Fuzzy mathematics approximation theory

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I.J. Intelligent Systems and Applications, 2015, 08, 9-17 Published Online July 2015 in MECS (http://www.mecs-press.org/) DOI: 10.5815/ijisa.2015.08.02 A Few Applications of Imprecise Matrices Sahalad Borgoyary Assistant Professor, Department of Mathematics, Central Institute of Technology Kokrajhar, Assam, India E-mail: s.borgoyarycit.ac.in, borgsahagmail.com Abstract-This article introduces generalized form of extension universal graph composition of any classical set tells definition of the Fuzzy set and its complement in the sense of element of the complement part of the set never be reference function namely in imprecise set and its complement. included in the set and reversely the element of the set Discuss Partial presence of element, Membership value of an cannot be included in the complement of its set. In the imprecise number in the normal and subnormal imprecise same way we claim that membership functions of a set numbers. Further on the basis of reference function define usual and its complement are not same that can be common in matrix into imprecise form with new notation. And with the both the sets. On this fact the article bring up examples so help of maximum and minimum operators, obtain some new that we can compare the complement definition of Zadeh matrices like reducing imprecise matrices, complement of reducing imprecise matrix etc. Along with discuss some of the and the extension definition called new complement classical matrix properties which are hold good in the imprecise definition of Baruah. matrix also. Further bring out examples of application of the This definition has been developed by different addition of imprecise matrices, subtraction of imprecise individuals to bring up many papers and articles in the matrices etc. in the field of transportation problems. various field of mathematics since 1965. We claim that the problems which are solved by the complement Index Terms- Imprecise Number, Partial Presence, definition of Zadeh occur error due to some partial Membership Value, Imprecise Matrix, Reducing Imprecise element is skipped in the complement definition. So Matrices, Imprecise Form. rectification would be done in the new complement definition of Baruah such that we can get more accurate result. Further to look into some practical problem in the I. INTRODUCTION field of transportation which is obtained in the form of Two Classical set theories Fuzzy matrices that can be written in the imprecise form by the help of definition of Baruah 2,3,4,5, this article comes out. and (1) The rest of article organized as follows: Section II deals with definition of imprecise number, normal where and are empty and universal set imprecise number and its complement, partial presence, respectively are not hold good in the definition of membership function, membership value. Section III Zadeh1. In the other word the universal set is the largest deals with the definition of imprecise matrices, new set of consideration set. Then how a one membership can matrix called reducing imprecise matrices, determinant, be common in both the set and its complement set or why transpose, identity, null of imprecise matrices and their the union of membership and non membership set do not complement with applications and examples. Section IV equal to universal set? We feel like there something is deals arithmetic operations of imprecise matrices and missing, because of that only it happen. The reason their existence of properties and applications. Finally behind such a claim can be contributed to the fact of section V goes to the conclusion. complement definition Zadeh for the fuzzy membership function defined by ( ) II. DEFINITIONS = ( ) (2) The articles of Baruah 2, 3, 4, 5 explain how the definition of Zadeh violates the theory of union and intersection of classical set and Boolean