Scholarly article on topic 'Interval-valued Fuzzy Matrices with Interval-valued Fuzzy Rows and Columns'

Interval-valued Fuzzy Matrices with Interval-valued Fuzzy Rows and Columns Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — Madhumangal Pal

Abstract Fuzzy matrix (FM) is a very important topic of fuzzy algebra. In FM, the elements belong to the unit interval [ 0 , 1 ] . When the elements of FM are the subintervals of the unit interval [ 0 , 1 ] , then the FM is known as interval-valued fuzzy matrix (IVFM). In IVFM, the membership values of rows and columns are crisp, i.e., rows and columns are certain. But, in many real life situations they are also uncertain. So to model these types of uncertain problems, a new type of interval-valued fuzzy matrices (IVFMs) are called interval-valued fuzzy matrices with interval-valued fuzzy rows and columns (IVFMFRCs). For these matrices, null, identity, equality, etc. are defined along with some binary operators. Complement and density of IVFMFRC are defined and several important properties are investigated. An application of IVFMFRC in image representation is also given.

Academic research paper on topic "Interval-valued Fuzzy Matrices with Interval-valued Fuzzy Rows and Columns"

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Fuzzy Information and Engineering

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ELSEVIER

ORIGINAL ARTICLE

Interval-valued Fuzzy Matrices with Interval-valued Fuzzy Rows and Columns

CrossMark

Madhumangal Pal

Received: 9 September 2014/ Revised: 28 April 2015/ Accepted: 21 July 2015/

Abstract Fuzzy matrix (FM) is a very important topic of fuzzy algebra. In FM, the elements belong to the unit interval [0,1]. When the elements of FM are the subinter-vals of the unit interval [0,1], then the FM is known as interval-valued fuzzy matrix (IVFM). In IVFM, the membership values of rows and columns are crisp, i.e., rows and columns are certain. But, in many real life situations they are also uncertain. So to model these types of uncertain problems, a new type of interval-valued fuzzy matrices (IVFMs) are called interval-valued fuzzy matrices with interval-valued fuzzy rows and columns (IVFMFRCs). For these matrices, null, identity, equality, etc. are defined along with some binary operators. Complement and density of IVFMFRC are defined and several important properties are investigated. An application of IVFMFRC in image representation is also given.

Keywords Fuzzy matrix • Interval-valued fuzzy matrix ■ Fuzzy rows and columns ■ Complement of fuzzy matrices • Density of fuzzy matrices

© 2015 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.

1. Introduction

Like classical (crisp) matrices, fuzzy matrices (FMs) are now a very rich topic in modeling uncertain situations occurred in science, automata theory, logic of binary

Madhumangal Pal (E3)

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India email: mmpalvu@gmail.com

Peer review under responsibility of Fuzzy Information and Engineering Branch of the Operations Research Society of China.

© 2015 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.

This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/jfiae.2015.09.006

relations, medical diagnosis, etc. In FMs, only the elements are uncertain, while rows and columns are certain. But, in many real life situations we observed that rows and columns may also be uncertain. Pal [29] has defined fuzzy matrices with fuzzy rows and columns (FMFRCs).

The elements of FMFRCs are non-negative proper fraction. But, when the elements are the subintervals of the unit interval [0,1], then the FM is known as IVFM. In IVFM, the rows and columns are considered as scrips, but we have seen that they may also be uncertain, i.e., rows and columns have some membership values.

For example, in interval-valued fuzzy graph, the vertices and edges are uncertain and their membership values are the subintervals of [0,1]. This graph can be represented as an IVFM. For each vertex there is a row and column. The ijth entry represents the membership value of the edge joining i and j. In this matrix, the rows and columns are uncertain.

Thomson [37] defined FMs for the first time in 1977. In this paper, he discussed about the convergence of the powers of a fuzzy matrix. The theories of fuzzy matrices were developed by Kim and Roush [22] as an extension of Boolean matrices. With max-min operation the fuzzy algebra and its matrix theory are considered by many authors [5,14, 20,26, 27, 34]. Hashimoto [15] studied the canonical form of a transitive fuzzy matrix. Xin [38] studied controllable fuzzy matrices. Hemashina et al. [18] investigated iterates of fuzzy circulant matrices. The transitivity of matrices over path algebra (i.e., additively idempotent semiring) is discussed by Hashimoto [15-17]. Generalized fuzzy matrices, matrices over an incline and some results about the transitive closure, determinant, adjoint matrices, convergence of powers and conditions for nilpotency are considered by Duan [13] and Lur et al. [23]. Dehghan et al. [12] give two ideas for finding the inverse of a fuzzy matrix viz. scenario-based and arithmetic-based.

There are some limitations in dealing with uncertainties by fuzzy sets. To overcome these difficulties, Atanassov [4] introduced the theory of intuitionistic fuzzy set in 1993 as a generalization of fuzzy sets. Based on this concept Pal et al. have defined intuitionistic fuzzy determinant in 2001 [26] and intuitionistic fuzzy matrices (IFMs) in 2002 [27]. Bhowmik and Pal [5] produced some results on IFMs, intuitionistic circulant fuzzy matrix and generalized intuitionistic fuzzy matrix [5-11]. Shyamal and Pal [33, 35] defined the distances between IFMs and hence defined a metric on IFMs. They also mentioned few applications of IFMs. In [25], the similarity relations, in-vertibility conditions and eigenvalues of IFMs are studied. Idempotent, regularity, permutation matrix and spectral radius of IFMs are also discussed. The parameter-izations tool of IFM enhances the flexibility of its applications. For other works on IFMs see [1-3,24, 30, 31, 34, 35].

The concept of IVFMs as a generalization of fuzzy matrix was introduced and developed in 2006 by Shaymal and Pal [36] by extending the max-min operation in fuzzy algebra. For more works on IVFMs see [28].

Combining IFMs and IVFMs, a new fuzzy matrix called interval-valued intuitionistic fuzzy matrices (IVIFMs) is defined [19]. For other works on IVIFMs, see [9, 11].

In [29], Pal has defined FMFRCs. In these matrices, rows and columns are also

fuzzy numbers, i.e., unlike fuzzy matrices they are also uncertain. Motivated from this paper and interval-valued fuzzy graph, we define IVFMFRCs. The complement, some binary operations, density, etc. are defined and some important properties are proved.

2. Preliminaries

Some basic operations on interval-valued fuzzy numbers are defined below.

Let 23 denote the set of all subintervals of the interval [0,1]. Let a = [aT, a+] and b = [b~, è+l be two elements of D. Then

1) a®b = [a- + b~ - a~ ■ b~,a+ + b+ - a* ■ b+],

2) a O b = Ia~,a+] O [b~,b+] = [a ■ b~,a* ■ b+],

3) a Ab= [a~,a+] A [b~,b+] = [a~ A b~,a+ A b+],

4) a V b = [a~,a+] V = [<T V b~,a* V 2>+],

6) ac = [1 - a+, 1 - a'].

The operators, "+", "-" and "■" used in extreme right are ordinary addition, subtraction and multiplication, respectively.

Two intervals [a~, a+] and [Zr, ¿+] are equal if and only if a~ = b~ and a+ = b+. We denote [0,0] and [1,1] as 0 and 1 respectively.

Now we define interval-valued fuzzy matrix whose rows and columns are certain as follows.

Definition 2.1 An interval-valued fuzzy matrix of order mxn is defined as, A = (aij)mxn, where ay = at] is the ijth element of A, represents the membership value. All the elements of an WFM are intervals and they are members ofD [36].

In IVFM, the elements are the membership grade of some attributes, they are not crisp number, so naturally some new operations are needed to handle such matrices.

In IVFMs, it is assumed that all the rows and columns are certain, but in our new concept we assumed that the rows and columns are also uncertain and their membership values are members of O. These type of matrices are defined in the next section.

3. New Definition of Interval-valued Fuzzy Matrices

LetA = [rA (¡)J [ca (/)] [% Jmxrc be an IVFMFRC of order mxn. Here i = 1,2, ■■■ ,m; j = 1,2, • • • ,n represents the ly'th element of A, (i) and ca(J) represent the membership values of ith row and yth column respectively for i = 1,2, ■ • ■ ,m\j= 1,2, ■ ■ ■ , n.

Ca(1) Ca(2) • • • cA(n) rA(l)r an a12 ■•• aln A = rA(2) ¿¿21 a22 ... a2n

rA(m) taml am2 '" "mí!

be a matrix, where rA(i), i = 1,2, ■ • • , m, cA(/), y = 1,2, ■ • • , n and ay, i = 1,2, ■ • • , m; j = 1,2, ■ ■ ■ ,», represent respectively the membership values of rows, columns and elements.

(i) When rA(f) = l,i = 1,2, ■■ • ,m; cA(J) = 1 ,j = 1,2,••■ ,n and a¡¡ e [0,1], i = 1,2, ■ ■ • ,m; j = 1,2, ■ ■ • , n, then A is an FM.

(ii) When rA(i) e [0,1], i = 1,2, • • • , m; cA(/) e [0,1], j = 1,2, • • • , n and a,¡ e [0,1], i = 1,2, • • • ,m; j = 1,2, • • • , n, then A is called a fuzzy matrix with fuzzy rows and columns (FMFRC).

(iii) When fA(í) e £>,i = 1,2," ■ ,m; cA(j) e D, j = 1,2,-■■ ,n and e 2), i = 1,2,■•■ ,m;j = 1,2,• •• ,n, then A is called an IVFMFRC. For this case, all

= ííT., a+1, rA(i) = [rA(i), r+(i)],cA(i) = [cA(i), c+(¡)l are members of 2). The membership values of the rows of an IVFMFRC A may be written as an interval fuzzy vector rA = [rA(l), rA{ 2), ■ • ■ , rA(m)] and similarly cA = [cA( 1), cA( 2), ■■ ■ ,cA («)]■

If the value of rA(i) or cA(J) is 0 for some i or j, then it implies that the ith row or jth column has no importance for the IVFMFRC A and in this case one can remove them from A. When rA(i) - 1 and cA(J) = 1 for all i, j, then IVFMFRC A becomes IVFM.

3.1. Equality of IVFMFRC

The equality of two IVFMFRCs can be defined in three different ways. Let A = [rA(i)][cAU)][a,j]mxr, and B = [rBmicBU)]lbij]mxn be two IVFMFRCs.

T^pe I: If y a (') = rB(j) and cA(j) = cB(J) for i = 1,2, •• • ,m\j= 1,2, •• • ,n, whatever may be the relation between a¡j and b¡¡, then we say that A and B are i?C-equal and it is denoted by A =RC B. If rA(i) ± rB(i) or cA(J) + cB(J) for at least one i or j, then we say that A =f= B in RC-equal sense. This is the weak equality between two IVFMFRCs and it occurs only in the IVFMFRCs and FMFRCs.

T^pe II: If a¡¡ = b¡¡ for all i and j, whatever may be the values of rA(i), rB(i), cA(J), cB(J), then A and B are £-equal and it is denoted by A =e B. This type of equality occurs in FMs also.

If a¡j í b¡¡ for at least one i or j, then we say A +e B or A + B \a £-equal sense.

T^pe III: If both A =rc B and A =e B, then we say that A and B ate equal and it is denoted as A = B. That is, if

(i) a¡j = bij for all i, j,

(ii) rA(i) = rB(i) for all /,

(iii) CA(j) = CB(J) for all j.

Then A is equal to B and is denoted as A = B. If A and B are not equal, then it is denoted by A £ B. That is, if A B and /or A B, then we write A + B.

3.2. Null IVFMFRC

Based on the membership values of rows, columns and elements, three types of null IVFMFRC are defined.

T^pe I: If rA(i) = 0, cA(j) = 0 and at} = 0 for all i and j, then IVFMFRC A is called p-null, denoted by 0P. For example

000 000 000 000

is a 3 x 3 order p-null IVFMFRC.

This concept is the same as FMs as well as classical matrices. This type of null matrix is called empty IVFMFRC.

T^pe II: If ay = 0 for all i and j, whatever may be the values of rA{i) and caO), then the IVFMFRC A is called is-null, and it is denoted by 0g. For example

[0.2,0.4] [0.1,0.5] [0.2,0.8]

[0.5,0.6] [0.0,0.1] [0.6,0.7] [0.0,0.0] [0.0,0.0] [0.0,0.0] [0.0,0.0] [0.0,0.0] [0.0,0.0] [0.0,0.0] [0.0,0.0] [0.0,0.0]

is a 3 x 3 order E-null IVFMFRC. This concept is also similar to FMs.

T^pe III: If rA(i) = 0, cA(j) = 0 for all i and j, whatever may be the values of then the IVFMFRC is called RC-null and it is denoted by 0RC. For example

[0.0,0.0] [0.0,0.0] [0.0,0.0]

[0.0,0.0] [0.0,0.0] [0.0,0.0] [0.0,0.0] [0.2,0.5] [0.0,0.1] [0.3,0.5] [0.10,0.15] [0.3,0.8] [0.6,0.8] [0.1,0.9] [0.45,0.85] [0.2,0.5] [0.2,0.7] [0.5,0.5] [0.56,0.71]

is a /fC-null IVFMFRC.

This type of null matrix is new and it is only defined for IVFMFRC and FMFRC.

3.3. Identity IVFMFRC

Two types of identity IVFMFRC are defined here.

Type I: A square IVFMFRC of order n x n is called p-identity IVFMFRC if rA(i) = 1 and ca(]) = 1 for all i and j and a« = 1, al;- = 0, i j for all /, j. It is denoted by Ip. For example

111 1 [100 1 010 1 10 01

is a 3 x 3 order p-identity IVFMFRC.

Type II: A square IVFMFRC of order nxn is called /-identity if an = 1, a,j = 0,i# j for all i, j, whatever may be the values of r,i(i) and c/SJ) and it is denoted by //. For example

[0.2,0.5] [0.3,0.7] [0.4,0.8] [0.2,0.7] [[1.0,1.0] [0.0,0.0] [0.0,0.0] [0.5,0.7] [0.0,0.0] [1.0,1.0] [0.0,0.0] [0.1,0.4] [0.0,0.0] [0.0,0.0] [1.0,1.0].

is a 3 x 3 order /-identity IVFMFRC. 4. Operators on IVFMFRCs

In this section, some operators, viz. V, A, O, © are defined and explained with numerical examples.

4.1. V Operator

Let A = [rA(i)] [cA(_/)] [ajj]mxn and B = [r£(i)][cB(;)][6,y]„,x„ be two IVFMFRCs. Then

A V B = D = [ro(i)][cD0)][rfi7]mx„,

rod) = rA(j) V rE(i) = [rA(i) v r~(i), r\(i) v r+(;)], cd(I) = cAU) V cBU) = lcA(j) V cB(j), c+AU) V 40)] and dij = a,j V bij = [or. v bjj, at V b+] for all i, j. Note that the order of A and B must be equal.

