FUZZY BINARY RELATIONS AND ITS SPECIAL TYPES

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www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science

FUZZY BINARY RELATIONS AND ITS SPECIAL TYPES

R. Rajalakshmi

*1

Assistant Professor, Department of Mathematics, Dhanalakshmi Srinivasan Engineering

College, Perambalur, Tamilnadu, India.

ABSTRACT

One of the most fundamental notations in pure and applied Science is the concept of a relation. Science has been described as the discovery of relations between objects, states and events. Fuzzy relations generalize the concept of relations in the same manner as fuzzy sets generalize the fundamental idea of sets .The aim of this paper is to present, in an unitary way, some special types of fuzzy binary relations. All these fuzzy binary relations are characterized and we established the inclusions these classes of fuzzy relations

Keywords: Fuzzy Relations; Operations Of Fuzzy Relations; Binary Relations; Equivalence Relations;

Inverse Relations; Ordering Relations; Compositions.

I.

INTRODUCTION

Nowadays fuzzy set theory is a flourishing field in mathematics. There are a lot of researches and paper presentations going on in this area. This field is very young. Fuzzy concepts were first proposed by Lotfi A.Zadeh and others in 1965. A crisp relation represents the occurrence of association, interaction or interconnectedness between the elements of two or more sets. This concepts can be generalized to allow for varies degree or strengths of association or interaction between elements. The paper is organized as follows. We introduce basic notation and concepts connected with fuzzy relation. Degrees of association can be represented by membership grades in a fuzzy family member in the same way as degrees of set membership are represent in the fuzzy set. This work is done with the aim of understanding the basic concepts of fuzzy set theory. We have so far seen the types of fuzzy binary relations sets in the form of an introduction.

II.

METHOD OF ANALYSIS

CARTESIAN PRODUCT

Definition :

The Cartesian product of two crisp sets X and Y, denoted by X×Y is the crisp set of all ordered pairs such that the first element in each pair is a member of x and the second element is a member of Y.

Formally,

X×Y = {(x,y) / x ∈ X and y ∈ X }. Note that if X ≠Y then X×Y ≠ Y×X.

The Cartesian product can be generalized for a family of crisp sets { xi / i∈ Nn} and denoted either by X1×X2×…Xn or by Xi∈ Nn Xi

Elements of the Cartesian product of n crisp sets are n-tuples. (x1,x2,…xn) such that xi ∈ Xi, i ∈ Nn. Thus,

Xi∈ NnXi ={(x1,x2,…xn)/xi ∈ Xi i ∈ Nn}

It is possible for all the sets Xi to be equal, that is to be a single set X. In this case, the Cartesian product of a set with itself n-times is usually denoted by Xn.

Definition:

A relation among crisp sets X1,X2,…Xn is a subset of the Cartesian product Xi∈ NnXi . It is denoted either by R(X1,X2,…Xn) or by the abbreviated form R(Xi∈Nn).

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Thus R(X1,X2,…Xn) X1×X2×….×Xn so that for relations among sets X1,X2,..Xn the Cartesian product X1×X2×….×Xn represents the universal set. Because a family member is itself a set, the basic set concepts such as containment or subset, union, intersection, and complement can be applied without modification to relations.

Each crisp relation R can be defined by a characteristic function that assigns a value of 1 to every tuple of the universal set belonging in the relation and a 0 to every tuple that does not belong thus

R(X1,X2,….,Xn) = 1, iff (x1,x2,…xn)∈ R 0,other wise

The association of a tuple in a relation signifies that the elements of the tuple are connected with one another. For instance let R represent the relation of marriages between the set of all men and women, then only those pair whose members are married to each other will be assigned a value of 1, indicating that they belong in this relation. A relation between the two sets is called binary; if three, four, or five sets are involved, these are called ternary, quaternary, or quinary respectively, In general, a relation defined on n sets is called n-arg or n-dimensional.