algebra and By the help of this complement definition and the justified them with counter examples and suggest to intersection and union of two fuzzy sets A and B defined rectify it some partial element is necessary for inclusion by, and exclusion to the complement definition of Zadeh. ( ) ( ) + A. Imprecise Number: Imprecise number is a closed and (3) interval , - is divided into closed sub-intervals ( ) ( ) +, with the partial presence of element in both the intervals. ( ) ( ) where and are the membership functions B. Partial presence: Partial presence of an element in of the fuzzy sets A and B really give rises to the result of an imprecise real number , - is described by the intersection and union of fuzzy set and its complement is present level indicator function ( ) which is counted not a null set and the universal set respectively. In the Copyright © 2015 MECS I.J. Intelligent Systems and Applications, 2015, 08, 9-17 10 A Few Applications of Imprecise Matrices from the reference function ( ) such that present level indicator for any x, , is ( ( ) ( )) , where ( ) ( ) 𝜌 (𝑥 ) 𝜑 (𝑥 ) C. Membership value: If an imprecise number Membership value of the , - is associated with a presence level indicator complementary fuzzy set function ( ) , where ( ) 𝜇 (𝑥 )=Membership value of fuzzy set ( ) ( ) (4) With a constant reference function 0 in the entire real Fig. 1. Complement of a normal fuzzy number line. Where ( ) is continuous and non-decreasing in the interval , - , and ( ) is a continuous and non- Membership value and the membership function are , - increasing in the interval with not same in the imprecise form of definition. However both are same in the usual set. For this reason the ( ) ( ) , intersection and union of imprecise form is defined in the then ( ( ) ( )) is called membership value of following: the indicator function ( ) Let ( ) ( ) ( ) + D. Normal Imprecise Number: A normal imprecise number , - is associated with a presence level and indicator function ( ) , where ( ) ( ) ( ) + ( ) be the two imprecise sets. Then the intersection and ( ) ( ) union is defined by, ( ) ( ) With a constant reference function 0 in the entire real ( ( ) ( )) (6) line. Where ( ) is continuous and non-decreasing in ( ( ) ( )) the interval , - and ( ) is a continuous and non- , - increasing in the interval with and ( ) ( ) ( ( ) ( )) ( ) ( ) (7) ( ( ) ( )) ( ) ( ) (5) We claim that the theory of intersection and union of Here, the imprecise number would be characterized by classical set are hold good by the definitions of imprecise ( ) + , R being the real line. number and its complement defined above. For this For any real line, ( ) ( ) normal and reason we consider a graph of surface having rectangular subnormal imprecise number will be characterized in shape of dram filled with half portion of water. So the common, ( ) ( ) +, where ( ) is called empty portion also becomes half. The filled of water and membership function measured from the reference the empty portions are the membership value and ( ) function ( ) and ( ( )) is called the reference value respectively. Here portion of dram filled membership value of the indicator function. with water is indicated by dark region and unfilled Here, the number is normal imprecise number when portion by white region. membership value of indicator function ( ) is equal to 1 otherwise subnormal if not equal to 1. Moreover it can 1 𝛾 be said that the universal set if the membership value of ( ) equal to 1 and null or empty set if equal to 0. E. Complement: For a normal imprecise number N= 𝛽 ( ) + as defined above, the complement 2 ( ) will have constant presence level indicator function equal to 1, the reference function 𝛼 0 being ( ) for Let us consider a curve defined in the closed universal Fig. 2. Complement of Sub-normal imprecise number (a) set Where the universal set comprises of membership Now according to Zadeh, membership value of the ( ) function and the non-membership functions fuzzy set is ( ) and ( ) called membership value of the complementary function shown in the Fig.1. Membership A= function of the complementary function is measured from the level up to what the level of curve is occupied. and membership value of its complement set is Copyright © 2015 MECS I.J. Intelligent Systems and Applications, 2015, 08, 9-17 A Few Applications of Imprecise Matrices 11 According to Zadeh, for the membership set or the fuzzy set 2 2 So, . / (8) +, which is a non-empty common region value of the the membership value of the layer of oil becomes fuzzy set and its complement set. And . / (not universal set) 2 (9) and the membership set of the complementary set is That is the combine region of the fuzzy set and its complement set. But the inclusion of empty portion in 2. / . /3 +, the set A and fill portion in the complement of set is which gives a membership value of the complementary meaningless as shown in the Fig.2. set nothing but portion of the dram not filled with oil But according to the definition of imprecise set, . ( ) ( ) + + This cannot be the membership value of the 2 complementary set as the usual value must be ( ) ( ) + and + . / . are the values of the imprecise set ( ) and the By the intersection and union definition of fuzzy set complement of imprecise set ( ) . So by the definition of union and intersection of imprecise set 2 . / . /3 +, (12) defined by Baruah we can obtained, which is not a null set as the membership value does not equal to zero. And ( ) ( ) ( ) ( ) 2 2 2 2 2 . / . /3 +, (13) 2 3, (10) which is not the universal set as the membership values is not equal to one. which gives membership value, ( ) =0 to form an But according to the definition of imprecise number, empty set. the membership value of the imprecise set And . / . / . / 2 3 (14) ( ) ( ) is . / . 2 2 + , (11) That is filled with oil. And the portions not filled with oil are the imprecise sets which gives membership value, 1-0=1 to form the universal set. ( ) Let us consider another example of dram of oil filled 2 2 with layer portion of oil. Since the oil floats on the and ( ) 2 3 , water and the dram not full of oil, there will be portion which is shown in the Fig.3. Complementary part is of dram as empty and portion will be layer of water is the combination of both imprecise sets. So the shown in the following figure: intersection of imprecise set and its complement are 1 ( ) ( ) 2 2 2 . / . /3 2 3 (15) 2 and . / . / ( ) ( ) 1 0 2 + (16) Fig. 3. Complement of Sub-normal imprecise number (b) Copyright © 2015 MECS I.J. Intelligent Systems and Applications, 2015, 08, 9-17 12 A Few Applications of Imprecise Matrices Which gives membership value 0 and 0 to form ( ) is the membership value, and are empty set. Union of two empty sets is again empty. So individually imprecise number. the intersection becomes empty. By the associativity law of sets union of imprecise sets, A. Definition. For any square of matrix ( . / . /) ( . / . /) becomes, 0 1 of order 2, an imprecise matrix would be represented ( ) ( ) and defined by 2 2 ( ) ( ) ( ) ( ) ( ) (18) 2 ( ) ( ) where all the elements of matrix are individually 2 imprecise numbers having membership values measured from the reference function of value zero. So, imprecise + (17) matrix of order 2 is denoted by ( ) ,( )- whose membership value is ( ) to form a In general ( ) ( ) is imprecise matrix universal set. of order , where m and n are the row and column Further it can be verified that the following properties of the imprecise matrix respectively. of classical sets are satisfied by the definition of Example- Suppose a bus is moving to provide daily imprecise sets A, B and C: necessary foods to the three different locations namely (1) Idempotent Laws: center 1, 2 and 3 respectively. Bus capacity of proving (i) services are such a way that total 100% and when at the (ii) centre 1 is 