But, in our new concept one can operate two IVFMFRCs with different orders. Suppose A = [rA(McA(M[aij\mxn and B = [rB(0]LcB0')J[Mpx<7be two IVFMFRCs. For the sack of simplicity, we assume that m<p and n<q.\im = p and n = q, then there is nothing new.

Otherwise, three different cases may arise:

(i) m < p, n< q,

(ii) m< p, n<q,

(iii) m < p, n < q.

In these cases, add p-m (may be 0) rows and q - n (may also be 0) columns at the end of rows and columns.

The elements of these rows and columns are taken as 0 and membership values of all rows and columns are taken as 0. After introduction of these rows and columns,

the IVFMFRC A becomes an IVFMFRC of order pxq. Now, the V operation can be performed as in previous case. Note that the order of A V B becomes pxq. To illustrate this new concept, let us consider two IVFMFRCs as

A = [0.1,0.5] [0.4,0.6]

[0.1,0.7] [0.7,1.0] [0.6,0.8] [0.0,0.1] [0.4,0.8] [0.3,0.7] [0.2,0.5] [0.6,0.7] [0.0,1.0]

[0.0,0.2] [0.6,0.8] [0.3,0.9]

[0.2,0.5] [0.1,0.6] [0.0,0.1] [0.0,0.2] [0.6,0.8] [0.3,0.9] [0.0,0.3] [0.1,0.7]

Note that the number of rows of A is less than that of B and the number of columns of B is less than that of A. The augmented matrices A0 and Ba are given by

[0.1,0.5] [0.4,0.6] [0.0,0.0]

[0.0,0.2] [0.6,0.8] [0.3,0.9]

[0.1,0.7] [0.7,1.0] [0.6,0.8] [0.0,0.1] [0.4,0.8] [0.3,0.7] [0.2,0.5] [0.6,0.7] [0.0,1.0] [0.0,0.0] [0.0,0.0] [0.0,0.0]

[0.2,0.5] [0.1,0.6] [0.0,0.0] [0.0,0.1] [0.0,0.2] [0.0,0.0] [0.6,0.8] [0.3,0.9] [0.0,0.0] [0.0,0.3] [0.1,0.7] [0.0,0.0]

Therefore, A V В = A„ V Ba =

[0.1,0.5] [0.6,0.8] [0.3,0.9]

[0.2,0.7] [0.7,1.0] [0.6,0.8] [0.0,0.1] [0.4,0.8] [0.3,0.7] [0.6,0.8] [0.6,0.9] [0.0,1.0] [0.0,0.3] [0.1,0.7] [0.0,0.0]

Note that the order of A and В are 2 x 3 and 3 x 2, but the order of A V В is 3 x 3. Observations: The following results are obvious:

(i) Op V A = A;

(ii) (a) 0£ V A # A,

(b) 0E VA ФЯСА,

(c) 0£ VA =e A;

(iii) (а) Оде V A* A,

(b) Оде V А фцс A,

(c) 0RCV A =E A.

4.2. A Operator

The A operation is similar to V operation. Let A = \rA(/)]\cA(j)]\aij]mxn and B = [rB(m[cBU)][bij]mM be two IVFMFRCs. Then

A a B = D = [rndWcniMd^,

rD(i) = rA(i) A rB(i) = [rA(i) A r~(i), r\(i) A 4(0], CdU) = cA(J) A cB0) = [cA(i) A cB(i), c+(i) A CJ(0]

dy = a,] A bij = [aT. A bT, at A for all i, j. If the orders of A and B are different, then this case can be handled as in case of V operation.

4.3. O and @ Operators

Let A = [rA(i)][cAa)]K-]mxn and B = [rB(0][eB0№;]mx„ be two IVFMFRCs. Then

A®B = D= [rD(i)][cD(j)][dij]mxn,

rD(i) = rA(i) ® rB(i) = [rA(i) + r~(i) - rA(i) ■ r~(i), r\(i) + r+(i) - r\(i) ■ r+(;)J,

cdO) = cA0')ffics0') = 40) + 40) "40)'40)L

dtj = otj ® btj = [aT + br. - aT ■ bT., at + bt - at ■ bt] for all i and j. The operator © is defined as

AoB = E = [rEmcEm\eii\mxn,

rE(i) = rA(i) O rB(i) - [r~(i) ■ r~(i), r+(i) ■ r+B(i)], cfXj) = cA(j) © cB(J) = lcA(j) ■ 40), 40) ■ 40)J. and e,j = aij O btj = [aT ■ bT., at ■ bt] for all i and j

A@B = F = [rFmcFmUij\m*n,

rF{i) = rA(f)@rB(i) = [|(r-(i) + r-(O), ¡(r+A(i) + 4(0)1, cfU) = caU)@cbU) = [^(40) + cB(j)), ±(40) + 40))], and fij - aij@b,j - [\{aT + bT.), I (a* + bt)] for all i and j.

5. g-IVFMFRC

In this section, a special type of IVFMFRC is defined along with other two types of IVFMFRCs.

Definition 5.1 If aT. < rA(i) A 40) and at < rA(i) A 40) for all i,j, then the IVFMFRC A = [rA(0][cA0)][ay]mx„ is called g-IVFMFRC.

[0.1,0.5] [0.2,0.8] [0.6,0.7]

B= [0.2,0.7] [0.3,0.8]

[0.1,0.2] [0.2,1.0] [0.5,0.8] [0.0,0.2] [0.1,0.4] [0.0,0.3] [0.1,0.2] [0.2,0.8] [0.5,0.7] [0.0,0.1] [0.1,0.6] [0.4,0.6]

[0.5,0.7] [0.1,0.6] [0.4,1.0] [0.2,0.6] [0.1,0.6] [0.0,0.7] [0.3,0.5] [0.1,0.5] [0.3,0.7]

both are g-IVFMFRCs.

Definition 5.2 If a~j = rA(i) A c~A(j) and at IVFMFRC A is called a complete IVFMFRC.

rA(i) A c\(J) for all i,j, then the

From the definition it is obvious that every complete IVFMFRC is g-IVFMFRC, but converse is not true. The IVFMFRC

A = [0.1,0.6] [0.3,0.4]

[0.2,0.5] [0.3,0.7] [0.5,0.9] [0.1,0.5] [0.1,0.6] [0.1,0.6] [0.2,0.4] [0.3,0.4] [0.3,0.4]

is a complete IVFMFRC.

Now we define another kind of IVFMFRC.

Definition 5.3 If a~- < r^(i') • cA(j) and at < rA(i) • cA(j) for all i,j, then the IVFMFRC A is called a dot IVFMFRC, where • denotes ordinary multiplication.

The IVFMFRC is

[0.6,0.9] [0.7,1.0] [0.2,0.5] [0.4,0.8] [0.2,0.7] [0.2,0.7] [0.0,0.4] [0.2,0.7] [0.1,0.5] [0.1,0.6] [0.0,0.2] [0.6,0.7] [0.3,0.6] [0.4,0.5] [0.1,0.3].

Lemma 5.1 Every dot IVFMFRC is a g-IVFMFRC.

Proof Since 0 < r-(i) < 1, 0 < r+(i) < 1 and 0 < cA(i) < 1, 0 < c+(i) < 1 r~A(i) • c"0) < r~A{i) A c~(j) and r*A(i) • c+(j) < r+(i) A c*(j) for all i, j.

If A is a dot IVFMFRC, then a7. < rA(i) • cA(j) and at < r\(i) • c\(j) for all i, j. Therefore, for all i, j, aTj < rA(i) • cA(j) < rA{i) A cA(j) and at < rA(i) • cA(j) < r\(i) A c*(j), i.e. dTj < r~A(i) A cA(j) and at < r*(i) A c+(j) for all i, j. Hence, A is a g-IVFMFRC.

6. Complement of IVFMFRC

In this section, two types of complements of an IVFMFRC are defined. Like fuzzy matrix, the complement of an IVFMFRC is defined below:

We define 1 - a, a e D as follows:

1 - a = [1,1] - [a~, a+] = [1 - a+, 1 - a"]. Note that 0 < 1 - a+ < 1 - a < 1 and hence 1-aefl.

Definition 6.1 Let A = [rA(i)][cA(j)][aij\ myn be an IVFMFRC. Its complement is denoted by Ac and it is defined as Ac = [1 - rA(j)\\l - cA(J)\{l - ay]mxn-

Definition 6.2 An IVFMFRC is called self complement if Ac = A.

Theorem 6.1 If A is cm IVFMFRC, then {A°J = A.

Proof Let B = Ac. Then rB(i) = 1 - rA(i),cB(J) = 1 - cA(j), bu = 1 - a0 = [1 - a+, 1 - a~l If D = B° = (Ac)c, then dtj = 1 - = [1 - bt., 1 - &-] = [l-(l-a-]),l-(l-atj)] = [a-rat]] = aij. Similarly, rD(i) = rA(i), cD{j) = cA(j) for all i, j. Hence, D = A, i.e. (ACJ = A.

Theorem 6.2 If an IVFMFRC A = [rA(i)][cA(j)][aij]m>ol is self complement, then r~A(i) + r+(i) = 1, c~(i) + c+(i) = 1, or. + at = \,for all i, j.

Proof By the definition of complement, a?. = [1,1]- [aT., at] = [1 - at, 1 - a7.]. Since Ac = A, [1 - at, 1 - a^] = [a~, at]. This gives a~- + at = 1 for all i, j. The other parts of the proof are similar.

Theorem 6.3 (De Morgan's laws) Let A = [^(Oirc/tOlirazjlmxn and B = [rB(i)][cB(J)] [bij]mxn be two IVFMFRCs. Then

(i) (A V B)c = Ac A B°,

(ii) (A A Bf = ACVBC.

Proof (i) Let D = A V B. Then rD{i) = rA(i) V rB(i), cD(J) = cA(J) V cB(j), dtj = aij V bij.

LetE = DF. Then rE(i) = l-rD(i) = l-rA(i)VrB{i),cE(j) = 1 -cD(J) = 1 -cA(j)V cB(j), eij = 1-dij = 1-a.jVh.j = [1,1 ]-[ar.Vfe-,a+.vi+.] = [l-a+.Vfe+, l-a7.vZ>7.]. Let F = AC A B°.

Therefore, ftJ = (1 - a,j) A (1 - b,j) = ([1,1] - [a^at]) A ([1,1] - [b^bt]) = [1 - at, 1 - aTj] A [1 - bt, 1 - bjj\ = [(1 - at) A (1 - bt), (1 - a7.) A (1 - bT.)\ = [l-atvbt,l-azvbz\ = eij.

Similarly, rF(i) = rE(i), cF(J) = cE(j) for all i, j. Hence, (A V B)c = Ac A B°. (ii) Proof is similar to (i).

A = [0.1,0.7] [0.2,0.5]

[0.3,0.6] [0.1,0.8] [0.2,0.7] [0.0,0.2] [0.4,0.5] [0.5,0.7] [0.7,0.9] [0.3,0.8] [0.3,0.6]

[0.2,0.7] [0.5,0.9] [0.2,0.4]

[0.0,0.2] [0.3,0.5] [0.3,0.4] [0.3,0.5] [0.0,0.3] [0.5,0.7] [0.3,0.5] [0.8,1.0]

Before going to such operations we augment A and B to make same size and let the augmented matrices be Aa and Ba corresponding to A and B. Then

A„y B„ =

[0.2,0.7] [0.5,0.9] [0.2,0.4]

[0.3,0.6] [0.3,0.8] [0.2,0.7] [0.3,0.4] [0.4,0.5] [0.5,0.7] [0.7,0.9] [0.5,0.8] [0.3,0.6] [0.3,0.5] [0.8,1.0] [0.0,0.0]

(Aa V Ba)c =

Again,

[0.3,0.8] [0.1,0.5] [0.6,0.8]

[0.4,0.7] [0.2,0.7] [0.3,0.8] [0.6,0.7] [0.5,0.6] [0.3,0.5] [0.1,0.3] [0.2,0.5] [0.4,0.7] . [0.5,0.7] [0.0,0.2] [1.0,1.0]

[0.4,0.7] [0.2,0.9] [0.3,0.8] [0.3,0.9] [0.8,1.0] [0.5,0.6] [0.3,0.5] [0.5,0.8] [0.1,0.3] [0.2,0.7] [0.4,0.7] [1.0,1.0] [1.0,1.0] [1.0,1.0] [1.0,1.0]

[0.8,1.0] [0.5,0.7] [1.0,1.0] [0.3,0.8] [0.6,0.7] [0.5,0.7] [1.0,1.0] [0.1,0.5] [0.7,1.0] [0.3,0.5] [1.0,1.0] [0.6,0.8] [0.5,0.7] [0.0,0.2] [1.0,1.0]

Ac„ A B'

[0.4,0.7] [0.2,0.7] [0.3,0.8] 1C _ [0.3,0.8] [0.6,0.7] [0.5,0.6] [0.3,0.5] [0.1,0.5] [0.1,0.3] [0.2,0.5] [0.4,0.7] [0.6,0.8] [0.5,0.7] [0.0,0.2] [1.0,1.0]

Hence, (A„ V Ba)c = Aca A B°a.

The above result is true for all IVFMFRCs, if both IVFMFRCs have same size. But, if their sizes are different, then the matrices should be augmented to make same size before computing complement.

Let us define a new operator 0 for the IVFMFRCs A = \rA (i)] \cA 0')1 \aij]mx„ and B = [rB(i)][cB0')]№;;Lxn as

AOB = (Ac A B) V (A A BT).

Theorem 6.4 For caty IVFMFRCs AandB, Ac0B° = AoB.

Proof By definition

Ac0B° = {(Ac)c A Bc) V \AC A (BT) = (A A Bc) V (Ac A B) = AoB.

7. Density of IVFMFRC

Sometimes we see that a matrix, it may be classical matrix or fuzzy matrix, contains many zeros. This type of matrix is called sparse matrix and special methods are used to store such matrix in computer. In contrast, if the number of non-zero elements is high then the matrix is called dense matrix. The density of an FM is defined below.

Definition 7.1 Let A = [ay]mx„ be an FM. The density of A is denoted as D(A) and is defined as

From definition it follows that 0 < D(A) < 1 for an FM A. Actually, D(A) represents the average membership of the elements in the FM A.

But, in FMFRC, rows and columns are not certain, and hence the density is to be re-defined for FMFRC. The definition is given below.

Definition 7.2 [29] Let A = [i-a(0][ca(/)] [%Lx* be an FMFRC. The density of A is denoted by D(A) and is defined as

For an IVFMFRC, au > 0 for all i, j. Thus, D(A) is non-negative for any IVFMFRC A. D(A) is zero only when all ay = 0. Higher value of D(A), indicates the matrix is more dense. If the value of D{A) is closed to zero, then the IVFMFRC is called sparse IVFMFRC with degree of sparsity D(A).