Membership array:

A connection of relation can be written as a set of ordered tuples. Another appropriate way of this relation R(X1,X2,…Xn) involves an n-dimensional membership array: MR [ i1,i2,…in]. Each component of the first dimensional it of this array corresponds to exactly one member of X1, each element of dimension i2 to exactly one member of X2 and so on if the n tuple (X1,X2,…Xn) ∈ X1×X2×….×Xn corresponds to the element

i1,i2,…in of MR, then

i1,i2,…in = 1, iff (x1,x2,…xn)∈ R 0,other wise

Definition :

Let X,Y R be universal sets; then ̃ ={[(x,y), ̃ (x,y)]/(x,y) ∈ X×Y}

is called a fuzzy relation on X×Y.

Example 1:

When ∪={1,2,3} then “approximately equal” is the binary fuzzy relation.

1/(1,1) +1/(2,2) +1/(3,3) +0.8/(1,2) +0.8/(2,3) +0.8/(2,1)+0.8/(3,2) +0.3/(1,3) +0.3/(3,1) ……..1

The membership function R of this relation can be described by when x=y

R = 0.8 when |x-y|=1 ………2

0.3 when|x-y|=2

In matrix notation this can be represented as Y

1 2 3

1 1 0.8 0.3

X 2 0.8 1 0.8

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Example 2:

Let ∪=[0,250] be the interval of height of persons, and suppose that 0 for x-y ≤0

R(x,y) = x-y/20 for 0<x-y<20 ……….1 1 for x-y≥20

That is R(x,y) = (x-y;0,20), then

R = ∫ (x,y)/(x,y) ………...2 It represents the notion “much taller than”

OPERATIONS ON FUZZY RELATION:

Two operations on fuzzy relations are the intersection and the union operations. They are defined as follows.

Definition:INTERSECTION

Let R and S be binary relations defined on X×Y the intersection of R and S defined by, (x,y) ∈ X×Y : R S(x,y) =min( R(x,y), S(x,y)) ……….1

Definition: UNION

The Union of R and S defined by,

(x,y) ∈ X×Y: R∪S(x,y) =max( R(x,y), S(x,y)) ……….2

Example 3

Let define two binary relations R and S that are defined as follows R= “x considerable larger than y”

y1 y2 y3 y4

x1 0.8 1 0.1 0.7

x2 0 0.8 0 0 ……….…………..1

x3 0.9 1 0.7 0.8

S= “y very close to x”

y1 y2 y3 y4

x1 0.4 0 0.9 0.6

x2 0.9 0.4 0.5 0.7 ……….2

x3 0.3 0 0.8 0.5

The intersection of the relations R and S is given by the relation.

y1 y2 y3 y4

x1 0.4 0 0.1 0.6

R∩S x2 0 0.4 0 0 ……….3

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The union of the relations R and S is given by the relation. y1 y2 y3 y4 x1 0.8 1 0.9 0.7 R S x2 0.9 0.8 0.5 0.7 ……….4 x3 0.9 1 0.8 0.8

Definition :

Let ̃={[(x,y),µ ̃(x,y)]/(x,y) ∈ X×Y} be a fuzzy binary relation. The first projection of ̃ is then defined as,

̃(1)_{= {(x, max(µ ̃(x,y))/(x,y) ∈ X×Y}} The second projection is defined as, ̃(2)_{= {(y, max(µ ̃(x,y))/(x,y) ∈ X×Y}} And the total projection as,

̃(T)_{=max max{µ ̃(x,y)/(x,y) ∈ X×Y}}

Example 4:

Let X={x1, x2, x3} and Y ={y1, y2, y3, y4}.

Consider the relation R =”x considerably larger than y”

y1 y2 y3 y4

x1 0.8 1 0.1 0.7

x2 0 0.8 0 0 ……….……….1

x3 0.9 1 0.7 0.8

Then the projection on X means that,

x1 is assigned the highest membership degree from tuples (x1, y1), (x2, y2), (x3, y3), and (x4, y4) (i.e.,) 1 (the maximum of the first row)

x2 is assigned the highest membership degree from the tuples (x2,y1), (x2,y2), (x2,y3),and (x2,y4) (i.e.,) 0.8 (the maximum of the second row)

x3 is assigned the highest membership degree from the tuples (x3,y1), (x3,y2), (x3,y3),and (x3,y4) (i.e.,) 1 ( the maximum of the third row)

So one obtains the fuzzy set

proj R on X=1/x1+0.8/x2+1/x3 ………...2

In the same way, the projection on Y can be taken by searching for the maxima of the four columns. This gives the fuzzy set

Proj R on Y = 0.9 /y1 +1 /y2 +.7 /y3 +0.8 /y4 ………3

The total projection of this relation can also be taken. It is equal to 1, the maximal membership degree in this relation. In table form this can be illustrated by,

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Definition: Cylindrical extension

If ̃ql X is the largest relation in x of which the projection is ̃q , ̃ql is then called the cylindrical extension of ̃q and ̃q is the base of ̃ql.