40% ,then its capacity of services at the center (2) Commutatively Laws: 2 and 3 are 30% and 30%, when at the center 2 is 30%, (i) then its capacity at the center 1 and 3 are 35% and 35%, (ii) when at the center 3 is 20% ,then its capacity at the (3) Associatively Laws: center 1 and 2 are 40% and 40% respectively. Which ( ) (i) ( ) nothing but a transportation problem and the conditions, (ii) ( ) ( ) allocation of services at the center 1, 2 and 3 are the (4) Distributive Laws: square imprecise matrix of order 3. ( ) ( ) (i) ( ) Here, (ii) ( ) ( ) ( ) (5) De Morgan’s Law: (i) ( ) (ii) ( ) (19) III. IMPRECISE MATRICES It is concerned that many of our problems are in the form of matrix that has been discussed with the definition 2 2 of classical matrices. But these are not overcome our real life problem as our day to day life problems are very which are membership values of the indicator function much related with fuzzy concept. Matrix with real entries at the centre 1, 2, and 3 are measured from the reference in 1, 0 and the matrix operations defined by fuzzy functions of value as 0. Thus the problem form the logical operations of reference functions are called following imprecise matrix, imprecise matrices. All the imprecise matrices are ( ) ( ) ( ) classical matrix but every classical matrix are not ( ) ( ) ( ) imprecise matrix. Fuzzy matrices are first time introduced by Thomson who discussed more details in ( ) ( ) ( ) the Convergence of Power of Fuzzy Matrices 9. Based ( ) ( ) ( ) on reference function Dhar has also proposed fuzzy ( ) ( ) ( ) (20) matrices in the articles 6, 7, 8. Here, all the elements ( ) ( ) ( 2 ) of imprecise matrix are imprecise number and are measured form 0 and the elements of its compliment are The logic proposed in the problem is an imprecise measured from 1. Though the imprecise number is form of diagram showing only the part of membership characterized by, ( ) +, for convenient functions at the centre 1, 2 and 3, which are measure of writing the elements of the imprecise matrix are from the reference functions of the centers. Graphically, written in the form of order pair ( ( ) ) , where Copyright © 2015 MECS I.J. Intelligent Systems and Applications, 2015, 08, 9-17 A Few Applications of Imprecise Matrices 13 membership functions and the reference functions of 0.3 0.35 each element of the matrix. 2 C. Reducing Imprecise Matrices: Reducing into 0.3 smaller form of imprecise matrix by making minimum 0.4 0.4 1 0.35 0.3 of all the elements of the matrix with respect to least element of that matrix is known as reducing matrix. It 0.4 3 would be represented and defined by, 0.2 ( ( ) ( ( )) , (23) ( )) Fig. 4. Imprecise matrix of order three. Where and are the smallest membership and Here the values of each path in the figure are the reference functions of the imprecise matrix ( ) membership values of the difference between respectively, membership functions and the reference functions of each element of the matrix. ( ) ( ) Example- If ( ) B. Definition: The complement of an imprecise matrix ( ) ( 2 ) ( ) ( ) ( ) Red ( ) ( 2) ( 2) of order would be represented and defined by ( ) ( ) ( 2) ( 2 2) ( ) ( ) , (21) ( ) ( ) where all the elements of complement of imprecise ( 2 ) ( 2 ) matrix are individually imprecise number measured from (24) ( 2 ) 2 1(one). For example So, its reducing imprecise becomes, ( ) ( ) ( ) ( 2) ( 2) ( ) ( ) ( 2) ( 2) is the complement of imprecise matrix of D. Multiplication of Imprecise Matrices: Multiplication of two imprecise matrices ( ) and ( ) ( ) ( ) , (22) ( ) will be represented and defined by, ( ) ( ) ( ) ( ) ( )+,(25) Which may be explained when a bus has not capable of providing service 70% at the center 1, there is 50% Provided two matrices are conformable for chances at the center 2. Similarly if bus not provides multiplication and the symbols have their meaning as service at the center 1 is 30%, their chance at the centre 