Theorem 7.1 [29] For a g-FMFRC, 0 < D{A) < 1.

Theorem 7.2 [29] If A is a complete g-IVFMFRC, then D(A) = 1.

Like classical sub-matrix, one can defined sub-IVFMFRC. Any portion, not necessarily consecutive, along with corresponding rows and columns membership values is called sub-IVFMFRC.

Like FMFRC, the density of an IVFMFRC is defined as D(A) = [D"(A), D+(A)], where

D(A) =

2 Ta{i) A CA(j)'

provided 2 rA (i) A cA ( j) 4- 0.

•uu a

and D+(A) =

D-(A) =

4(0 A 40)

4(0 A 40)'

Since atj < at for all i, j, therefore it is easy to verify that D (A) < D+(A).

A= [0.2,0.5] [0.3,0.8]

B= [0.6,0.8] [0.7,0.9]

[0.6,0.9] [0.0,1.0] [0.5,0.6] [0.8,1.0] [0.3,0.7] [0.6,0.8]

[0.6,0.9] [0.5,1.0] [0.5,0.6] [0.5,0.7] [0.6,0.9] [0.4,0.8]

D~(A) =

0.5 + 0.8 + 0.3 + 0.6 11 _ 0.6 + 1.0 + 0.7 + 0.8

0.5 + 0.5 + 0.8 + 0.8 ~ 13 M ( } ~ 0.5 + 0.5 + 0.8 + 0.8

31 26'

Thus density of A, D(A) = [11/13,31/26]. Also,

, , 0.5 + 0.5 + 0.6 + 0.4 10 , 0.6 + 0.7 + 0.9 + 0.8

D~(B) =-= — and D+(B) =-= —

v ' 0.8 + 0.8 + 0.9 + 0.9 17 v ' 0.8 + 0.8 + 0.9 + 0.9 17

Therefore, D{B) ■■

10 15 17' 17

Note that B is a g-IVFMFRC, and in this case the density belongs to O. In general, there is no upper bound of D(A) if A is any IVFMFRC, but it is bounded when it is g-IVFMFRC.

Theorem 7.3 If A is a g-IVFMFRC, then D(A) 6 V.

Proof It is obvious that D~(A) < D+(A). We shall prove that D*(A) < 1. Since A is a g-IVFMFRC, at < r+(i) A c+A(j) for all i, j. Therefore,

That is,

Hence D+(A) < 1. Thus, D(A) e D.

Definition 7.3 An IVFMFRC A is called balanced ifD~(S) < D~(A) and D+(S) < D+(A)for all sub-IVFMFRCs S of A.

Let A = [0.6,0.9] [0.7,1.0]

[0.8,1.0] [0.5,0.6] [0.5,0.5] [0.5,0.8] [0.7,0.9] [0.3,0.4]

[0.8,1.0]

[0.6,0.9] [[0.5,0.5]],

[0.5,0.6] 5 2 " [0.6,0.9] [ [0.5,0.8]], [0.8,1.0]

53 " [0.7,1.0] [ [0.7,0.9]],

[0.5,0.6]

54 " [0.7,1.0] [ [0.3,0.4]], be the sub-IVFMFRCs of A.

DCSi) = [5/9,5/9], D(S2) = [5/6,4/3], D(S3) = [7/10,9/10], D(S4) = [1/2,2/3], D(A) = [20/31,26/31]. Note that D~(S2) > D~(A), that is, A is not a balanced IVFMFRC. Now, we define a particular type of balanced IVFMFRC.

Definition 7.4 An IVFMFRC A is called strictly balanced ifD~(S) = D~(A) and D+(S) - D+(A)for all sub-IVFMFRCs S of A.

Let B ■■

[0.2,0.6] [0.5,0.7]

[0.4,0.8] [0.2,0.5] [3/25,3/5] [1/10,1/2] [7/50,7/10] [1/10,1/2]

[0.4,0.8] [0.2,0.6] [ [3/25,3/5]],

[0.4,0.8] [0.5,0.7] [ [7/50,7/10]] be the sub-IVFMFRCs of B.

Now, D(Si) = [1/5,1], D(S2) = D(B) = [1/5,1].

Since D(Si) = D(S2) = D(S3) = D(S4) strictly balanced IVFMFRC.

[0.2,0.5] [0.2,0.6] [[1/10,1/2]],

[0.2,0.5] [0.5,0.7] [[1/10,1/2]]

[1/5,1], D(S3) = [1/5,1], D(S4) = [1/5,1], D(S) = [1/5,1], therefore B is a

Theorem 7.4 If A is a complete g-TVFMFRC, then D~(A) < D+(A) and D+(A) = 1.

Proof Let A be a complete g-IVFMFRC. Therefore, ay = rA(i) A ca(j), i.e., a^ = r~A{i) A c~A<J)311(1 atj = rA^ A C1U)for aU U J-Now,

D+(A) =

Zijr^AcXU)

and D"(A) =

Thus, D+(A) = 1 and obviously, ZT(A) < D+(A).

[0.2,0.5] [0.4,0.6]

Let A =

[0.3,0.7] [0.1,0.7]

[0.2,0.5] [0.3,0.6] [0.1,0.5] [0.1,0.6]

[0.2,0.5]

[0.3,0.7] [[0.2,0.5]],

Fuzzy Inf. Eng. (2015)7: 335-368_349

[0.4,0.6] S2~ [0.3,0.7] [[0.3,0.6]], [0.2,0.5] [0.1,0.7] [[0.1,0.5]], [0.4,0.6] 54 ~ [0.1,0.7] [[0.1,0.6]]. Then D(S{) = [2/5,1], D(S2) = [1/2,1], D(53) = [1/5,1], D(S4) = [1/6,1], D(A) = [7/22,1].

Note that A is a complete g-IVFMFRC and D+(St) = D+(A) = 1 for all i = l,2,3,4andD"(S,)< 1.

Theorem 7.5 Let A = [rA (¡')J [c,i (y')J [ay JmXn be an IVFMFRC and Ac be its complement. Then D+(A) + D+(AC) > 1.

Zat 2(1 - ajj) Proof By definition D*(A) = 'J-— and D+(AC) = 'J

Zr+A(i)AC+AU) V 7 £(1 - r-(i)) A (1 - C~(j))

Now, rA(i) a cA(j) < 1 for a fixed i and j.

Therefore, £ r+A(i) A cA(j) < mn. hi

Similarly, 2(1 - A (1 - c"(;)) < mn.

+ (1 " «"))

Thus, D+(A) + D+(AC) > -— + --= --> 1.

mn mn mn

8. Application of IVFMFRC

In this section, an application of IVFMFRC is described. IVFMFRC can be successfully used to represent an image. Also, IVFMFRC is used to image contraction. Any plane image can be treated as a matrix. The grey value of a pixel is consider as an element of the corresponding matrix. The grey value depends on the colour of a pixel and it can be represented within the unit interval [0,1]. Thus, the value of each element of the matrix lies on [0,1]. Sometimes, it is difficult to measure the grey value of a pixel due to hesiness of the image or defect of the instrument or improper snap, etc. In this case, the grey value can be represented as a sub-interval of [0,1]. Again, certain portion of the entire image may not be significant, i.e., that portion contains only background colour. While the other portion may be highly significant and some another portion is less significant, etc. Depending on this analogy, one can grade each row and each column of the matrix corresponding to the given image. Thus, the rows and columns are also uncertain and hence they have some membership values. This type of image can be represented as an IVFMFRC. Table 1 represents the IVFMFRC for the image shown in Fig.l. In this table, the first row and first column represent the membership values of the IVFMFRC.

Fig. 1 A sample image

Table 1: IVFMFRC for the image of Figure 1.

Columns 1 through 4

[0.99,0.99] [0.99,0.99] [0.99,0.99] [0.98,0.99]

[0 99, 0 99] [0 04 0 05] [0 02 0 03] [0 03 0 04] [0 00 0 01]

[0 28, 0 30] [0 02 0 03] [0 01 0 02] [0 00 0 00] [0 00 0 01]

[0 25, 0 27] [0 05 0 06] [0 00 0 01] [0 02 0 03] [0 06 0 07]

[0 24, 0 25] [0 01 0 02] [0 00 0 01] [0 94 0 95] [1 00 1 00]

[0 20, 0 22] [0 03 0 04] [0 13 0 14] [0 98 0 99] [0 98 0 99]

[0 11, 0 12] [0 09 0 10] [0 03 0 04] [1 00 1 00] [0 98 0 99]

[0 10, 0 11] [0 01 0 02] [0 03 0 04] [0 91 0 92] [0 96 0 97]

[0 08, 0 10] [0 00 0 01] [0 00 0 01] [0 98 0 99] [0 95 0 96]

[0 07, 0 08] [0 01 0 02] [0 00 0 00] [0 99 1 00] [0 93 0 94]

[0 05, 0 06] [0 01 0 02] [0 00 0 01] [0 95 0 96] [0 95 0 96]

[0 04, 0 05] [0 01 0 02] [0 00 0 01] [0 97 0 98] [0 91 0 92]

[0 02, 0 04] [0 01 0 02] [0 01 0 02] [0 92 0 93] [0 12 0 13]

[0 02, 0 05] [0 00 0 01] [0 00 0 00] [0 87 0 87] [0 07 0 08]

[0 02, 0 04] [0 03 0 04] [0 00 0 00] [0 83 0 84] [0 11 0 12]

[0 02, 0 04] [0 00 0 01] [0 03 0 04] [0 89 0 90] [0 04 0 05]

[0 02, 0 03] [0 01 0 02] [0 00 0 00] [0 90 0 91] [0 14 0 15]

[0 00, 0 02] [0 03 0 04] [0 02 0 03] [0 96 0 97] [0 90 0 91]

[0 00, 0 00] [0 00 0 01] [0 00 0 01] [0 98 0 99] [0 94 0 95]

[0 00, 0 00] [0 01 0 02] [0 02 0 03] [0 98 0 99] [0 91 0 92]

[0 00, 0 00] [0 00 0 01] [0 00 0 00] [0 92 0 93] [0 96 0 97]

[0 01, 0 01] [0 01 0 02] [0 00 0 00] [0 98 0 99] [0 94 0 95]

[0 01, 0 02] [0 00 0 00] [0 00 0 00] [0 93 0 94] [0 99 1 00]

[0 01, 0 02] [0 00 0 01] [0 00 0 01] [1 00 1 00] [0 98 0 99]

[0 03, 0 04] [0 00 0 00] [0 02 0 03] [0 98 0 99] [0 99 1 00]

[0 02, 0 05] [0 00 0 00] [0 00 0 00] [0 98 0 99] [0 97 0 98]

[0 15, 0 20] [0 00 0 01] [0 00 0 00] [0 98 0 99] [0 96 0 97]

[0 25, 0 30] [0 00 0 01] [0 00 0 00] [0 99 1 00] [0 99 1 00]

[0 32, 0 40] [0 01 0 02] [0 00 0 01] [0 99 1 00] [0 99 1 00]

[0 40, 0 43] [0 00 0 01] [0 00 0 01] [0 99 1 00] [0 99 1 00]

[0 44, 0 48] [0 00 0 01] [0 00 0 00] [1 00 1 00] [1 00 1 00]

[0 52, 0 54] [0 00 0 01] [0 00 0 00] [0 99 1 00] [0 99 1 00]

[0 66, 0 70] [0 00 0 00] [0 00 0 00] [0 99 1 00] [0 99 1 00]

[0 78, 0 82] [0 00 0 00] [0 02 0 03] [0 98 0 99] [1 00 1 00]

[0 84, 0 85] [0 01 0 02] [0 00 0 00] [0 09 0 10] [0 00 0 00]

[0 88, 0 90] [0 02 0 03] [0 07 0 08] [0 00 0 00] [0 00 0 00]

[0 85, 0 88] [0 00 0 00] [0 03 0 04] [0 00 0 00] [0 01 0 02]

[0 86 0 87]|[0 07 0 08] [0 02 0 03] [0 00, 0 01] [0 00, 0 00]

[0 99 0 99] | [1 00 1 00] [0 91 0 92] [1 00, 1 00] [0 99, 1 00]

Columns 5 through 8

[0 98 0 99] [0 97 0 98] [0 97, 0 98] [0 58, 0 60]

[0 99 0 99] [0 03 0 04] [0 02 0 03] [0 03, 0 04] [0 00, 0 01]

[0 28 0 30] [0 04 0 05] [0 02 0 03] [0 00, 0 01] [0 00, 0 01]

[0 25 0 27] [0 02 0 03] [0 02 0 03] [0 00, 0 00] [0 00, 0 01]

[0 24 0 25] [0 99 1 00] [0 98 0 99] [0 92, 0 93] [0 94, 0 95]

[0 20 0 22] [0 97 0 98] [0 96 0 97] [0 90, 0 91] [0 85, 0 86]

[0 11 0 12] [0 96 0 97] [0 83 0 84] [0 83, 0 84] [0 08, 0 09]

[0 10 0 11] [0 89 0 90] [0 83 0 84] [0 05, 0 06] [0 01, 0 02]

[0 08 0 10] [0 88 0 89] [0 04 0 05] [0 05, 0 06] [0 00, 0 01]

[0 07 0 08] [0 89 0 90] [0 08 0 09] [0 00, 0 01] [0 01, 0 02]

[0 05 0 06] [0 06 0 07] [0 05 0 06] [0 07, 0 08] [0 04, 0 05]

[0 04 0 05] [0 15 0 16] [0 12 0 13] [0 00, 0 00] [0 01, 0 02]

[0 02 0 04] [0 06 0 07] [0 01 0 02] [0 00, 0 01] [0 01, 0 02]

[0 02 0 05] [0 04 0 05] [0 03 0 04] [0 02, 0 03] [0 02, 0 03]

[0 02 0 04] [0 01 0 02] [0 00 0 01] [0 00, 0 00] [0 06, 0 07]

[0 02 0 04] [0 08 0 09] [0 12 0 13] [0 04, 0 05] [0 00, 0 01]

[0 02 0 03] [0 04 0 05] [0 04 0 05] [0 04, 0 05] [0 06, 0 07]

[0 00 0 02] [0 10 0 11] [0 08 0 09] [0 00, 0 00] [0 06, 0 07]

[0 00 0 00] [0 10 0 11] [0 09 0 10] [0 01, 0 02] [0 01, 0 02]

[0 00 0 00] [0 89 0 90] [0 00 0 01] [0 12, 0 13] [0 01, 0 02]

[0 00 0 00] [0 90 0 91] [0 12 0 13] [0 01, 0 02] [0 01, 0 02]

[0 01 0 01] [0 93 0 94] [0 90 0 91] [0 01, 0 02] [0 07, 0 08]