Example 5:

Consider the fuzzy set

A = projection of R on X=1/x1+0.8/x2+1/x3 ……….1 That was derived in the above example in the equation 2

It’s cylindrical extension on the domain X×Y is given by

y1 y2 y3 y4

x1 1 1 1 1

Ce(A) = x2 0.8 0.8 0.8 0.8 ……….2

x3 1 1 1 1

Consider the fuzzy set

B= proj of R on Y =.9/y1+ 1/y2+.7/y3+.8/y4…………3

That was derived in the above example in equation 3. It’s cylindrical extension on the domain X×Y is given by y1 y2 y3 y4 x1 0.9 1 0.7 0.8 Ce(B) = x2 0.9 1 0.7 0.8 ………4 x3 0.9 1 0.7 0.8

BINARY RELATIONS:

Any relation connecting two sets X and Y is well-known as a binary relation. It is usually denote by R(X,Y). when X ≠ Y, binary relations R(X,Y) are often referred to as bipartite graphs; when x=y, they are called directed graphs or bipartite graphs. Membership matrices and sagittal diagrams are two useful representations of binary relations R(X, Y).

Sets X,Y is represented by a set of nodes in the illustration; nodes parallel to one set are clearly distinguished from nodes representing the other set. Elements of X×Y with non zero membership grades

y1 y2 y3 y4 Proj on X

x1 0.8 1 0.1 0.7 1

x2 0 0.8 0 0 0.8 ………..4

x3 0.9 1 0.7 0.8

1

Proj on Y _{0.9 1} _0.7 _0.8 _{Total projection}

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in R(X,Y) are represented in the diagram by lines connecting the respective nodes. These lines are assigned with the value of the membership grade (X,Y) = (X,Y)

Examples of two convenient representations of a fuzzy binary relation. A. sagittal diagram B. membership matrix.

SOME DEFINITIONS IN FUZZY BINARY RELATIONS:

Definition :

The domain of a crisp binary relation is dom R(X,Y). It is defined as the crisp subset of X whose numbers involved in the relation.

Formally,dom R(X, Y) = { x/x ϵ X, (x, y) ϵ R for some y ϵ Y}

If R(X,Y) is a fuzzy relation, its domain is the fuzzy set dom R(X,Y) whose membership function is µ dom R(X) =max µR(x , y) for each x ϵ X.

Definition :

Each element of set X in the domain of R to the degree equal to the strength of its relation to any member of set Y. The range of a crisp binary R(X,Y) is denoted by ran R(X,Y) and is defined as the subset of Y whose members involved in the relation.

Thus,

ran R(X,Y) = {y/y ϵ Y, (x,y) ϵ R, for some x ϵ X}.

When R(x,y) is a fuzzy relation, its range is a fuzzy set ran R(X,Y) whose functions defined by µran R(y) =Max µR (x,y) for each y ϵ Y.

Therefore, the relation that each element of Y has to an element of X is equal to the degree of that elements membership in the range of R.

Definition:

The larger of a fuzzy relation R is a number h(R) defined by, h(R)=Max y ϵY Max x ϵ X R(x,y).

That is the maximized membership position assigned by any pair (X,Y) in R.

Definition:

If h(R) =1 then the relation is known as normal; otherwise it is known as subnormal.

The concepts of domain and range provide a basis for defining the following classifications of binary relations. In the fuzzy relations, however, references to the domain and the range of the relation are assumed to refer to the support of the domain and range respectively.

y1 y2 y3 y4 y5 x1 .9 1 0 0 0 x2 0 .4 0 0 0 x3 0 0 1 .2 0 x4 0 0 0 0 .4 x5 0 0 0 0 .5 x6 0 0 0 0 .2

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Definition :

Let R is binary relation on sets X and Y. If the domain of R is similar to the support of the set X, then the relation is known as “completely specified” otherwise it is known as “incompletely specified”. If the range of R is similar to the support of set Y, then R is called a relation from x onto Y; other than it is called a relation from X into Y.

