybad 

linguisticmatrixoforder9  4withentriesfromB.

a) FindMt andprove(Mt)t =M.

b) FindalllinguisticsubmatricesofM.

c) FindalllinguisticcolumnsubmatricesofM.

d) A= fair best veryverybad

     ,

Is A a linguistic submatrix of M? Justify your claim.

e) IsAt alinguisticsubmatrixofM?Justifyyour claim.

f) Findmin{M,Mt}andmax{M,Mt}.

g) Findmin{min{M,Mt}}andmax{max{M,Mt}}. h) Findmin{max{M,Mt}}andmaxmin{M,Mt}}.

i) Findmin{max{Mt,M}}andmaxmin{Mt,M}}.

j) Calculatemin{min{Mt,M}}and maxmax{Mt,M}}.

k) Determinemin{Mt,M}andmax{M,Mt}.

l) Doesanyonetheoperationsmentionedinthe problem(a)to(k)yieldalinguisticsymmetric matrix?Justifyyourclaim.

10. LetT= 76

                 

goodbadbestworstfair bestgoodjustbad verygoodgoodbadbest verybadbadbadbestbad justbadjustgoodgoodbad badbadgoodbad badbadbadgood 

bealinguisticmatrixoforder7  6.Withentriesfromthe linguisticsetB.

i) Show the transpose of all linguistic row submatrices is a linguistic column submatrix of T.

ii) Prove in general the transpose of a linguistic column submatrix of T is not a linguistic row submatrixofT.

iii) ProveTisnotasymmetriclinguisticmatrix.

iv) J=

goodbad best verygood verybadbad jstbadjustgood badbad

             

be a linguistic submatrix of T. Is Jt a linguistic submatrixofT?Justifyyourclaim.

v) Prove or disprove the transpose of every linguistic square submatrix of T is a linguistic squaresubmatrixofT.

vi) Will the claim in (v) be true for all linguistic square submatrices of W and W any linguistic matrix?

vii) Characterize all those linguistic submatrices of T for which their transpose is a linguistic submatrix.

viii) C can these claims in (i) to (vii) be true for any linguisticmatrix?

11. Find any special feature associated with linguistic symmetricmatrices.

12. Let

V=

goodbadgoodfair fairbad veryfairfairverybad verygoodbestverybad bestveryfair bestbadverybadgood

               

bealinguisticsquarematrixoforder6  6.

i) Will min min {A, At} and min min {At, A} be only linguistic empty term square matrix of order6  6.

ii) What can we say about max{max{A, At}} and max{max{At,A}}?

iii) ClearlyP=

          

good fair veryfair verygood best

is a linguistic submatrix of V will Pt be a linguisticsubmatrixofV?

iv) Let W = (good, bad, , good, fair) be the linguisticsubmatrixofV.

Can W be a linguistic submatrix of V? Justify yourclaim

v) A= bad bad     

is a linguistic submatrix of V. It At a linguistic submatrixofV?Justifyyourclaim.

vi) B= verybad verybad      and C= veryfair good     

arelinguisticsubmatricesofV.ProvebothBt and Ct arelinguisticsubmatricesofV.

vii) What is specialty about A, B and C to be linguistic submatrices of V whose transpose are alsolinguisticsubmatricesofV?

viii) T = [bad, ] is a linguistic submatrix of V prove Tt isalsoalinguisticsubmatrixofV.

ix) L = verybad       

is linguistic submatrix of V proveLt alsoalinguisticsubmatrixofV.

x) Find all linguistic submatrices of V which are such that their transpose is also a linguistic submatrices.

xi) fair     is a linguistic submatrix prove [fair, ] is alsoalinguisticsubmatrixofV.

xii) Let X = (very good, , best) be the linguistic submatrixofV.IsXt alinguisticsubmatrixofV?

13. Does there exist a linguistic matrix Y such that for every linguisticsubmatrixofYitstransposeisagainalinguistic submatrix?

14. Let E = goodbadbest badgoodfair bestfairgood

     be a linguistic square matrixwhichissymmetric.

Prove every linguistic submatrix of E is such that its transposeisalinguisticsubmatrixofE.

15. Let F= goodbadbest badjustfairfair bestfairjustgood

    

bealinguisticsymmetricsquarematrix.

Is the transpose of every linguistic submatrix again a linguisticsubmatrixofF?

16. Prove all symmetric n  n linguistic matrices M are scuh that every linguistic submatrices of M is such that their transposearealsoalinguisticsubmatrices.

(Exploit the fact transpose of a symmetric linguistic matrixMisitself,thatisM=Mt).

17. Let S be a symmetric linguistic matrix of order 7 with entriesfromthelinguisticsetBgiveninproblem9. S=

very goodbadbestgood bad badbestfairbad very bestgood bad veryvery bestjustfair badfair very fairgoodfair fair very goodfairbestgood bad just goodbadgoodbad fair

.

   

                           

a)

i) Find all linguistic submatrices of S and prove the transpose of these linguistic submatricesare againlinguisticsubmatricesofS.

ii) Find the transpose of the following linguistic submatricesofS.