[0 01 0 02] [0 98 0 99] [0 95 0 96] [0 92, 0 93] [0 13, 0 14]

[0 01 0 02] [0 94 0 95] [0 97 0 98] [0 98, 0 99] [0 93, 0 94]

[0 03 0 04] [0 99 1 00] [0 99 1 00] [0 98, 0 99] [0 98, 0 99]

[0 02 0 05] [0 95 0 96] [0 95 0 96] [0 95, 0 96] [0 94, 0 95]

[0 15 0 20] [0 94 0 95] [0 93 0 94] [0 94, 0 95] [0 94, 0 95]

[0 25 0 30] [0 97 0 98] [0 99 1 00] [0 98, 0 99] [0 99, 1 00]

[0 32 0 40] [0 98 0 99] [1 00 1 00] [0 98, 0 99] [0 99, 1 00]

[0 40 0 43] [0 98 0 99] [0 99 1 00] [0 98, 0 99] [1 00, 1 00]

[0 44 0 48] [0 98 0 99] [0 99 1 00] [0 99, 1 00] [0 99, 1 00]

[0 52 0 54] [0 98 0 99] [1 00 1 00] [0 99, 1 00] [0 99, 1 00]

[0 66 0 70] [0 98 0 99] [0 99 1 00] [0 98, 0 99] [1 00, 1 00]

[0 78 0 82] [1 00 1 00] [0 98 0 99] [0 99, 1 00] [0 98, 0 99]

[0 84 0 85] [0 00 0 00] [0 00 0 00] [0 03, 0 04] [0 01, 0 02]

[0 88 0 90] [0 03 0 04] [0 00 0 00] [0 00, 0 00] [0 00, 0 00]

[0 85 0 88] [0 02 0 03] [0 00 0 00] [0 00, 0 00] [0 03, 0 04]

[0 86 0 87] [0 00 0 00] [0 00 0 00] [0 01, 0 02] [0 00, 0 01]

[0 99 0 99] [1 00 1 00] [1 00 1 00] [1 00, 1 00] [0 98, 0 99]

Columns 9 through 12

[0 58 0 60] [0 55 0 57] [0 55, 0 56] [0 55, 0 56]

[0 99 0 99] [0 00 0 01] [0 00 0 00] [0 03, 0 04] [0 01, 0 02]

[0 28 0 30] [0 00 0 01] [0 02 0 03] [0 01, 0 02] [0 01, 0 02]

[0 25 0 27] [0 00 0 00] [0 00 0 00] [0 01, 0 02] [0 00, 0 00]

[0 24 0 25] [0 96 0 97] [0 95 0 96] [0 91, 0 92] [0 14, 0 15]

[0 20 0 22] [0 89 0 90] [0 04 0 05] [0 15, 0 16] [0 05, 0 06]

[0 11, 0 12] I [0 15 0 16] [0 05 0 06] [0 07 0 08] [0 09 0 10]

[0 10, 0 11] 1 [0 00 0 00] [0 07 0 07] [0 00 0 01] [0 06 0 07]

[0 08, 0 10] 1 [0 02 0 03] [0 03 0 04] [0 06 0 07] [0 00 0 00]

[0 07, 0 08] I [0 02 0 03] [0 00 0 01] [0 00 0 01] [0 01 0 02]

[0 05, 0 06] | [0 04 0 05] [0 02 0 03] [0 01 0 02] [0 02 0 03]

[0 04, 0 05] | [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

[0 02, 0 04] | [0 04 0 05] [0 02 0 03] [0 03 0 04] [0 03 0 04]

[0 02, 0 05] I [0 01 0 02] [0 00 0 01] [0 01 0 02] [0 03 0 04]

[0 02, 0 04] | [0 01 0 02] [0 02 0 03] [0 07 0 08] [0 13 0 14]

[0 02, 0 04] I [0 01 0 02] [0 01 0 02] [0 05 0 06] [0 07 0 08]

[0 02, 0 03] I [0 01 0 02] [0 02 0 03] [0 07 0 08] [0 10 0 11]

[0 00, 0 02] | [0 01 0 02] [0 03 0 04] [0 04 0 05] [0 11 0 12]

[0 00, 0 00] | [0 00 0 00] [0 02 0 03] [0 02 0 03] [0 22 0 23]

[0 00, 0 00] I [0 03 0 04] [0 03 0 04] [0 04 0 05] [0 06 0 07]

[0 00, 0 00] 1 [0 00 0 01] [0 00 0 00] [0 06 0 07] [0 14 0 15]

[0 01, 0 01] | [0 04 0 05] [0 05 0 06] [0 05 0 06] [0 14 0 15]

[0 01, 0 02] I [0 10 0 11] [0 00 0 00] [0 16 0 17] [0 15 0 16]

[0 01, 0 02] | [0 94 0 95] [0 21 0 22] [0 07 0 08] [0 18 0 19]

[0 03, 0 04] | [0 97 0 98] [0 94 0 95] [0 94 0 95] [1 00 1 00]

[0 02, 0 05] I [0 93 0 94] [0 94 0 95] [0 97 0 98] [1 00 1 00]

[0 15, 0 20] I [0 93 0 94] [0 96 0 97] [1 00 1 00] [1 00 1 00]

[0 25, 0 30] I [0 98 0 99] [0 99 1 00] [1 00 1 00] [1 00 1 00]

[0 32, 0 40] | [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 40, 0 43] I [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 44, 0 48] I [0 99 1 00] [0 98 0 99] [1 00 1 00] [1 00 1 00]

[0 52, 0 54] | [0 98 0 99] [0 97 0 98] [0 98 0 99] [1 00 1 00]

[0 66, 0 70] I [0 98 0 99] [0 96 0 97] [0 98 0 99] [0 62 0 63]

[0 78, 0 82] | [0 98 0 99] [0 98 0 99] [0 99 1 00] [1 00 1 00]

[0 84, 0 85] I [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 01 0 02]

[0 88, 0 90] | [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 01]

[0 85, 0 88] I [0 00 0 01] [0 00 0 01] [0 00 0 01] [0 00 0 01]

[0 86, 0 87] | [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

[0 99, 0 99] | [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

Columns 1!: through 16

[0 50 0 52] [0 48 0 49] [0 47 0 50] [0 40 0 44]

[0 99, 0 99] I [0 01 0 02] [0 00 0 00] [0 00 0 00] [0 01 0 02]

[0 28, 0 30] 1 [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

[0 25, 0 27] | [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

[0 24, 0 25] I [0 10 0 11] [0 07 0 08] [0 01 0 02] [0 11 0 12]

[0 20, 0 22] | [0 03 0 04] [0 04 0 05] [0 13 0 14] [0 02 0 03]

[0 11, 0 12] I [0 02 0 03] [0 00 0 00] [0 07 0 08] [0 03 0 04]

[0 10, 0 11] 1 [0 00 0 00] [0 00 0 00] [0 07 0 08] [0 14 0 15]

[0 08, 0 10] I [0 05 0 06] [0 06 0 07] [0 03 0 04] [0 06 0 07]

[0 07, 0 08] I [0 00 0 00] [0 02 0 03] [0 05 0 06] [0 05 0 06]

[0 05, 0 06] | [0 01 0 02] [0 02 0 03] [0 02 0 03] [0 01 0 02]

[0 04, 0 05] I [0 04 0 05] [0 05 0 06] [0 05 0 06] [0 05 0 06]

[0 02, 0 04] | [0 14 0 15] [0 10 0 11] [0 09 0 10] [0 09 0 10]

[0 02, 0 05] | [0 24 0 25] [0 22 0 23] [0 22 0 23] [0 23 0 24]

[0 02, 0 04] I [0 43 0 44] [0 41 0 42] [0 41 0 42] [0 43 0 44]

[0 02, 0 04] | [0 37 0 38] [0 27 0 28] [0 21 0 22] [0 19 0 20]

[0 02, 0 03] I [0 41 0 42] [0 27 0 28] [0 16 0 17] [0 11 0 12]

[0 00 0 02] | [0 42, 0 43] [0 27, 0 28] [0 19, 0 20] [0 06, 0 07]

[0 00 0 00] | [0 44, 0 45] [0 25, 0 26] [0 09, 0 10] [0 08, 0 09]

[0 00 0 00] I [0 42, 0 43] [0 29, 0 30] [0 16, 0 17] [0 04, 0 05]

[0 00 0 00] I [0 43, 0 44] [0 21, 0 22] [0 07, 0 08] [0 04, 0 05]

[0 01 0 01] 1 [0 32, 0 33] [0 34, 0 35] [0 14, 0 15] [0 04, 0 05]

[0 01 0 02] | [0 38, 0 39] [0 26, 0 27] [0 19, 0 20] [0 07, 0 08]

[0 01 0 02] | [0 44, 0 45] [0 24, 0 25] [0 20, 0 21] [0 21, 0 22]

[0 03 0 04] | [0 49, 0 50] [0 52, 0 53] [0 29, 0 30] [0 37, 0 38]

[0 02 0 05] | [0 68, 0 69] [0 78, 0 79] [0 77, 0 78] [0 79, 0 80]

[0 15 0 20] I [0 77, 0 78] [0 91, 0 92] [0 92, 0 93] [0 94, 0 95]

[0 25 0 30] I [0 79, 0 80] [0 94, 0 95] [0 96, 0 97] [0 98, 0 99]

[0 32 0 40] | [0 80, 0 81] [0 95, 0 96] [0 99, 1 00] [1 00, 1 00]

[0 40 0 43] | [0 79, 0 80] [0 94, 0 95] [0 98, 0 99] [0 98, 0 99]

[0 44 0 48] I [0 77, 0 78] [0 92, 0 93] [0 96, 0 97] [0 97, 0 98]

[0 52 0 54] | [0 72, 0 73] [0 88, 0 89] [0 91, 0 92] [0 93, 0 94]

[0 66 0 70] | [0 74, 0 75] [0 76, 0 77] [0 78, 0 79] [0 77, 0 78]

[0 78 0 82] I [1 00, 1 00] [1 00, 1 00] [1 00, 1 00] [1 00, 1 00]

[0 84 0 85] I [0 01, 0 02] [0 02, 0 03] [0 02, 0 03] [0 01, 0 02]

[0 88 0 90] | [0 00, 0 01] [0 00, 0 01] [0 00, 0 01] [0 00, 0 01]

[0 85 0 88] I [0 00, 0 01] [0 00, 0 01] [0 00, 0 01] [0 00, 0 01]

[0 86 0 87] | [0 00, 0 00] [0 00, 0 00] [0 00, 0 00] [0 00, 0 00]

[0 99 0 99] I [1 00, 1 00] [1 00, 1 00] [1 00, 1 00] [1 00, 1 00]

Columns 17 through 20

[0 35, 0 40] [0 30, 0 32] [0 27, 0 31] [0 27, 0 30]

[0 99 0 99] I [0 00, 0 00] [0 03, 0 04] [0 00, 0 00] [0 05, 0 06]

[0 28 0 30] I [0 00, 0 00] [0 00, 0 00] [0 03, 0 04] [0 00, 0 00]

[0 25 0 27] | [0 02, 0 03] [0 00, 0 00] [0 00, 0 00] [0 05, 0 06]

[0 24 0 25] | [0 03, 0 04] [0 85, 0 86] [0 88, 0 89] [0 87, 0 88]

[0 20 0 22] | [0 10, 0 11] [0 09, 0 10] [0 03, 0 04] [0 83, 0 84]

[0 11 0 12] I [0 03, 0 04] [0 03, 0 04] [0 03, 0 04] [0 05, 0 06]

[0 10 0 11] 1 [0 07, 0 08] [0 11, 0 12] [0 04, 0 05] [0 00, 0 00]

[0 08 0 10] I [0 10, 0 11] [0 05, 0 06] [0 03, 0 04] [0 02, 0 03]

[0 07 0 08] I [0 08, 0 09] [0 03, 0 04] [0 05, 0 06] [0 00, 0 00]

[0 05 0 06] | [0 09, 0 10] [0 03, 0 04] [0 05, 0 06] [0 00, 0 00]

[0 04 0 05] | [0 05, 0 06] [0 03, 0 04] [0 10, 0 11] [0 06, 0 07]

[0 02 0 04] | [0 12, 0 13] [0 10, 0 11] [0 12, 0 13] [0 12, 0 13]

[0 02 0 05] I [0 26, 0 27] [0 27, 0 28] [0 22, 0 23] [0 25, 0 26]

[0 02 0 04] | [0 34, 0 35] [0 43, 0 44] [0 39, 0 40] [0 38, 0 39]

[0 02 0 04] I [0 11, 0 12] [0 14, 0 15] [0 16, 0 17] [0 11, 0 12]

[0 02 0 03] I [0 09, 0 10] [0 01, 0 02] [0 08, 0 09] [0 09, 0 10]

[0 00 0 02] | [0 05, 0 06] [0 06, 0 07] [0 07, 0 08] [0 05, 0 06]

[0 00 0 00] I [0 05, 0 06] [0 03, 0 04] [0 04, 0 05] [0 05, 0 06]

[0 00 0 00] I [0 03, 0 04] [0 00, 0 01] [0 00, 0 01] [0 04, 0 05]

[0 00 0 00] I [0 03, 0 04] [0 01, 0 02] [0 02, 0 03] [0 05, 0 06]

[0 01 0 01] | [0 04, 0 05] [0 05, 0 06] [0 06, 0 07] [0 10, 0 11]

[0 01 0 02] | [0 09, 0 10] [0 14, 0 15] [0 18, 0 19] [0 26, 0 27]

[0 01 0 02] | [0 30, 0 31] [0 37, 0 38] [0 44, 0 45] [0 56, 0 57]

[0 03 0 04] | [0 61, 0 62] [0 65, 0 66] [0 72, 0 73] [0 83, 0 84]

[0 02 0 05] I [0 83, 0 84] [0 86, 0 87] [0 90, 0 91] [0 92, 0 93]

[0 15 0 20] I [0 95, 0 96] [0 96, 0 97] [0 97, 0 98] [0 98, 0 99]

[0 25 0 30] I [0 98, 0 99] [0 98, 0 99] [0 98, 0 99] [0 98, 0 99]

[0 32,0 40] I [0 98 0 99] [0 98 0 99] [0 98 0 99] [0 97 0 98]

[0 40,0 43] | [0 97 0 98] [0 97 0 98] [0 97 0 98] [0 97 0 98]

[0 44,0 48] I [0 95 0 96] [0 94 0 95] [0 96 0 97] [0 96 0 97]

[0 52,0 54] | [0 92 0 93] [0 92 0 93] [0 94 0 95] [0 94 0 95]

[0 66,0 70] | [0 78 0 79] [0 79 0 80] [0 79 0 80] [0 79 0 80]

[0 78,0 82] | [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 84,0 85] I [0 01 0 02] [0 02 0 03] [0 02 0 03] [0 02 0 03]