If each part of the domain of a binary relation R, presents exactly once in r, then the relation is known as a mapping or a function. When at least one member of the domain is associated to more than one element of the range, the relation is not a mapping and is as an alternative called one too many. If R(x,y) is a mapping , let this main property be denoted by R(x y).if µR(x y) (x,y)>0 then y is called the image of x in R.

If R be a mapping and in addition, atleast one of the members of its range appears more than once in R, then the relation is called many to one. This name refers to the fact that many elements from the domain map to a single element of the range instead, if each element of the range appears exactly once in the mapping it is called one to one relation.

INVERSE RELATIONS

Definition:

Inverse of a crisp connection R(X,Y) is written as R-1_{(X,Y) and also it is a subset of Y×X such that} R-1_{(X,Y) ={(y, x) / (x, y) ϵ R } where x ϵ X and y ϵ Y.}

Clearly, dom R(X, Y) = ran R -1_{(X, Y) and} dom R-1_{(X, Y) = ran R(X, Y).}

Definition:

Let fuzzy relation R(X,Y) the inverse fuzzy connection R-1_{(X, Y) is defined by,} R-1_{(Y,X) = R(x,y) for all (x, y) ϵ X×Y}

A membership matrix R-1 _{representing R}-1_{(X,Y) is the transpose of the matrix. This matrix R for R(X,Y) that} is the rows of R-1 _{equal to the columns of R and the columns of R}-1 _{equal to the rows R.}

Clearly, ( R-1₎-1 _{=R , for any binary relation R.}

Example 6:

Let R(X,Y) be a fuzzy connection on X={x,y,z} and Y= {a,b} such that,

A B

X 0.3 0.2

MR = Y 0 0.6

Z 0.6 0.4

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X Y Z

A 0.3 0 0.6

R-1 =RT =

B 0.2 1 0.4

When RT_{denotes the transpose of R}_.

COMPOSITIONS:

Definition:

Two crisp binary relations P(X,Y) and Q(X,Y) defined with a regular set Y. These two relations is denoted by

R(X, Z) = P(X, Y) ⁰ Q(Y, Z)

and subset R(X, Z) of X×Z such that (X, Z) ϵ R if and only if there exists at least one y ϵ Y such that (x, y) ϵ P and (y, z) ϵ Q.

The following three properties are satisfied for binary relations, P, Q, R. P Q ≠ Q o R

(P Q)-1_{= Q}_{-1 o}_P-1 (P Q) R = P o (Q o R).

Now as the classical set operations such as union and intersection have a mixture of generalizations for fuzzy sets the composition operation for fuzzy associations can take numerous forms the most regular of these is the max-min composition.

Definition:

The max-min composition for fuzzy relations denoted again by P(X,Y) Q(Y.Z) is defined by, (P o Q) (X, Z) = max y Y min[P(X,Y), o Q(Y, Z)] x ϵ X and z ϵ Z

This process satisfies the similar three properties for the composition of crisp relationships.

Example 7

Let P1(x,y) and P2(y,z) be defined by the following relational matrices.

y1 y2 y3 y4 y5 x 1 0.1 0.2 0 1 0.7 P1 : x 2 0.3 0.5 0 0.2 1 x 3 0.8 0 1 0.4 0.3 z1 z2 z3 z4 y1 0.9 0 0.3 0.4 y2 0.2 1 0.8 0 P2 : y3 0.8 0 0.7 1 y4 0.4 0.2 0.3 0 y5 0 1 0 0.8

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www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science The determination for x=x1, z=z1 .