2 discuss above. is 40%, where the conditions and the allocations of the ( ) ( ) center 1nd 2 are imprecise form of square matrix of order If ( ) ( ) ( 2 ) 2. The values of the allocations or the elements of the imprecise matrix are the membership value of the ( ) ( ) ( ) imprecise number measured from the reference function and ( 2 ) ( 2 ) of value 0 and the membership value of the complement imprecise matrix is measured from the reference function be the two imprecise matrices, then their multiplication of value 1.Graphically, is, ( ) 0.5 ( ) ( ) ( ) ( ) ( 2) ( 2) 2 0.6 1 0.5 0.3 ( ) ( ) 0.7 ( ) ( ) ( ) ( ) Fig. 5. Complement of imprecise matrix of order two. ( 2) ( ) ( ) ( ) ( 2 2) ( 2 2) 0.5 ( ) ( ) ( ) ( ) ( ) ( ) 2 0.7 1 0.4 ( ) ( ) 0.3 (26) ( ) ( ) Fig. 6. Imprecise matrix of order two. E. Identity Imprecise Matrices: Multiplicative identity Here, Fig.5. and Fig.6. are compliment to each other. imprecise matrix will be represented and defined by, And the values of each path in the figure are the ( ) ( ) , (27) membership values of the difference between Copyright © 2015 MECS I.J. Intelligent Systems and Applications, 2015, 08, 9-17 14 A Few Applications of Imprecise Matrices where ( ( )) ( ( ) ( )) ( ) ( ) (31) ( ) Example- Matrix, ( ( ) ( )) ( ) ( ) Where ( ) and ( ) are not is an identity of square imprecise matrix of order 2. always equal, however for equal value of max can be F. Null Imprecise Matrix: Null matrix will be expressed in common max ( ) and the represented and defined by, elements of determinant are the imprecise numbers ( ) , - ( ) . (28) measured from the reference function of value 0. Thus Determinant of imprecise matrix of order three is ( ) ( ) Example- Matrix, ( ) written by, ( ) ( ) ( ) ( ) ( ) is a null square matrix imprecise of order 2. ( ( )) ( ) ( ) ( ) G. Transpose of Imprecise Matrices: Transpose of ( ) ( ) ( ) ( ) imprecise matrix ,( )- would be represented and defined by and defined by, ( ( )) ,( )- . (29) ( ) ( ) ( ( ( ) ( ))) Graphically, ( ( ( ) ( )))+ 0.5 ( ( ( ) ( ))) 2 0.6 1 0.4 ( ( ( ) ( )))+ 0.3 ( ( ( ) ( ))) Fig.7. Imprecise matrix of order two. ( ( ) ( )) ( ( )+, (32) 0.5 where final value of determinant can obtained taking 1 2 0.6 0.4 maximum operator between two additive number and 0.3 minimum operator between two subtractive numbers Fig. 8. Transpose of imprecise matrix. respectively. ( ) ( ) The figure shows that changes of directions of activity ( ) Example: If ( ) ( 2 ) is the formation of transpose of an imprecise matrix, where the conditions and allocations at the center 1 and 2 be square matrix of order two, then are the imprecise form and the value of the elements are ( ) ( ) measured from the reference functions. And the values of ( ) ( ) ( ) ( ) 2 each path in the figure are the membership values of the difference between membership functions and the ( ( 2) ( )) reference functions of each element of the matrix. Thus it ( ( ) ( )) be obtained imprecise matrix, ( ) ( )+ ( ), (33) ( ) ( ) ( ) which is an imprecise number of membership value ( ) ( ) 0.3 and the transpose of imprecise matrix, I. Proposition: Complement of imprecise matrix is again an imprecise matrix. ( ) ( ) ( ) (30) ( ) ( ) ( 2 ) ( ) Example: If ( ) ( ) ( 2 ) Fig.7. and Fig.8. are transpose to each other. H. Determinant of Imprecise Matrices: If be a square matrix of order 2, then its complement becomes ( 2) ( ) is a determinant of order 2, then the determinant of ( ) (34) ( ) ( 2) imprecise matrix would be represented by, This is also an imprecise matrix of order two measured ( ) ( ) ( ) ( ) from the reference function of value 1. ( ) ( ) and defined by IV. ARITHMETIC OPERATIONS OF IMPRECISE MATRICES Copyright © 2015 MECS I.J. Intelligent Systems and Applications, 2015, 08, 9-17 A Few Applications of Imprecise Matrices 15 Arithmetic operations of imprecise matrix are the So we have, operations of addition, subtraction, multiplication and ( ) ( ) ( ) ( ) (37) division. These operations are expressed by the maximum and minimum operators of fuzzy sets which B. Proposition: Addition of complement of two are introduced by M. Z. Ragab and E.G. Emam in their imprecise matrices is commutative. article 11. Here, also these are discussed with new ( ) ( ) Proof: That is to prove that if and are notation and definition to obtain possible definition and imprecise complement matrices of ( ) and ( ) , their properties with applications so as to study easier. then A. Addition: Addition of two imprecise matrices ( ) ( ) ( ) ( ) ( ) (38) and ( ) would be denoted and defined by Since the complement of imprecise matrix is again a ( ) imprecise matrix. So by the proposition 4.1 ( ) ( ) , (35) ( ) commutativity law hold good. C. Substraction: Substraction of two imprecise provided ( ) and ( ) are conformable for matrices ( ) and ( ) would be defined by addition i.e. they are in same type, Where , , are membership functions and reference ( ) ( ) ( ) (39) ( ) ( ) functions of imprecise matrix and ( ) respectively. provided both the matrices are conformable for Thus the maximum possibility that can provide substraction. Where , and , are services by the two objects same at different locations is the resultant of sum of the two matrices. membership and reference functions of imprecise matrices ( ) and ( ) respectively. ( ) ( ) Example- If ( ) Thus the minimum possibility that can provide ( ) ( ) services by the two objects same at different locations is ( 2 ) ( ) the resultant of subtraction of the two matrices. and ( ) ( ) ( ) ( 2 ) ( ) Example- If ( ) then, ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( 2) ( ) ( 2 ) ( ) ( ) ( ) then, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) (36) ( ) ( ) ( ) ( ) ( ) ( ) 2 In physical significance, it is a transportation problem ( ) ( ) ( ) ( ) where two buses and are providing services ( 2 ) ( ) at the same centre 1 and 2, in such a way that bus ( ) (40) can provide service 40% at 2,when 50% at the centre 1 ( 2 ) ( ) and 10% at 2, when 30% at the centre 1. Similarly bus In physical significance, it is a transportation problem ( ) can provide service 30% at 2 when 20% at the ( ) ( ) where two buses and are providing services centre 1 and 70% at 2 when 30% at the centre 1. The ( ) at the same centre 1 and 2, in such a way that bus maximum possibility of services at the centre 1 and 2 are can provide service 30% at 2,when 20% at the centre 1 40% at 2 when 50% at 1 and 70% at 2 when30% at 1. and 50% at 2, when 40% at the centre 1. Similarly bus B. Proposition: Addition of two imprecise matrices is ( ) can provide service 40% at 2 when 30% at the commutative. centre 1 and 10% at 2 when 20% at the centre 1. The ( ) ( 2 ) minimum possibility of services at the centre 1 and 2 are Example- If ( ) ( ) ( ) 30% at 2 when 20% at 1 and 10% at 2 when20% at 1. D. Proposition: Subtraction of two imprecise matrices ( 2 ) ( ) and ( ) is commutative. ( ) ( ) ( ) ( 2 ) then Example- If ( ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( 2 ) ( ) and ( ) ( ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( 2 ) and ( ) ( ) ( ) ( ) ( ) ( ) then, ( ) ( ) Copyright © 2015 MECS I.J. Intelligent Systems and Applications, 2015, 08, 9-17 16 A Few Applications of Imprecise Matrices ( ) ( 2 ) intersection of imprecise sets as are also defined in the and ( ) ( ) ( ) ( ) form of maximum and minimum operators. So we have, ( ) ( ) ( ) ( ) (41) V. CONCLUSION ( ) ( ) Our real life problem is complex in nature. It is very F. Proposition: If and are Complement ( ) ( ) much related with fuzziness. Imprecise form is the of imprecise matrices and , then feasible definition of fuzziness. So, in this article ( ) ( ) ( ), (42) definition of fuzzy concept and its complement are discussed in the imprecise form with examples. Various where ( ) is complement of sum of two orders of imprecise matrices are expressed in the above imprecise