[0 88,0 90] I [0 00 0 01] [0 00 0 01] [0 00 0 01] [0 00 0 01]

[0 85,0 88] | [0 00 0 01] [0 00 0 01] [0 00 0 01] [0 00 0 01]

[0 86,0 87] I [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

[0 99,0 99] I [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

Columns 21 through 24

[0 25 0 27] [0 20 0 22] [0 12 0 16] [0 10 0 12]

[0 99,0 99] | [0 02 0 03] [0 06 0 07] [0 04 0 05] [0 04 0 05]

[0 28,0 30] I [0 02 0 03] [0 03 0 04] [0 04 0 05] [0 04 0 05]

[0 25,0 27] | [0 00 0 00] [0 02 0 03] [0 05 0 06] [0 02 0 03]

[0 24,0 25] | [0 94 0 95] [0 97 0 98] [0 98 0 99] [0 97 0 98]

[0 20,0 22] I [0 88 0 89] [0 93 0 94] [0 96 0 97] [0 99 1 00]

[0 11,0 12] I [0 07 0 08] [0 85 0 86] [0 92 0 93] [0 98 0 99]

[0 10,0 11] 1 [0 08 0 09] [0 02 0 03] [0 89 0 90] [0 95 0 96]

[0 08,0 10] I [0 00 0 00] [0 03 0 04] [0 11 0 12] [0 89 0 90]

[0 07,0 08] I [0 03 0 04] [0 01 0 02] [0 12 0 13] [0 92 0 93]

[0 05,0 06] I [0 01 0 02] [0 03 0 04] [0 00 0 00] [0 20 0 21]

[0 04,0 05] | [0 03 0 04] [0 10 0 11] [0 10 0 11] [0 07 0 08]

[0 02,0 04] I [0 05 0 06] [0 06 0 07] [0 16 0 17] [0 17 0 18]

[0 02,0 05] I [0 25 0 26] [0 20 0 21] [0 21 0 22] [0 22 0 23]

[0 02,0 04] | [0 43 0 44] [0 43 0 44] [0 41 0 42] [0 44 0 45]

[0 02,0 04] | [0 09 0 10] [0 12 0 13] [0 14 0 15] [0 24 0 25]

[0 02,0 03] I [0 05 0 06] [0 07 0 08] [0 10 0 11] [0 22 0 23]

[0 00,0 02] | [0 03 0 04] [0 07 0 08] [0 12 0 13] [0 24 0 25]

[0 00,0 00] | [0 03 0 04] [0 05 0 06] [0 17 0 18] [0 37 0 38]

[0 00,0 00] I [0 05 0 06] [0 10 0 11] [0 32 0 33] [0 59 0 60]

[0 00,0 00] I [0 10 0 11] [0 23 0 24] [0 49 0 50] [0 73 0 74]

[0 01,0 01] 1 [0 22 0 23] [0 42 0 43] [0 67 0 68] [0 81 0 82]

[0 01,0 02] | [0 43 0 44] [0 64 0 65] [0 84 0 85] [0 91 0 92]

[0 01,0 02] | [0 70 0 71] [0 83 0 84] [0 94 0 95] [0 96 0 97]

[0 03,0 04] | [0 90 0 91] [0 90 0 91] [0 96 0 97] [0 97 0 98]

[0 02,0 05] | [0 95 0 96] [0 97 0 98] [0 99 1 00] [0 98 0 99]

[0 15,0 20] I [0 99 1 00] [0 99 1 00] [1 00 1 00] [0 99 1 00]

[0 25,0 30] I [0 98 0 99] [0 98 0 99] [0 98 0 99] [0 99 1 00]

[0 32,0 40] | [0 97 0 98] [0 97 0 98] [0 98 0 99] [0 99 1 00]

[0 40,0 43] I [0 96 0 97] [0 97 0 98] [0 98 0 99] [1 00 1 00]

[0 44,0 48] I [0 94 0 95] [0 95 0 96] [0 97 0 98] [0 97 0 98]

[0 52,0 54] | [0 94 0 95] [1 00 1 00] [0 94 0 95] [0 93 0 94]

[0 66,0 70] | [0 79 0 80] [1 00 1 00] [0 79 0 80] [0 78 0 79]

[0 78,0 82] | [1 00 1 00] [0 00 0 01] [1 00 1 00] [1 00 1 00]

[0 84,0 85] I [0 02 0 03] [0 00 0 00] [0 02 0 03] [0 02 0 03]

[0 88,0 90] | [0 00 0 01] [0 00 0 01] [0 00 0 01] [0 00 0 01]

[0 85,0 88] I [0 00 0 01] [0 00 0 00] [0 00 0 01] [0 00 0 01]

[0 86,0 87] | [0 00 0 00] [1 00 1 00] [0 00 0 00] [0 00 0 00]

[0 99,0 99] I [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

Columns 25 through 28

[0.10,0.12] [0.09,0.10] [0.08,0.10] [0.07,0.09]

[0.99, 0.99] [0.28,0.30] [0.25,0.27] [0.24,0.25] [0.20,0.22] [0.11,0.12] [0.10,0.11] [0.08,0.10] [0.07,0.08] [0.05,0.06] [0.04,0.05] [0.02,0.04] [0.02,0.05] [0.02,0.04] [0.02,0.04] [0.02,0.03] [0.00,0.02] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.01,0.01] [0.01,0.02] [0.01,0.02] [0.03,0.04] [0.02,0.05] [0.15,0.20] [0.25,0.30] [0.32,0.40] [0.40,0.43] [0.44,0.48] [0.52,0.54] [0.66,0.70] [0.78,0.82] [0.84,0.85] [0.88,0.90] [0.85,0.88] [0.86,0.87] [0.99, 0.99]

[0.99, 0.99] [0.28,0.30] [0.25,0.27] [0.24,0.25] [0.20,0.22] [0.11,0.12] [0.10,0.11]

[0.04,0.05] [0.03,0.04] [0.00,0.01] [0.99,1.00] [0.99,1.00] [0.99,1.00] [0.98,0.99] [0.97,0.98] [0.97,0.98] [0.96,0.97] [0.94,0.95] [0.20,0.21] [0.33,0.34] [0.49,0.50] [0.38,0.39] [0.40,0.41] [0.66,0.67] [0.73,0.74] [0.80,0.81] [0.85,0.86] [0.89,0.90] [0.90,0.91] [0.91,0.92] [0.93,0.94] [0.97,0.98] [0.98,0.99] [0.99,1.00] [0.99,1.00] [1.00,1.00] [0.96,0.97] [0.90,0.91] [0.75,0.76] [1.00,1.00] [0.01,0.02] [0.00,0.01] [0.00,0.01] [0.00,0.00] [1.00,1.00]

[0.00,0.00] [0.00,0.01] [0.00,0.00] [0.94,0.95] [0.96,0.97] [0.97,0.98] [0.97,0.98]

[0.04,0.05] [0.03,0.04] [0.00,0.00] [0.96, 0.97] [0.97, 0.98] [0.99, 1.00] [0.99, 1.00] [0.99, 1.00] [0.99, 1.00] [0.99, 1.00] [0.99, 1.00] [1.00,1.00] [1.00,1.00] [0.61,0.62] [0.65, 0.66] [0.76,0.77] [0.82,0.83] [0.87,0.88] [0.90,0.91] [0.91,0.92] [0.90,0.91] [0.91,0.92] [0.90,0.91] [0.92,0.93] [0.95, 0.96] [0.96, 0.97] [0.98, 0.99] [0.98, 0.99] [0.98, 0.99] [0.95, 0.96] [0.89, 0.90] [0.74,0.75] [1.00,1.00] [0.01,0.02] [0.00,0.01] [0.00,0.01] [0.00,0.00] [1.00,1.00]

[0.04,0.06]

[0.00,0.00] [0.00,0.00] [0.00,0.00] [0.97, 0.98] [0.98, 0.99] [0.99, 1.00] [0.98, 0.99]

[0.02,0.03] [0.01,0.02] [0.00,0.00] [0.94,0.95] [0.95,0.96] [0.97,0.98] [0.99,1.00] [0.99,1.00] [0.96,0.97] [1.00,1.00] [1.00,1.00] [1.00,1.00] [1.00,1.00] [0.61,0.62] [0.83,0.84] [0.87,0.88] [0.93,0.94] [0.96,0.97] [0.97,0.98] [0.97,0.98] [0.96,0.97] [0.96,0.97] [0.95,0.96] [0.96,0.97] [0.97,0.98] [0.98,0.99] [0.96,0.97] [0.96,0.97] [0.96,0.97] [0.95,0.96] [0.90,0.91] [0.76,0.77] [1.00,1.00] [0.02,0.03] [0.00,0.01] [0.00,0.01] [0.00,0.00] [1.00,1.00]

[0.02,0.04]

[0.00,0.00] [0.00,0.01] [0.00,0.00] [0.99,1.00] [0.98,0.99] [0.99,1.00] [0.99,1.00]

[0.01,0.02] [0.01,0.02] [0.00,0.00] [0.92,0.93] [0.94,0.95] [0.96,0.99] [0.97,0.98] [0.99,1.00] [1.00,1.00] [0.99,1.00] [1.00,1.00] [1.00,1.00] [1.00,1.00] [0.69,0.70] [0.85,0.86] [0.95,0.96] [0.97,0.98] [0.98,0.99] [0.98,0.99] [0.98,0.99] [0.98,0.99] [0.97,0.98] [0.97,0.98] [0.98,0.99] [0.98,0.99] [0.97,0.98] [0.95,0.96] [0.95,0.96] [0.95,0.96] [0.93,0.94] [0.90,0.91] [0.76,0.77] [1.00,1.00] [0.02,0.03] [0.00,0.01] [0.00,0.01] [0.00,0.00] [1.00,1.00]

[0.01,0.02]

[0.00,0.00] [0.01,0.02] [0.01,0.02] [1.00,1.00] [0.99,1.00] [0.99,1.00] [0.99,1.00]

Columns 2 9 through 32 [0.05,0.06]

[0 08,0 10] 1 [0 98 0 99] [0 97 0 98] [0 98 0 99] [0 98 0 99]

[0 07,0 08] 1 [1 00 1 00] [0 98 0 99] [1 00 1 00] [0 97 0 98]

[0 05,0 06] 1 [0 98 0 99] [1 00 1 00] [0 96 0 97] [1 00 1 00]

[0 04,0 05] 1 [1 00 1 00] [1 00 1 00] [0 94 0 95] [0 98 0 99]

[0 02,0 04] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [0 92 0 93]

[0 02,0 05] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 02,0 04] 1 [0 74 0 75] [0 64 0 65] [0 50 0 51] [1 00 1 00]

[0 02,0 04] 1 [0 81 0 82] [0 80 0 81] [0 65 0 66] [1 00 1 00]

[0 02,0 03] 1 [0 94 0 95] [0 85 0 86] [0 70 0 71] [1 00 1 00]

[0 00,0 02] 1 [0 96 0 97] [0 85 0 86] [0 70 0 71] [1 00 1 00]

[0 00,0 00] 1 [0 97 0 98] [0 85 0 86] [0 70 0 71] [1 00 1 00]

[0 00,0 00] 1 [0 96 0 97] [0 84 0 85] [0 69 0 70] [1 00 1 00]

[0 00,0 00] 1 [0 96 0 97] [0 83 0 84] [0 69 0 70] [1 00 1 00]

[0 01,0 01] 1 [0 96 0 97] [0 84 0 85] [0 68 0 69] [1 00 1 00]

[0 01,0 02] 1 [0 96 0 97] [0 84 0 85] [0 69 0 70] [1 00 1 00]

[0 01,0 02] 1 [0 96 0 97] [0 85 0 86] [0 69 0 70] [1 00 1 00]

[0 03,0 04] 1 [0 97 0 98] [0 85 0 86] [0 70 0 71] [1 00 1 00]

[0 02,0 05] 1 [0 97 0 98] [0 85 0 86] [0 70 0 71] [1 00 1 00]

[0 15,0 20] 1 [0 96 0 97] [0 85 0 86] [0 70 0 71] [1 00 1 00]

[0 25,0 30] I [0 95 0 96] [0 83 0 84] [0 69 0 70] [1 00 1 00]

[0 32,0 40] 1 [0 94 0 95] [0 82 0 83] [0 70 0 71] [1 00 1 00]

[0 40,0 43] I [0 93 0 94] [0 81 0 82] [0 70 0 71] [1 00 1 00]

[0 44,0 48] 1 [0 92 0 93] [0 81 0 82] [0 69 0 70] [1 00 1 00]

[0 52,0 54] 1 [0 90 0 91] [0 79 0 80] [0 68 0 69] [1 00 1 00]

[0 66,0 70] 1 [0 77 0 78] [0 68 0 69] [0 60 0 61] [1 00 1 00]

[0 78,0 82] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 84,0 85] I [0 01 0 02] [0 01 0 02] [0 01 0 02] [0 00 0 01]

[0 88,0 90] 1 [0 00 0 01] [0 00 0 01] [0 00 0 00] [0 00 0 00]

[0 85,0 88] I [0 00 0 01] [0 00 0 01] [0 00 0 01] [0 00 0 01]

[0 86,0 87] 1 [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 01]

[0 99,0 99] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

Columns 33 through 36

[0 00 0 01] [0 08 0 09] [0 11 0 13] [0 14 0 15]

[0 99,0 99] 1 [0 01 0 02] [0 00 0 01] [0 00 0 00] [0 00 0 00]

[0 28,0 30] 1 [0 00 0 01] [0 00 0 01] [0 07 0 08] [0 02 0 03]

[0 25,0 27] 1 [0 05 0 06] [0 00 0 01] [0 00 0 00] [0 05 0 06]

[0 24,0 25] 1 [0 99 1 00] [1 00 1 00] [1 00 1 00] [0 95 0 96]

[0 20,0 22] 1 [1 00 1 00] [1 00 1 00] [0 96 0 97] [1 00 1 00]

[0 11,0 12] 1 [1 00 1 00] [0 98 0 99] [1 00 1 00] [0 98 0 99]

[0 10,0 11] 1 [0 99 1 00] [0 98 0 99] [1 00 1 00] [1 00 1 00]

[0 08,0 10] 1 [0 98 0 99] [1 00 1 00] [0 96 0 97] [1 00 1 00]

[0 07,0 08] 1 [0 99 1 00] [0 99 1 00] [1 00 1 00] [1 00 1 00]

[0 05,0 06] 1 [1 00 1 00] [0 99 1 00] [1 00 1 00] [1 00 1 00]

[0 04,0 05] 1 [0 99 1 00] [0 99 1 00] [1 00 1 00] [1 00 1 00]

[0 02,0 04] 1 [1 00 1 00] [0 99 1 00 [1 00 1 00] [1 00 1 00]