Let x=x1,z=z1 and y=yi; i=1,2,…5;

min{=P1 (x1,y1), =P2(y1,z1)} =min{.1,.9} = .1 min{=P1 (x1,y2), =P1(y2,z1)} =min{.2,.2} = .2 min{=P1 (x1,y3),=P1(y3,z1)} =min{0,.8} = 0 min{ =P1 (x1,y1),=R2 (y1,z1)} =min{.1,.9} = .1 min{=P1 (x1,y4),=R2 (y4,z1)} =min{1,.4} = .4 min{=P1 (x1,y5), ̃ (y5,z1)} =min{.7,0} = 0 =P1 ⁰ P2 (x1,z1) = ((x1, z1), max{.1,.2,0,.4,0}) = ((x1, z1), .4)

In analogy to the above computation we now decide the grades of membership for all pairs (xi, zi), i=1,…3, j=1,…4 and arrive at ,

z1 z2 z3 z4 x1 0.4 0.7 0.3 0.7 (P1⁰ P2) x2 0.3 1 0.5 0.8 x3 0.8 0.3 0.7 1

Equivalence Relations:

A Crisp binary relation R(X,Y) that is reflexive,Symmetric, and transitive is called equivalence relation. For each elements x in X, we define Ax = {y/(x,y) ϵ R(x,Y)}

Ax is clearly a subset of X. The elements x is itself contained in Ax due to the reflexivity of R; because r is transitive and symmetric, each member of Ax is related to all the other members of Ax.

Example:8

X={1,2,…10} , the Cartesian product XxY contains 100 members (1,1),(2,2),(1,3)…(10,10). R(x,x)={(x,y)/ x and y have the same remainder when divided by 3}. The relation is easily shown to be reflexive, symmetric, and transitive and is an equivalence relation on X. The three correspondence classes define by this relation are:

A1=A4=A7=A10={1,4,7,10} A2=A5=A8={2,5,8} A3=A6=A9={3,6,9}

Ordering Relations:

Binary relation R(X,Y) is reflexive, anti symmetric and transitive is called a partial ordering. The common symbol ≤ is indicative of the relations. Thus x≤y denotes (x,y)ϵR and signifies that x precedes y. The inverse partial ordering R-1_{(x,x) is recommended by the ≥, If y≥x, indicating (y,x) ϵ R}-1_{, then we declare} that y succeeds x.

Some primary concepts associated with partial orderings:

1) If xϵX and x≤Y for every yϵx. Then x is called the first member (or minimum) of X with respect to the relation denoted by ≤

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2) If x ϵX and Y≤x for every yϵx,then x is called the last member (or maximum) of X with repect to the partial ordering relation.

3) If xϵX and y≤x implies x=y, then x is called a minimal member of X with respect to the relation. 4) If x ϵX and x≤y implies x=y, then x is called a maximal member of X with respect to the relation.

Sup-i compositions of fuzzy relations:

Sup-i compositions of binary fuzzy relations, where i refer to a t-norm, generalize the standard max-min composition.

Given a challenging t-norm i and two fuzzy relations P(x,y) and Q(y,z) , the sup-i composition of P and Q is a fuzzy relation (Pi _{⁰ Q) on X×Z defined by}

(Pi _{⁰ Q) (x, z) = sup}_YϵY_{i[P(x, y), Q(y, z)] ………1} For all x ϵ X, z ϵ Z.

Fuzzy relations P(X,Y), Pj(X,Y), Q(Y,Z), Qj(Y,Z) and R(Z,Y), where j takes values in an index set J, the following are basic properties of the sup-i composition under the regular fuzzy union and intersection.

1) (P

i_o_Q)i_o_{R = P}i_o_(Qi_o_R),

2)

Pi o ( Qj) = (

P

io Qj )

3)

Pi o ( Qj) (Pio Qj)

4)

( Pj) i o

Q =

(P

jio

Q)

5) ( P

j

)

o Q (Pjo Q

)

6)

(Pi o Q)-1

= Q

-1i o P-1

Definition:

The relation R on X2_{is i-transitive iff}

R(x,z) ≥ i [R(x,y),R(y,z)] ………....1 For all x,y,z ϵ X. a fuzzy relation R on x2_{is i-transitive iff}

Ri_o_{R R ……….2}

When a relation R is not i-transitive, we define its i-transitive closure as a relation RT (i) that is the smallest i-transitive relation containing R.

Properties of the i-transitive closure, let

R(n)_{= R}_{i o}_R(n-1)_{for n=2, 2, ….……….3} Where R is a fuzzy relation on X2_{and R}(1)=_R.

Inf – wi compositions of fuzzy relations

Given a continuous t-norm i, let

wi(a, b) = sup {x [0,1] / i(a, x) ≤ b } for every a, b ≤ [0,1].