matrices. with graphical representation. Arithmetic operations of ( ) ( ) matrix are defined by the maximum and minimum Example- If ( ) ( ) ( ) operators. And these operators are the main tools to obtain properties of the above mentioned imprecise ( 2 ) ( ) and ( ) matrices. Further applications of addition of two ( ) ( ) imprecise matrices, Subtraction of two imprecise ( ) ( ) matrices and their complement are also discussed in the ( ) then, ( ) ( ) field of transportation problems. Study of more dimensions of imprecise numbers is the future prospect ( 2) ( ) ( ) and of research. ( ) ( ) ( ) ( ) ( ) ( ) And ( ) ( ) ACKNOWLEDGMENT ( ) ( ) The author would like to thanks the anonymous ( ) ( ) ( ) reviewers for their careful reading of this article and for their helpful comments which have improved this work. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) So, (43) REFERENCES ( ) ( ) G. Proposition: If and are Complement 1 L.A. Zadeh, Fuzzy sets, Inform. And Control, 1965, 8: 338-53 of imprecise matrices ( ) and ( ), Then 2 H. K. Baruah, In Search of the Roots of fuzziness: The ( ) ( ) ( ) (44) Measure meaning of Partial Presence, Annals of Fuzzy mathematics and Informatics, 2(1), 2011, 57-68 where ( ) is complement of difference of two 3 H. K. Baruah., Theory of fuzzy sets: Beliefs and Realities, imprecise matrices. I. J. Energy Information and Communications. 2(2), (2011), 1-22 ( ) ( ) Example- If ( ) and 4 H. K. Baruah., Construction of Membership Function of a ( ) ( 2 ) Fuzzy Number, ICIC Express Letters 5(2), (2011), 545- 549 ( ) ( ) ( ) 5 H. K. Baruah., An introduction Theory of Imprecise Sets: ( ) ( ) The Mathematics of partial presence, J. Math. Computer ( ) ( ) Science, 2(2), (2012), 110-124 then, ( ) 6 M. Dhar., Theory of Fuzzy Sets: An Overview, I.J. ( ) ( 2) Information Engineering and Electronic Business, 2013, 3, ( ) ( ) 22-33 ( ) and ( ) ( ) 7 M. Dhar, A Revisit to Probability- Possibility Consistency Principles, I.J. Intelligent Systems and Applications, 2013, ( ) ( ) 04, 90-99 ( ) ( ) And ( ) ( 2 ) 8 M. Dhar, A Note on Determinant and Adjoint of Fuzzy Square Matrix, I.J. Intelligent Systems and Applications, ( ) ( ) 2013, 05, 58-67 ( ) ( ) ( 2) 9 M.G. Thomson, Convergence of Powers of a Fuzzy Matrix, J. Math. Anal. Appl., 57, 476-480. Elsevier, 1977 ( ) ( ) ( ) ( ) 10 J. B. Kim, Determinant Theory for Fuzzy and Boolean ( ) ( 2) Matrices, Congressus Numerantium Utilitus Mathematica Pub. (1978), 273-276 ( ) ( ) ( ) So we have (45) 11 M. Z. Ragab and E.G. Emam, The determinant adjoint of The definition of addition and subtraction of imprecise a square fuzzy matrix, Fuzzy Sets and Systems, 61 (1994) matrices are almost same with the definition of union and 297-307 Copyright © 2015 MECS I.J. Intelligent Systems and Applications, 2015, 08, 9-17 A Few Applications of Imprecise Matrices 17 12 M. Z. Ragab and E.G. Emam, On the min-max composition of fuzzy matrices, Fuzzy Sets and Systems , 75 (1995) 83-82 13 L.J. Xin, controllable Fuzzy Matrices, Fuzzy Sets and Systems, 45, 1992, 313-319 14 L.J. Xin, Convergence of Powers of Controllable Fuzzy Matrices, Fuzzy Sets and Systems, 63, 1994, 83-88 15 T. J. Neog and D. K. Sut, An Introduction to the Theory of Imprecise soft sets, I.J. Intelligent Systems and Applications, 2012, 11, 75-83 16 M. Dhar and H.K. Baruah, The Complement of Normal Fuzzy Numbers: An Exposition, I.J. Intelligent Systems and Applications, 2013, 08, 73-82 Authors’ Profiles Sahalad Borgoyary Currently Pursuing Ph. D. in the Bodoland University, Kokrajhar. He is an Assistant Professor in the department of Mathematics, Central Institute of Technology Kokrajhar, BTAD, Assam, India. He received master degree from Gauhati University, Kokrajhar Campus. His Research interests are included as Fuzzy Mathematics, Operations Research. Copyright © 2015 MECS I.J. Intelligent Systems and Applications, 2015, 08, 9-17

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