[0 02,0 05] 1 [1 00 1 00] [0 99 1 00] [1 00 1 00] [1 00 1 00]

[0 02,0 04] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 02,0 04] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 02,0 03] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 00,0 02] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 00,0 00] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 00 0 00] I [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 00 0 00] | [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 01 0 01] 1 [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 01 0 02] | [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 01 0 02] | [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 03 0 04] | [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 02 0 05] I [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 15 0 20] I [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 25 0 30] | [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 32 0 40] I [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 40 0 43] I [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 44 0 48] | [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 52 0 54] | [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 66 0 70] I [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 78 0 82] | [1 00 1 00] [1 00 1 00] [0 98,0 99] [0 95,0 96]

[0 84 0 85] | [0 02 0 03] [0 00 0 00] [0 00,0 00] [0 05,0 06]

[0 88 0 90] I [0 00 0 00] [0 01 0 02] [0 04,0 05] [0 00,0 00]

[0 85 0 88] I [0 00 0 00] [0 03 0 04] [0 00,0 00] [0 00,0 00]

[0 86 0 87] | [0 03 0 04] [0 01 0 02] [0 00,0 00] [0 02,0 03]

[0 99 0 99] I [0 97 0 98] [0 96 0 97] [1 00,1 00] [1 00,1 00]

Columns 37 through 40

[0 19 0 20] [0 29 0 30] [0 58,0 60] [0 60,0 62]

[0 99 0 99] I [0 02 0 03] [0 04 0 05] [0 00,0 00] [0 00,0 00]

[0 28 0 30] | [0 00 0 00] [0 00 0 00] [0 01,0 02] [0 02,0 03]

[0 25 0 27] I [0 00 0 00] [0 05 0 06] [0 07,0 08] [0 00,0 00]

[0 24 0 25] I [0 02 0 03] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 20 0 22] | [0 00 0 01] [0 06 0 07] [0 00,0 00] [0 01,0 02]

[0 11 0 12] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 10 0 11] 1 [0 01 0 02] [0 00 0 00] [0 03,0 04] [0 02,0 03]

[0 08 0 10] I [0 00 0 00] [0 02 0 03] [0 00,0 00] [0 00,0 01]

[0 07 0 08] | [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 05 0 06] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 04 0 05] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 02 0 04] | [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 02 0 05] | [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 02 0 04] | [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 02 0 04] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 02 0 03] | [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 00 0 02] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 00 0 00] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 00 0 00] | [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 00 0 00] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 01 0 01] 1 [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 01 0 02] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 01 0 02] | [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 03 0 04] | [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 02 0 05] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 15 0 20] | [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 25 0 30] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 32 0 40] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 40 0 43] I [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0.44,0.48]I [0.52,0.54]| [0.66,0.70]I [0.78,0.82]| [0.84,0.85]| [0.88,0.90]| [0.85,0.88]I [0.86,0.87]| [0.99,0.99]|

[0.00,0.00] [0.00,0.00] [0.00,0.00] [0.03,0.04] [0.00,0.00] [0.02,0.03] [0.02,0.03] [0.00,0.00] [0.96,0.97]

[0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.01] [0.03,0.04] [0.00,0.00] [0.03,0.04] [0.04,0.05] [0.97,0.98]

[0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.04,0.05] [0.00,0.00] [0.00,0.00] [0.00,0.00] [1.00,1.00]

[0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.01] [0.00,0.00] [0.01,0.02] [0.04,0.05] [0.00,0.00] [1.00,1.00]

If the membership value of a row or a column is zero, then that row and column may be removed to reduce the size of the matrix/image.

The complement of the image of Figure 1 is shown in Figure 2. This is obtained by finding the complement of the IVFMFRC shown in Table 1.

Fig. 2 Complement image Table 2: IVFMFRC for the image of Figure 2.

Columns 1 through 4

[0.01,0.01] [0.01,0.01] [0.01,0.01] [0.01,0.02]

[0 01, 0 01] [0 95, 0 96] [0 97, 0 98] [0 96, 0 96] [0 99, 0 99]

[0 70, 0 72] [0 97, 0 98] [0 98, 0 99] [1 00, 1 00] [0 99, 1 00]

[0 73, 0 75] [0 94, 0 95] [0 99, 1 00] [0 97, 0 97] [0 93, 0 93]

[0 75, 0 76] [0 97, 0 98] [0 99, 1 00] [0 05, 0 05] [0 00, 0 00]

[0 78, 0 80] [0 96, 0 96] [0 86, 0 87] [0 01, 0 02] [0 01, 0 02]

[0 88, 0 89] [0 90, 0 91] [0 96, 0 96] [0 00, 0 00] [0 01, 0 01]

[0 89, 0 90] [0 98, 0 99] [0 96, 0 96] [0 08, 0 08] [0 03, 0 03]

[0 90, 0 92] [0 99, 0 99] [0 99, 1 00] [0 01, 0 02] [0 04, 0 04]

[0 92, 0 93] [0 98, 0 99] [1 00, 1 00] [0 00, 0 01] [0 06, 0 06]

[0 94, 0 95] [0 98, 1 00] [0 99, 1 00] [0 04, 0 05] [0 04, 0 05]

[0 95, 0 96] [0 98, 0 99] [0 99, 0 99] [0 02, 0 03] [0 08, 0 09]

[0 96, 0 98] [0 98, 0 99] [0 98, 0 99] [0 07, 0 07] [0 87, 0 87]

[0 95, 0 98] [0 99, 1 00] [1 00, 1 00] [0 12, 0 13] [0 92, 0 93]

[0 96, 0 98] [0 96, 0 96] [1 00, 1 00] [0 16, 0 16] [0 88, 0 89]

[0 96, 0 98] [0 99, 1 00] [0 96, 0 97] [0 10, 0 12] [0 95, 0 95]

[0 97, 0 98] [0 98, 0 98] [1 00, 1 00] [0 09, 0 09] [0 85, 0 85]

[0 98, 1 00] [0 96, 0 97] [0 97, 0 97] [0 03, 0 04] [0 09, 0 09]

[1 00, 1 00] [0 99, 0 99] [0 99, 1 00] [0 01, 0 02] [0 05, 0 06]

[1 00, 1 00] [0 98, 0 98] [0 97, 0 97] [0 01, 0 02] [0 08, 0 09]

[1 00, 1 00] [0 99, 1 00] [1 00, 1 00] [0 07, 0 08] [0 03, 0 04]

[0. .99, 0. .99] [0. . 98, .0. .99] [1. .00, .1. .00] [0. .01, .0, .01] [0. .05, .0, .05]

[0. .98, 0. .99] [1. .00, .1. .00] [1. .00, .1. .00] [0. .06, .0, .06] [0. .00, ,0, .01]

[0. .98, 0. .99] [0, ■ 99, .0. .99] [0, ■ 99, .0. .99] [0. .00, .0, .00] [0. .01, .0, .02]

[0. .96, 0. .97] [1. .00, .1. .00] [0. ■ 97, .0. .98] [0. .01, .0, .02] [0. .00, .0, .00]

[0. .95, 0. .98] [1. .00, .1. .00] [1. .00, .1. .00] [0. .01, ,0. .02] [0. .02, ,0. .02]

[0. .80, 0. .85] [0. ■ 99, .1. .00] [1. .00, .1. .00] [0. .01, .0, .02] [0. .03, .0, .04]

[0. .70, 0. .75] [0. ■ 99, .0. .99] [1. .00, .1. .00] [0. .00, .0, .00] [0. .00, .0, .00]

[0. .60, 0. .68] [0. . 98, .0. .99] [0. ■ 99, .0. .99] [0. .00, .0, .01] [0. .00, .0, .00]

[0. .57, 0. .60] [0. ■ 99, .0. .99] [0. .99, .0. .99] [0. .00, .0, .00] [0. .00, ,0, .00]

[0. .52, 0. .56] [0, ■ 99, .0. .99] [1. .00, .1. .00] [0. .00, .0, .00] [0. .00, .0, .00]

[0. .46, 0. .48] [0. ■ 99, .1. .00] [1. .00, .1. .00] [0. .00, .0, .01] [0. .00, .0, .01]

[0. .30, 0. .34] [1. .00, .1. .00] [1. .00, .1. .00] [0. .00, .0, .01] [0. .00, ,0, .01]

[0. .18, 0. .22] [1. .00, .1. .00] [0, ■ 97, .0. .98] [0. .01, .0, .01] [0. .00, .0, .00]

[0. .15, 0. .16] [0. . 98, .0. .98] [1. .00, .1. .00] [0. .90, .0, .91] [1. .00, . 1, .00]

[0. .10, 0. .12] [0. ■ 97, .0. .98] [0. .92, .0. .93] [1. .00, .1. .00] [1. .00, .1. .00]

[0. .15, 0. .18] [1. .00, .1. .00] [0. .96, .0. .97] [1. .00, .1, .00] [0. .98, ,0, .98]

[0. .13, 0. .14] [0, . 92, .0. .93] [0, .97, .0. .97] [0. .99, .0, .99] [1. .00, .1. .00]

[0. .01, 0. .01] [0. .00, .0. .00] [0. .08, .0. .08] [0. .00, .0, .00] [0. .00, .0, .00]

Columns 5 through 8

[0.01,0.02] [0.02,0.03] [0.02,0.03] [0.40,0.42]

[0 01 0 01] [0 96 0 97] [0 97 0 98] [0 960.96] [0 99, 0 99]

[0 70 0 72] [0 95 0 96] [0 97 0 97] [0 99, 0 99] [0 99, 0 99]

[0 73 0 75] [0 97 0 97] [0 97 0 97] [1 00, 1 00] [0 99, 0 99]

[0 75 0 76] [0 00 0 00] [0 01 0 02] [0 07, 0 07] [0 05, 0 05]

[0 78 0 80] [0 02 0 03] [0 03 0 04] [0 09, 0 09] [0 14, 0 14]

[0 88 0 89] [0 03 0 04] [0 16 0 16] [0 16, 0 16] [0 91, 0 91]

[0 89 0 90] [0 10 0 11] [0 16 0 16] [0 94, 0 95] [0 98, 0 98]

[0 90 0 92] [0 11 0 11] [0 95 0 96] [0 94, 0 95] [0 99, 0 99]

[0 92 0 93] [0 10 0 11] [0 91 0 91] [0 99, 0 99] [0 98, 0 98]

[0 94 0 95] [0 93 0 93] [0 94 0 94] [0 92, 0 93] [0 95, 0 96]

[0 95 0 96] [0 84 0 85] [0 87 0 88] [1 00, 1 00] [0 98, 0 99]

[0 96 0 98] [0 93 0 94] [0 98 0 99] [0 99, 1 00] [0 98, 0 98]

[0 95 0 98] [0 95 0 96] [0 96 0 96] [0 97, 0 97] [0 97, 0 98]

[0 96 0 98] [0 98 0 98] [0 99 1 00] [1 00, 1 00] [0 93, 0 93]

[0 96 0 98] [0 91 0 92] [0 87 0 88] [0 95, 0 95] [0 99, 1 00]

[0 97 0 98] [0 95 0 95] [0 95 0 95] [0 95, 0 95] [0 93, 0 93]

[0 98 1 00] [0 89 0 90] [0 91 0 92] [1 00, 1 00] [0 93, 0 94]

[1 00 1 00] [0 89 0 89] [0 90 0 91] [0 98, 0 98] [0 98, 0 98]

[1 00 1 00] [0 10 0 11] [0 99 1 00] [0 87, 0 87] [0 98, 0 98]

[1 00 1 00] [0 09 0 09] [0 87 0 87] [0 98, 0 98] [0 98, 0 98]

[0 99 0 99] [0 06 0 06] [0 09 0 09] [0 98, 0 98] [0 92, 0 92]

[0 98 0 99] [0 01 0 02] [0 04 0 43] [0 07, 0 07] [0 86, 0 86]

[0 98 0 99] [0 05 0 05] [0 02 0 02] [0 01, 0 02] [0 06, 0 06]

[0 96 0 97] [0 00 0 00] [0 00 0 00] [0 01, 0 02] [0 01, 0 02]

[0 95 0 98] [0 04 0 05] [0 04 0 04] [0 04, 0 05] [0 05, 0 05]

[0 80 0 85] [0 05 0 06] [0 06 0 06] [0 05, 0 06] [0 05, 0 05]

[0 70 0 75] [0 02 0 03] [0 00 0 00] [0 01, 0 01] [0 00, 0 01]

[0 60 0 68] [0 01 0 02] [0 00 0 00] [0 01, 0 01] [0 00, 0 00]

[0 57 0 60] [0 01 0 02] [0 00 0 00] [0 01, 0 01] [0 00, 0 00]

[0 52 0 56] [0 01 0 02] [0 00 0 00] [0 00, 0 01] [0 00, 0 00]

[0 46 0 48] [0 01 0 02] [0 00 0 00] [0 00, 0 01] [0 00, 0 01]

[0 30, 0 34] 1 [0 01 0 01] [0 00 0 00] [0 01 0 01] [0 00 0 00]

[0 18, 0 22] 1 [0 00 0 00] [0 01 0 02] [0 00 0 00] [0 01 0 02]

[0 15, 0 16] 1 [1 00 1 00] [1 00 1 00] [0 96 0 97] [0 98 0 99]

[0 10, 0 12] 1 [0 96 0 96] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 15, 0 18] 1 [0 97 0 98] [1 00 1 00] [1 00 1 00] [0 96 0 97]

[0 13, 0 14] 1 [1 00 1 00] [1 00 1 00] [0 98 0 99] [0 99 0 99]

[0 01, 0 01] 1 [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 01 0 02]

Columns 9 through 12

[0 40 0 42] [0 43 0 45] [0 44 0 45] [0 44 0 45]

[0 01, 0 01] 1 [0 99 0 99] [1 00 1 00] [0 96 0 96] [0 98 0 99]

[0 70, 0 72] 1 [0 99 1 00] [0 97 0 97] [0 98 0 99] [0 98 0 98]

[0 73, 0 75] 1 [1 00 1 00] [1 00 1 00] [0 98 0 98] [1 00 1 00]

[0 75, 0 76] 1 [0 03 0 04] [0 04 0 05] [0 08 0 08] [0 85 0 85]

[0 78, 0 80] 1 [0 10 0 12] [0 95 0 96] [0 84 0 84] [0 94 0 94]

[0 88, 0 89] 1 [0 84 0 85] [0 94 0 94] [0 92 0 93] [0 90 0 91]

[0 89, 0 90] 1 [1 00 1 00] [0 92 0 93] [0 99 0 99] [0 93 0 94]

[0 90, 0 92] 1 [0 97 0 97] [0 96 0 96] [0 93 0 94] [1 00 1 00]

[0 92, 0 93] 1 [0 97 0 97] [0 99 1 00] [0 99 1 00] [0 98 0 98]