                  

goodbad badbest best P bestverybad fair verybadgood goodbad

b)W=

c)Q=

         

bestfairbad bestverybadgood verybadveryfairjustfair fairveryfairgoodfair

goodbadbest badbestfair bestverybad bestverybadveryfair fairveryfairgood

            

d)R=

             

best verybad veryfair justfair

e)N=(verybad, ,good, ,fair,best,good_

f)L= verybadveryfair veryfairgood     

18. LetM= good bad very good just fair just good best bad worst

                        

bealinguisticdiagonalmatrixoforder8  8withentries fromthelinguisticsetBgiveninproblem9.

i) LetN=

verygood justfair                

bealinguisticsubmatrixofM.ProveNt isalsoa linguisticsubmatrixofM.

ii) LetP=

         

justfair justgood best bad

bealinguisticsubmatrixofM,showP=Pt .

19. Can there be any other linguistic matrix other than the linguistic symmetric matrix (which includes, diagonal, linguistic matrices) which are such that the transpose of everylinguisticsubmatrixisagainalinguisticsubmatrix?

20. Prove in case of the row linguistic matrix R none of its linguistic submatrices whose transpose is a linguistic submatrix of R (of course we do not take trivial 1  1 linguisticmatricesintoconsideration).

21. Study (20) by replacing linguistic row matrix by the linguisticcolumnmatrixC.

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INDEX

A

Columnlinguisticmatrices,48-59 D

Decreasingchain,13-5

Decreasingorder,13-5

Diagonallinguisticmatrix,69-78

E

Emptylinguisticmatrix,56-66

Emptylinguisticterm,22-7 I

Identitylinguisticmatrices,56-60

Increasingchain,13-5

Increasingorder,13-5

L

Linguisticcontinuum,8-25

Linguisticinterval,8-20

Linguisticmatrices,48-55

Linguisticorderedset,35-42

Linguisticpowerset,32-9

Linguisticsubinterval,8-20

Linguisticsubmatrices,75-84

Linguisticsubsets,31-40

Linguisticvariable,7-20

Linguistic/term/word/set,8-20

Lowertriangularlinguisticmatrices,90-5

O

Orderedlinguisticset,35-42

P

Partiallyorderedset,15-21

R

Rectangularlinguisticmatrices,48-59 Rowlinguisticmatrices,48-50

S

Squarelinguisticmatrices,48-58 Symmetriclinguisticmatrices,55-60

T

Totallyorderedset,15-21

U

Uppertriangularlinguisticmatrix,90-6

ABOUT THE AUTHORS

Dr.W.B.Vasantha Kandasamy is a Professor in the School of ComputerScienceandEngg,VITUniversity,India.Inthepastdecade she has guided 13 Ph.D. scholars in the different fields of nonassociative algebras, algebraic coding theory, transportation theory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. She has to her credit 694 research papers. She has guided over 100 M.Sc. and M.Tech. projects. She has worked in collaboration projects with the Indian Space Research Organization and with the Tamil Nadu State AIDS Control Society. She is presently working on a research project funded by the Board of Research in Nuclear Sciences, Government of India. This is her 133rd book. On India's 60th Independence Day, Dr.Vasantha was conferred the Kalpana Chawla Award for Courage and Daring Enterprise by the State Government of Tamil Nadu in recognition of her sustained fight for social justice in the Indian Institute of Technology (IIT) Madras and for her contribution to mathematics. The award, instituted in the memory of IndianAmerican astronaut Kalpana Chawla who died aboard Space Shuttle Columbia, carried a cash prize of five lakh rupees (the highest prizemoneyforanyIndianaward)andagoldmedal. Shecanbecontactedatvasanthakandasamy@gmail.com WebSite:http://www.wbvasantha.in

Dr. K. Ilanthenral is Associate Professor in the School of Computer Science and Engg, VIT University, India. She can be contacted at ilanthenral@gmail.com

Dr. Florentin Smarandache is a Professor of mathematics at the University of New Mexico, USA. He got his MSc in Mathematics and Computer Science from the University of Craiova, Romania, PhD in Mathematics from the State University of Kishinev, and Postdoctoral in Applied Mathematics from Okayama University of Sciences, Japan. He is the founder of neutrosophy (generalization of dialectics), neutrosophic set, logic, probability and statistics since 1995 and has published hundreds of papers on neutrosophic physics, superluminal and instantaneous physics, unmatter, quantum paradoxes, absolute theoryofrelativity,redshiftandblueshiftduetothemediumgradient andrefractionindexbesidestheDopplereffect,paradoxism,outerart, neutrosophyasanewbranchofphilosophy,LawofIncludedMultipleMiddle, multispace and multistructure, degree of dependence and independence between neutrosophic components, refined neutrosophic set, neutrosophic over-under-offset, plithogenic set, neutrosophic triplet and duplet structures, quadruple neutrosophic structures, DSmT and so on to many peerreviewed international journals and many books and he presented papers and plenary lectures to many international conferences around the world. http://fs.unm.edu/FlorentinSmarandache.htm Hecanbecontactedatsmarand@unm.edu