Theorem 1:

For any a, aj, b, d [0,1], where j takes values from an index set J, operation wi has the following properties:

1. i (a, b) ≤ d iff wi(a, d) ≥ b 2. wi [wi(a, b), b ] ≥ a

3. wi [i(a, b), d] = wi [a, wi(b, d)]

4. a ≤ b implies wi(a, d) ≥ wi (b, d) and wi(d, a) ≤ wi(d, b) 5. i [wi(a, b),wi(b, d)] ≤ wi(a, d)

6. wi(inf aj, b) ≥ sup wi(aj, b) 7. wi(sup aj,b) = inf wi (aj,b) 8. wi [b, sup aj] ≥ inf wi(b,aj)

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www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science 9. wi[b, inf aj] = inf wi(b,aj)

10. i[a, wi(a, b)] ≤ b

Proof:

We demonstrate the full proof by proving only properties 1and 7.

Proof of property 1

If i(a, b) ≤ d, then b {x/i(a, x) ≤d } and

Consequently, b ≤ sup{ x / i(a, x) ≤ d } = wi(a, d) (by definition) Hence, wi(a, d) ≥ b.

Converse part:

If b ≤ wi(a, d), then i (a, b) ≤ i[a,wi(a, d)]

=i[a, sup{x/i(a, x) ≤ d}] = sup{i(a, x) / i(a, x) ≤ d} ≤ d (since i is continuous) Hence , i(a, b) ≤ d

Proof of property 7:

Let S= sup aj. Then aj ≤ s and

wi(s, b) ≤ wi(aj, b) for any j J ……….1 [by property 4, a ≤ b wi(b, d) ≤ wi(a, d) ]

Hence, wi (s, b) ≤ inf wi(aj, b) ( here S= inf aj) ………..2 On the other hand, since,

Inf wi(aj, b) ≤ wi (aj, b) for all j J ………..3 ( by property 1, i(a, b) ≤ d iff wi(a, d) ≥ b) we have

i (aj, inf wi(aj, b)) ≤ b for all j J. ………...4 Thus, i(S,inf wi (aj,b)) =sup (aj0,inf wi(aj,b)≤ b ( using 4 ) Again property 1, we have

wi(s, b) ≥ inf wi(aj, b) [i(a, b) ≤ d iff wi(a, d) ≥ b] ………..5 Consequently,

wi(sup aj,b) = wi(s, b) = inf wi(aj, b) Hence the proof.

Theorem 2:

Let P(X, Y), Q1(Y, Z) and Q2(Y, Z) and R(Z, Y) be fuzzy relations. If Q1 Q2 then Pwi_{⁰ Q}₁_Pwi_{⁰ Q}₂ and Q1wi ⁰ R Q2 wi ⁰ R Proof: If Q1 Q2 Q1 Q2 = Q1 ………….1 and Q1 ∪ Q2 = Q2 …………..2 Hence, (P wi _{⁰ Q}₁_{) (P}wi_{⁰ Q}₂_{) = P}wi_{⁰ (Q}₁_Q₂₎ =P wi _{⁰ Q}₁_{(using 1)}

[We know that, P wi_{⁰ (}

Qj) = (P wi ⁰ Qj)] P wi_{⁰ Q}₁_Pwi_{⁰ Q}₂_{[ using Q}₁_Q₂_] Similarly,

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[We know that (Pjwi ⁰ Q) = ( Pj) wi ⁰ Q ] =Q2wi ⁰ R [using 2]

Which implies that, Q1 wi ⁰ R Q2wi ⁰ R Hence the proof.

III.

CONCLUSION

Fuzzy binary relations generalize the concept of fuzzy sets to multidimensional universes and introduction the Notation of association between the elements of some universe of discourse. Fuzzy binary relations generalize the concept of relations in the same manner as Fuzzy sets generalize the fundamental idea of sets. Operations with fuzzy binary relations are important to process fuzzy model constructed via fuzzy relations. Relations are associations and remain at the very basis of most methodological approaches of science and Engineering fuzzy relations are more general constructs than function; they allow dependencies between several variables to be captured without necessarily committing to any particular direction of the variable being involved. These fuzzy relations can proven to be a powerful tool for decision making.

IV.

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