[0 94, 0 95] 1 [0 95 0 96] [0 97 0 98] [0 98 0 99] [0 97 0 98]

[0 95, 0 96] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [0 99 0 99]

[0 96, 0 98] 1 [0 95 0 96] [0 97 0 97] [0 96 0 96] [0 96 0 96]

[0 95, 0 98] 1 [0 98 0 98] [0 99 1 00] [0 98 0 98] [0 96 0 96]

[0 96, 0 98] 1 [0 98 0 98] [0 97 0 97] [0 92 0 93] [0 86 0 87]

[0 96, 0 98] 1 [0 98 0 99] [0 98 0 98] [0 94 0 95] [0 92 0 92]

[0 97, 0 98] 1 [0 98 0 98] [0 97 0 97] [0 92 0 92] [0 89 0 89]

[0 98, 1 00] 1 [0 98 0 98] [0 96 0 96] [0 95 0 96] [0 88 0 88]

[1 00, 1 00] 1 [1 00 1 00] [0 97 0 98] [0 97 0 97] [0 77 0 78]

[1 00, 1 00] 1 [0 96 0 97] [0 96 0 96] [0 95 0 96] [0 93 0 93]

[1 00, 1 00] 1 [0 99 1 00] [1 00 1 00] [0 93 0 93] [0 85 0 86]

[0 99, 0 99] 1 [0 95 0 96] [0 94 0 95] [0 94 0 95] [0 85 0 86]

[0 98, 0 99] 1 [0 89 0 89] [1 00 1 00] [0 83 0 84] [0 84 0 84]

[0 98, 0 99] 1 [0 05 0 05] [0 78 0 79] [0 92 0 92] [0 81 0 82]

[0 96, 0 97] 1 [0 02 0 03] [0 05 0 05] [0 05 0 05] [0 00 0 00]

[0 95, 0 98] 1 [0 06 0 06] [0 05 0 05] [0 02 0 02] [0 00 0 00]

[0 80, 0 85] 1 [0 06 0 07] [0 03 0 04] [0 00 0 00] [0 00 0 00]

[0 70, 0 75] 1 [0 01 0 02] [0 00 0 00] [0 00 0 00] [0 00 0 00]

[0 60, 0 68] I [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

[0 57, 0 60] 1 [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

[0 52, 0 56] 1 [0 00 0 01] [0 01 0 01] [0 00 0 00] [0 00 0 00]

[0 46, 0 40] 1 [0 01 0 02] [0 02 0 03] [0 01 0 02] [0 00 0 00]

[0 30, 0 34] 1 [0 01 0 02] [0 03 0 03] [0 01 0 02] [0 00 0 00]

[0 18, 0 22] I [0 01 0 01] [0 01 0 02] [0 00 0 00] [0 00 0 00]

[0 15, 0 16] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [0 99 1 00]

[0 10, 0 12] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 15, 0 18] 1 [0 99 0 99] [0 99 0 99] [0 99 0 99] [0 99 0 99]

[0 13, 0 14] 1 [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 01, 0 01] 1 [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

Columns 13 through 16 [0.48,0.50]

[0.49,0.52]

[0.50,0.53]

[0.56,0.60]

[0 01, 0 01] [0 98 0 99] [1 00, 1 00] [1 00, 1 00] [0 98, 0 99]

[0 70, 0 72] [1 00 1 00] [1 00, 1 00] [0 99, 0 99] [1 00, 1 00]

[0 73, 0 75] [1 00 1 00] [1 00, 1 00] [1 00, 1 00] [1 00, 1 00]

[0 75, 0 76] [0 89 0 89] [0 92, 0 92] [0 98, 0 98] [0 88, 0 87]

[0 78, 0 80] [0 96 0 96] [0 95, 0 96] [0 86, 0 86] [0 97, 0 97]

[0 88, 0 89] [0 97 0 98] [0 99, 0 99] [0 92, 0 93] [0 96, 0 96]

[0 89, 0 90] [1 00 1 00] [1 00, 1 00] [0 92, 0 93] [0 85, 0 85]

[0 90, 0 92] [0 94 0 95] [0 93, 0 94] [0 96, 0 96] [0 93, 0 93]

[0 92, 0 93] [1 00 1 00] [0 97, 0 97] [0 94, 0 95] [0 94, 0 95]

[0 94, 0 95] [0 98 0 99] [0 97, 0 97] [0 97, 0 98] [0 98, 0 99]

[0 95, 0 96] [0 95 0 96] [0 94, 0 95] [0 94, 0 95] [0 94, 0 95]

[0 96, 0 98] [0 85 0 86] [0 89, 0 89] [0 90, 0 9.] [0 90, 0 90]

[0 95, 0 98] [0 75 0 75] [0 77, 0 78] [0 77, 0 77] [0 76, 0 77]

[0 96, 0 98] [0 56 0 56] [0 58, 0 58] [0 58, 0 58] [0 56, 0 56]

[0 96, 0 98] [0 62 0 63] [0 72, 0 72] [0 78, 0 79] [0 80, 0 81]

[0 97, 0 98] [0 58 0 58] [0 72, 0 72] [0 83, 0 84] [0 88, 0 88]

[0 98, 1 00] [0 57 0 57] [0 72, 0 72] [0 80, 0 80] [0 93, 0 93]

[1 00, 1 00] [0 55 0 56] [0 74, 0 75] [0 90, 0 90] [0 91, 0 91]

[1 00, 1 00] [0 57 0 57] [0 70, 0 71] [0 83, 0 83] [0 95, 0 95]

[1 00, 1 00] [0 56 0 56] [0 78, 0 78] [0 92, 0 93] [0 95, 0 95]

[0 99, 0 99] [0 67 0 67] [0 65, 0 65] [0 85, 0 86] [0 95, 0 96]

[0 98, 0 99] [0 61 0 61] [0 73, 0 73] [0 80, 0 81] [0 92, 0 93]

[0 98, 0 99] [0 55 0 55] [0 75, 0 75] [0 80, 0 80] [0 78, 0 78]

[0 96, 0 97] [0 50 0 51] [0 47, 0 48] [0 70, 0 71] [0 62, 0 63]

[0 95, 0 98] [0 31 0 31] [0 21, 0 21] [0 22, 0 22] [0 20, 0 21]

[0 80, 0 85] [0 22 0 23] [0 08, 0 08] [0 07, 0 07] [0 05, 0 05]

[0 70, 0 75] [0 20 0 21] [0 05, 0 06] [0 03, 0 04] [0 01, 0 02]

[0 60, 0 68] [0 20 0 20] [0 04, 0 04] [0 00, 0 01] [0 00, 0 00]

[0 57, 0 60] [0 20 0 21] [0 05, 0 06] [0 01, 0 02] [0 01, 0 01]

[0 52, 0 56] [0 22 0 23] [0 07, 0 07] [0 03, 0 04] [0 02, 0 03]

[0 46, 0 40] [0 27 0 27] [0 11, 0 12] [0 08, 0 08] [0 06, 0 07]

[0 30, 0 34] [0 37 0 37] [0 25, 0 26] [0 23, 0 23] [0 21, 0 22]

[0 18, 0 22] [0 00 0 00] [0 00, 0 00] [0 00, 0 00] [0 00, 0 00]

[0 15, 0 16] [0 98 0 99] [0 98, 0 98] [0 97, 0 98] [0 97, 0 98]

[0 10, 0 12] [0 99 1 00] [0 99, 1 00] [0 99, 1 00] [0 99, 1 00]

[0 15, 0 18] [0 99 0 99] [0 99, 0 99] [0 99, 0 99] [0 99, 0 99]

[0 13, 0 14] [1 00 1 00] [1 00, 1 00] [1 00, 1 00] [1 00, 1 00]

[0 01, 0 01] [0 00 0 00] [0 00, 0 00] [0 00, 0 00] [0 00, 0 00]

Columns 17 through 20

[0 60 0 65] [0 68, 0 70] [0 69, 0 73] [0 70, 0 73]

[0 01, 0 01] [0 99 0 99] [0 96, 0 96] [0 99, 0 99] [0 94, 0 95]

[0 70, 0 72] [1 00 1 00] [1 00, 1 00] [0 96, 0 96] [0 99, 1 00]

[0 73, 0 75] [0 97 0 98] [1 00, 1 00] [1 00, 1 00] [0 94, 0 95]

[0 75, 0 76] [0 96 0 96] [0 14, 0 15] [0 11, 0 12] [0 12, 0 13]

[0 78, 0 80] [0 89 0 89] [0 90, 0 91] [0 96, 0 96] [0 16, 0 16]

[0 88, 0 89] [0 96 0 96] [0 96, 0 96] [0 96, 0 96] [0 94, 0 95]

[0 89, 0 90] [0 92 0 93] [0 88, 0 88] [0 95, 0 95] [0 99, 0 99]

[0 90, 0 92] [0 89 0 89] [0 94, 0 94] [0 96, 0 96] [0 97, 0 98]

[0 92, 0 93] [0 91 0 92] [0 96, 0 97] [0 94, 0 95] [1 00, 1 00]

[0 94, 0 95] [0 90 0 90] [0 96, 0 96] [0 94, 0 94] [1 00, 1 00]

[0 95, 0 96] [0 94 0 94] [0 96, 0 96] [0 89, 0 90] [0 93, 0 94]

[0 96, 0 98] I [0 87 0 88] [0 89 0 90] [0 87 0 87] [0 87 0 87]

[0 95, 0 98] | [0 73 0 74] [0 72 0 73] [0 77 0 78] [0 74 0 75]

[0 96, 0 98] I [0 65 0 65] [0 56 0 57] [0 60 0 61] [0 61 0 62]

[0 96, 0 98] I [0 88 0 89] [0 85 0 85] [0 83 0 83] [0 88 0 89]

[0 97, 0 98] | [0 90 0 90] [0 98 0 98] [0 91 0 92] [0 90 0 90]

[0 98, 1 00] | [0 94 0 94] [0 93 0 94] [0 92 0 93] [0 94 0 95]

[1 00, 1 00] I [0 94 0 95] [0 96 0 97] [0 95 0 96] [0 94 0 95]

[1 00, 1 00] I [0 96 0 97] [0 99 1 00] [0 99 0 99] [0 95 0 96]

[1 00, 1 00] | [0 96 0 96] [0 98 0 98] [0 97 0 97] [0 94 0 94]

[0 99, 0 99] I [0 95 0 96] [0 94 0 94] [0 93 0 93] [0 89 0 89]

[0 98, 0 99] I [0 90 0 90] [0 85 0 86] [0 81 0 81] [0 73 0 73]

[0 98, 0 99] | [0 69 0 70] [0 62 0 63] [0 55 0 55] [0 43 0 44]

[0 96, 0 97] | [0 38 0 39] [0 34 0 35] [0 27 0 27] [0 16 0 16]

[0 95, 0 98] I [0 16 0 16] [0 13 0 13] [0 09 0 10] [0 07 0 07]

[0 80, 0 85] I [0 04 0 04] [0 03 0 03] [0 02 0 02] [0 01 0 02]

[0 70, 0 75] | [0 01 0 02] [0 01 0 02] [0 01 0 02] [0 01 0 02]

[0 60, 0 68] I [0 01 0 01] [0 01 0 02] [0 01 0 02] [0 02 0 02]

[0 57, 0 60] I [0 02 0 02] [0 02 0 03] [0 02 0 03] [0 02 0 03]

[0 52, 0 56] | [0 04 0 04] [0 05 0 05] [0 03 0 04] [0 03 0 04]

[0 46, 0 40] I [0 07 0 08] [0 07 0 07] [0 05 0 06] [0 05 0 06]

[0 30, 0 34] | [0 22 0 22] [0 21 0 22] [0 20 0 21] [0 20 0 20]

[0 18, 0 22] | [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

[0 15, 0 16] I [0 98 0 98] [0 98 0 98] [0 97 0 98] [0 97 0 98]

[0 10, 0 12] | [0 99 1 00] [0 99 1 00] [0 99 1 00] [0 99 1 00]

[0 15, 0 18] I [0 99 0 99] [0 99 0 99] [0 99 0 99] [0 99 0 99]

[0 13, 0 14] I [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00]

[0 01, 0 01] 1 [0 00 0 00] [0 00 0 00] [0 00 0 00] [0 00 0 00]

Columns 2] through 24

[0 73 0 75] [0 78 0 80] [0 84 0 88] [0 88 0 90]

[0 01, 0 01] 1 [0 97 0 98] [0 93 0 93] [0 95 0 96] [0 95 0 95]

[0 70, 0 72] | [0 97 0 98] [0 96 0 96] [0 95 0 96] [0 95 0 95]

[0 73, 0 75] I [0 99 0 99] [0 97 0 98] [0 94 0 95] [0 97 0 97]

[0 75, 0 76] I [0 05 0 05] [0 02 0 02] [0 01 0 01] [0 02 0 03]

[0 78, 0 80] | [0 11 0 11] [0 06 0 07] [0 03 0 04] [0 00 0 00]

[0 88, 0 89] | [0 92 0 93] [0 14 0 14] [0 07 0 08] [0 01 0 02]

[0 89, 0 90] I [0 91 0 92] [0 97 0 97] [0 10 0 11] [0 04 0 04]

[0 90, 0 92] | [1 00 1 00] [0 96 0 96] [0 88 0 89] [0 10 0 10]

[0 92, 0 93] | [0 96 0 96] [0 98 0 98] [0 87 0 87] [0 07 0 08]

[0 94, 0 95] I [0 98 0 98] [0 96 0 96] [1 00 1 00] [0 80 0 80]

[0 95, 0 96] I [0 96 0 96] [0 89 0 89] [0 89 0 89] [0 92 0 93]

[0 96, 0 98] | [0 94 0 95] [0 93 0 93] [0 83 0 83] [0 82 0 83]

[0 95, 0 98] I [0 74 0 75] [0 80 0 80] [0 78 0 78] [0 77 0 77]

[0 96, 0 98] I [0 56 0 56] [0 56 0 56] [0 58 0 58] [0 55 0 55]

[0 96, 0 98] I [0 90 0 91] [0 87 0 87] [0 85 0 85] [0 75 0 75]

[0 97, 0 98] | [0 94 0 95] [0 92 0 92] [0 89 0 89] [0 77 0 77]

[0 98, 1 00] I [0 96 0 97] [0 92 0 93] [0 87 0 88] [0 75 0 76]

[1 00, 1 00] I [0 96 0 97] [0 94 0 95] [0 82 0 82] [0 62 0 62]

[1 00, 1 00] | [0 94 0 95] [0 89 0 90] [0 67 0 67] [0 40 0 41]

[1 00, 1 00] I [0 89 0 89] [0 76 0 77] [0 50 0 51] [0 26 0 26]

[0 99, 0 99] I [0 77 0 78] [0 57 0 58] [0 32 0 33] [0 18 0 18]

[0 98, 0 99] I [0 56 0 57] [0 35 0 36] [0 15 0 16] [0 08 0 08]

[0 98 0 99] [0 29, 0 29] [0 16 0 17] [0 05,0 06] [0 03,0 03]

[0 96 0 97] [0 09, 0 09] [0 09 0 09] [0 03,0 03] [0 02,0 03]

[0 95 0 98] [0 04, 0 04] [0 02 0 03] [0 00,0 00] [0 01,0 01]

[0 80 0 85] [0 00, 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 70 0 75] [0 01, 0 02] [0 01 0 02] [0 01,0 01] [0 00,0 00]

[0 60 0 68] [0 02, 0 03] [0 02 0 02] [0 01,0 01] [0 00,0 00]

[0 57 0 60] [0 03, 0 03] [0 02 0 03] [0 01,0 01] [0 00,0 00]

[0 52 0 56] [0 05, 0 05] [0 04 0 04] [0 02,0 03] [0 02,0 02]

[0 46 0 40] [0 05, 0 06] [0 05 0 06] [0 05,0 05] [0 06,0 07]

[0 30 0 34] [0 20, 0 21] [0 20 0 21] [0 20,0 21] [0 21,0 22]

[0 18 0 22] [0 00, 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 15 0 16] [0 97, 0 98] [0 97 0 98] [0 97,0 98] [0 97,0 98]

[0 10 0 12] [0 99, 1 00] [0 99 1 00] [0 99,1 00] [0 99,1 00]

[0 15 0 18] [0 99, 0 99] [0 99 0 99] [0 99,0 99] [0 99,0 99]

[0 13 0 14] [1 00, 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 01 0 01] [0 00, 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

Columns 25 through 28

[0 88, 0 90] [0 89 0 91] [0 90,0 92] [0 91,0 93]

[0 01 0 01] [0 95, 0 95] [0 95 0 96] [0 97,0 97] [0 98,0 99]

[0 70 0 72] [0 96, 0 96] [0 96 0 97] [0 98,0 98] [0 98,0 99]

[0 73 0 75] [0 99, 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 75 0 76] [0 00, 0 00] [0 03 0 04] [0 05,0 06] [0 07,0 07]

[0 78 0 80] [0 00, 0 00] [0 02 0 02] [0 04,0 04] [0 05,0 05]

[0 88 0 89] [0 00, 0 00] [0 00 0 00] [0 02,0 03] [0 03,0 04]

[0 89 0 90] [0 01, 0 01] [0 00 0 00] [0 00,0 00] [0 02,0 02]

[0 90 0 92] [0 02, 0 03] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 92 0 93] [0 02, 0 02] [0 00 0 00] [0 03,0 04] [0 00,0 00]

[0 94 0 95] [0 03, 0 04] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 95 0 96] [0 05, 0 05] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 96 0 98] [0 79, 0 80] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 95 0 98] [0 66, 0 67] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 96 0 98] [0 50, 0 51] [0 38 0 38] [0 38,0 38] [0 30,0 51]

[0 96 0 98] [0 61, 0 62] [0 34 0 34] [0 16,0 16] [0 14,0 15]

[0 97 0 98] [0 60, 0 60] [0 23 0 24] [0 12,0 13] [0 04,0 04]

[0 98 1 00] [0 33, 0 34] [0 17 0 18] [0 06,0 07] [0 02,0 03]

[1 00 1 00] [0 26, 0 27] [0 12 0 13] [0 03,0 04] [0 01,0 02]

[1 00 1 00] [0 20, 0 20] [0 09 0 09] [0 02,0 03] [0 01,0 01]

[1 00 1 00] [0 14, 0 15] [0 08 0 09] [0 02,0 03] [0 01,0 02]

[0 99 0 99] [0 10, 0 11] [0 09 0 09] [0 03,0 03] [0 01,0 02]

[0 98 0 99] [0 09, 0 09] [0 08 0 09] [0 03,0 04] [0 02,0 03]

[0 98 0 99] [0 08, 0 08] [0 09 0 09] [0 04,0 04] [0 02,0 03]

[0 96 0 97] [0 06, 0 06] [0 07 0 07] [0 03,0 04] [0 01,0 02]

[0 95 0 98] [0 02, 0 03] [0 04 0 05] [0 02,0 03] [0 01,0 02]

[0 80 0 85] [0 01, 0 00] [0 03 0 03] [0 01,0 02] [0 02,0 03]

[0 70 0 75] [0 00, 0 00] [0 01 0 02] [0 03,0 03] [0 04,0 04]

[0 60 0 68] [0 00, 0 00] [0 01 0 02] [0 03,0 03] [0 04,0 05]

[0 57 0 60] [0 00, 0 00] [0 01 0 02] [0 03,0 03] [0 04,0 05]

[0 52 0 56] [0 03, 0 03] [0 04 0 04] [0 04,0 05] [0 06,0 07]

[0 46 0 40] [0 09, 0 09] [0 10 0 11] [0 09,0 09] [0 09,0 09]

[0 30 0 34] [0 24, 0 24] [0 25 0 26] [0 23,0 24] [0 23,0 23]

[0 18 0 22] [0 00, 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0.15,0.16] [0.10,0.12] [0.15,0.18] [0.13,0.14] [0.01,0.01]

[0.01,0.01] [0.70,0.72] [0.73,0.75] [0.75,0.76] [0.78,0.80] [0.88,0.89] [0.89,0.90] [0.90,0.92] [0.92,0.93] [0.94,0.95] [0.95,0.96] [0.96,0.98] [0.95,0.98] [0.96,0.98] [0.96,0.98] [0.97,0.98] [0.98,1.00] [1.00,1.00] [1.00,1.00] [1.00,1.00] [0.99,0.99] [0.98,0.99] [0.98,0.99] [0.96,0.97] [0.95,0.98] [0.80,0.85] [0.70,0.75] [0.60,0.68] [0.57,0.60] [0.52,0.56] [0.46,0.40] [0.30,0.34] [0.18,0.22] [0.15,0.16] [0.10,0.12] [0.15,0.18] [0.13,0.14] [0.01,0.01]

Columns 33

[0.98,0.98] [0.99,1.00] [0.99,0.99] [1.00,1.00] [0.00,0.00]

[1.00,1.00] [0.99,1.00] [1.00,1.00] [0.05,0.05] [0.03,0.04] [0.02,0.03] [0.02,0.03] [0.01,0.02] [0.00,0.00] [0.01,0.02] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.25,0.25] [0.18,0.18] [0.05,0.05] [0.03,0.04] [0.02,0.03] [0.03,0.03] [0.03,0.04] [0.03,0.04] [0.03,0.04] [0.03,0.03] [0.02,0.03] [0.02,0.03] [0.03,0.03] [0.04,0.04] [0.05,0.05] [0.06,0.06] [0.07,0.07] [0.09,0.09] [0.22,0.22] [0.00,0.00] [0.98,0.98] [0.99,1.00] [0.99,0.99] [1.00,1.00] [0.00,0.00]

through 36 [0.99,1.00]

[0.98,0.98] [0.99,1.00] [0.99,0.99] [1.00,1.00] [0.00,0.00]

[0.94,0.96]

[1.00,1.00] [1.00,1.00] [1.00,1.00] [0.02,0.02] [0.01,0.01] [0.00,0.00] [0.01,0.02] [0.02,0.03] [0.01,0.02] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.35,0.35] [0.19,0.19] [0.14,0.14] [0.14,0.15] [0.14,0.15] [0.15,0.16] [0.16,0.16] [0.15,0.16] [0.15,0.16] [0.14,0.15] [0.14,0.14] [0.14,0.14] [0.14,0.15] [0.16,0.16] [0.17,0.17] [0.18,0.18] [0.18,0.19] [0.20,0.20] [0.31,0.32] [0.00,0.00] [0.98,0.98] [0.99,1.00] [0.99,0.99] [1.00,1.00] [0.00,0.00]

[0.91,0.92]

[0.97,0.98] [0.99,1.00] [0.99,0.99] [1.00,1.00] [0.00,0.00]

[0.96,0.98]

[1.00,1.00] [0.99,0.99] [1.00,1.00] [0.00,0.00] [0.01,0.02] [0.00,0.00] [0.00,0.00] [0.01,0.01] [0.00,0.00] [0.03,0.03] [0.05,0.05] [0.00,0.00] [0.00,0.00] [0.49,0.49] [0.34,0.35] [0.29,0.29] [0.29,0.29] [0.29,0.30] [0.30,0.30] [0.30,0.31] [0.31,0.31] [0.30,0.31] [0.30,0.30] [0.29,0.30] [0.29,0.29] [0.29,0.29] [0.30,0.30] [0.29,0.30] [0.29,0.30] [0.30,0.30] [0.31,0.32] [0.39,0.40] [0.00,0.00] [0.98,0.99] [1.00,1.00] [0.99,1.00] [1.00,1.00] [0.00,0.00]

[0.87,0.89]

[0.97,0.98] [0.99,1.00] [0.99,0.99] [1.00,1.00] [0.00,0.00]

[0.98,0.99]

[1.00,1.00] [0.98,0.99] [0.98,0.99] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.01,0.01] [0.02,0.02] [0.00,0.00] [0.01,0.01] [0.07,0.07] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.00,0.00] [0.99,1.00] [1.00,1.00] [0.99,1.00] [0.99,1.00] [0.02,0.03]

[0.85,0.86]

Columns 2 9 through 32 [0.94,0.95]

[0.01,0.01]I[0.98,0.98] [0.99,0.99] [1.00,1.00] [1.00,1.00] [0.70,0.72]I[0.99,1.00] [0.99,1.00] [0.92,0.92] [0.97,0.97]

[0 73, 0 75] [0 94 0 95] [0 99 1 00] [1 00,1 00] [0 94,0 94]

[0 75, 0 76] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 04,0 04]

[0 78, 0 80] [0 00 0 00] [0 00 0 00] [0 03,0 03] [0 00,0 00]

[0 88, 0 89] [0 00 0 00] [0 01 0 01] [0 00,0 00] [0 01,0 01]

[0 89, 0 90] [0 00 0 00] [0 01 0 02] [0 00,0 00] [0 00,0 00]

[0 90, 0 92] [0 01 0 01] [0 00 0 00] [0 03,0 03] [0 00,0 00]

[0 92, 0 93] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 94, 0 95] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 95, 0 96] [0 00 0 00] [0 00 0 01] [0 00,0 00] [0 00,0 00]

[0 96, 0 98] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 95, 0 98] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 96, 0 98] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 96, 0 98] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 97, 0 98] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 98, 1 00] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[1 00, 1 00] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[1 00, 1 00] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[1 00, 1 00] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 99, 0 99] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 98, 0 99] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 98, 0 99] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 96, 0 97] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 95, 0 98] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 80, 0 85] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 70, 0 75] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 60, 0 68] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 57, 0 60] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 52, 0 56] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 46, 0 40] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 30, 0 34] [0 00 0 00] [0 00 0 00] [0 00,0 00] [0 00,0 00]

[0 18, 0 22] [0 00 0 00] [0 00 0 00] [0 01,0 02] [0 04,0 05]

[0 15, 0 16] [0 97 0 98] [1 00 1 00] [1 00,1 00] [0 94,0 94]

[0 10, 0 12] [1 00 1 00] [0 98 0 99] [0 95,0 96] [1 00,1 00]

[0 15, 0 18] [1 00 1 00] [0 96 0 96] [1 00,1 00] [1 00,1 00]

[0 13, 0 14] [0 96 0 97] [0 98 0 98] [1 00,1 00] [0 97,0 97]

[0 01, 0 01] [0 02 0 03] [0 03 0 04] [0 00,0 00] [0 00,0 00]

Columns 37 through 40

[0 80 0 81] [0 70 0 71] [0 40,0 42] [0 38,0 40]

[0 01, 0 01] [0 97 0 98] [0 95 0 96] [1 00,1 00] [1 00,1 00]

[0 70, 0 72] [1 00 1 00] [1 00 1 00] [0 98,0 98] [0 97,0 97]

[0 73, 0 75] [1 00 1 00] [0 94 0 95] [0 92,0 93] [1 00,1 00]

[0 75, 0 76] [0 97 0 98] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 78, 0 80] [0 99 1 00] [0 93 0 94] [1 00,1 00] [0 98,0 98]

[0 88, 0 89] [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 89, 0 90] [0 98 0 99] [1 00 1 00] [0 96,0 97] [0 97,0 98]

[0 90, 0 92] [1 00 1 00] [0 97 0 98] [1 00,1 00] [0 99,1 00]

[0 92, 0 93] [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 94, 0 95] [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 95, 0 96] [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 96, 0 98] [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 95, 0 98] [1 00 1 00] [1 00 1 00] [1 00,1 00] [1 00,1 00]

[0 96, 0 98] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 96, 0 98] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 97, 0 98] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 98, 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[1 00, 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[1 00, 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[1 00, 1 00] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 99, 0 99] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 98, 0 99] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 98, 0 99] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 96, 0 97] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 95, 0 98] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 80, 0 85] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 70, 0 75] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 60, 0 68] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 57, 0 60] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 52, 0 56] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 46, 0 40] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 30, 0 34] [1 00 1 00] [1 00 1 00] [1 00 1 00] [1 00, 1 00]

[0 18, 0 22] [0 96 0 96] [0 99 1 00] [1 00 1 00] [0 99, 0 99]

[0 15, 0 16] [1 00 1 00] [0 96 0 97] [0 95 0 95] [1 00, 1 00]

[0 10, 0 12] [0 97 0 97] [1 00 1 00] [1 00 1 00] [0 98, 0 99]

[0 15, 0 18] [0 97 0 97] [0 96 0 96] [1 00 1 00] [0 95, 0 96]

[0 13, 0 14] [1 00 1 00] [0 95 0 95] [1 00 1 00] [1 00, 1 00]

[0 01, 0 01] [0 03 0 04] [0 02 0 03] [0 00 0 00] [0 00, 0 00]

9. Conclusion

In this article, very new kind of interval-valued fuzzy matrix has been introduced. In this new approach, the rows and columns are taken as uncertain, whereas in IVFM they are certain. These types of fuzzy matrices can be used to handel images, fuzzy graphs, etc.

Besides, null-IVFMFRC, equality of IVFMFRCs and identity IVFMFRC are defined. Also, some binary operators are defined. Two types of complements and density of IVFMFRC are defined and studied several properties. But, product of two IVFMFRCs is not defined. At present product of IVFMFRCs is being researched with their properties. The power convergence, nilpotency, etc. can be investigated after suitable definition of multiplication. Other several operations can also be defined for IVFMFRC.

Acknowledgments

The author is thankful to the anonymous reviewers and the area editor for their comments to improve the presentation of the paper.

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