3. Crisp and Fuzzy Relations

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3.1. Introduction

A relation is of fundamental importance in all engineering fields.Relations can be also be used to represent similarity. Relations areinvolved in logic, classification, pattern recognition, and control

Some relations concern elements within the same universe: onemeasurement is larger than another, one event occurred earlier thananother, one element resembles another, etc.

Other relations concern elements from disjoint universes: themeasurement is large and its rate of change is positive, the x-coordinate is large and the y-coordinate is small, for example.

These examples are relationships between two objects, but inprinciple we can have relationships which hold for any number ofobjects.

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3.2. Crisp RelationA Crisp Relation R from a set A to a set B assigns to each

ordered pair exactly one of the following statements:(i)’’a is related to b’’ or (ii) ’’a is not related to b’’

The Cartesian Product AxB is the set of all possiblecombinations of the items of A and B. For example when:A = {a1,a2,a3} and B = {b1,b2}

The Cartesian product yieldsthe shown figure.

Which means:AxB = {(a1,b1),(a1,b2),(a2,b1),(a2,b2),(a3,b1),(a3,b2)}2012

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example 1: “Owning Cars”-Crisp RelationX = {Aly, Baher, Kamel}Y = {BMW, Chrysler, Ford, Mazda, Fiat}

BMW Chrysler Ford Mazda FiatAly 1 0 0 0 1Baher 1 0 1 1 0Kamel 0 1 0 0 0

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example 2: “Close to”-Crisp RelationX = Y = {1, .......,8}

1 2 3 4 5 6 7 81 1 1 0 0 0 0 0 02 1 1 1 0 0 0 0 03 0 1 1 1 0 0 0 04 0 0 1 1 1 0 0 05 0 0 0 1 1 1 0 06 0 0 0 0 1 1 1 07 0 0 0 0 0 1 1 18 0 0 0 0 0 0 1 1

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3.3. Fuzzy Relation:Fuzzy relations map elements of one universe, say U, to those of

another universe, say V, through the Cartesian product of the twouniverses. However, the ‘‘strength’’ of the relation between orderedpairs of the two universes is measured with a membership functionexpressing various ‘‘degrees’’ of strength of the relation on the unitinterval [0,1].

As an example a fuzzy relation “Friend” describes the degree of friendshipbetween two persons (in contrast to either being friend or not being friend inclassical relation!)

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example 3: Fuzzy relation “Similarity” U = V = {1, . . . , 8}

1 2 3 4 5 6 7 81 1 0.5 0 0 0 0 0 02 0.5 1 0.5 0 0 0 0 03 0 0.5 1 0.5 0 0 0 04 0 0 0.5 1 0.5 0 0 05 0 0 0 0.5 1 0.5 0 06 0 0 0 0 0.5 1 0.5 07 0 0 0 0 0 0.5 1 0.58 0 0 0 0 0 0 0.5 1

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example 4: Let A be a fuzzy set defined on a universe of three discreteemperatures, X = {x1,x2,x3}, and B be a fuzzy set defined on a universeof two discrete pressures, Y = {y1,y2} Fuzzy set A represents the“ambient” temperature and fuzzy set B the “near optimum” pressure fora certain heat exchanger, and the Cartesian product might representthe conditions (temperature-pressure pairs) of the exchanger that areassociated with “efficient” operations. For example, let:

A = 0.2/x1 + 0.5/x2 + 1/x3 and B = 0.3/y1 + 0.9/y2

y1 y2Then AxB = R = x1 0.2 0.2

x2 0.3 0.5x3 0.3 0.9

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Because the fuzzy relation is one kind of fuzzy sets. Therefore wecan apply operations of fuzzy set to the relations (e.g. Union,Intersection, Complement,..).

example 4:For the given 2 fuzzy relations:

We get

and

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Also -cut can be calculated, but the result is a crisprelation. So for example

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3.4. Composition of Fuzzy Relations:

A fuzzy relation R is defined on sets A, B and anotherfuzzy relations S is defined on sets B,C.

That is, R ⊆ A x B, S ⊆ B x C.The composition S • R = SR of the two relations R and Sexpresses the relation from A to C.

This composition is defined by an inner product. The innerproduct is similar to an ordinary matrix (dot) product,except Multiplication is replaced by Minimum andSummation by Maximum. Thus this composition is defined bythe following

µ S • R (a, c) = max [ min (µR (a, b), µS (b, c))]

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example:Consider the fuzzy sets A, B and C to represent sets ofevents.The relation R ⊆ AxB, gives the possibility of occurrence ofB after A, and the relation S ⊆ BxC gives the possibility ofoccurrence of C after B.

For example, by the relation R, the possibility of b1 to occur after a1is 0.1. And by the relation S, the possibility of occurrence of c1 afterb1 is 0.9.

R b1 b2 b3 b4a1 0.1 0.2 0.0 1.0a2 0.3 0.3 0.0 0.2a3 0.8 0.9 1.0 0.4

S c1 c2 c3b1 0.9 0.0 0.3b2 0.2 1.0 0.8b3 0.8 0.0 0.7b4 0.4 0.2 0.3

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Now we want to get the possibility of occurrence of C when A has occurred.So our main job now will be the obtaining the composition S • R ⊆ AxC.The following matrix M S • R represents this composition and is obtainedfrom the product of MR and MS..

and it is also depicted in the following figure:

S • R c1 c2 c3a1 0.4 0.2 0.3a2 0.3 0.3 0.3a3 0.8 0.9 0.8

R b1 b2 b3 b4a1 0.1 0.2 0.0 1.0a2 0.3 0.3 0.0 0.2a3 0.8 0.9 1.0 0.4

S c1 c2 c3b1 0.9 0.0 0.3b2 0.2 1.0 0.8b3 0.8 0.0 0.7b4 0.4 0.2 0.3

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So, the possibility of occurrence of C when A has occurredhas been induced from the composition rule S • R . Thismanner is named as an “inference” which is a processproducing new information.

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Application: Computer EngineeringIn computer engineering, different logic families are often compared on thebasis of their power-delay product. Consider the fuzzy set F of logicfamilies, the fuzzy set D of delay times (ns), and the fuzzy set P of powerdissipations (mw).If F = {NMOS,CMOS,TTL,ECL,JJ}, D = {0.1,1,10,100},P = {0.01,0.1,1,10,100}Suppose R1 = D x F and R2 = F x P

then we can computeR3= R1 o R2 R3= R1 o R2

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There is a very interesting physical analogy for the max–mincomposition operator, the figure below illustrates several chainsplaced together in parallel. If we were to take one of these chainsout of the system, we would find that the chain would break at itsweakest link. Hence, the strength of one chain is equal to thestrength of its weakest link; in other words, the minimum (∩)strength of all the links in the chain governs the strength of theoverall chain.Now, if we take the entire chain system, we would find that theweaker chains would break at first until the strongest chain wasleft alone; in other words, the maximum (∪) strength of all thechains in the chain system would govern the overall strength ofthe chain system. Each chain in the system is analogous to themin operation in the max–min composition, and the overall chainsystem strength is analogous to the max operation in the max–min composition.

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Assume

Which can be represented by the graph

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from which

Ending with the result

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3.5. Properties of Fuzzy Relations:the properties of commutativity, associativity, distributivity,involution, idempotency and De Morgan’s principles all hold forfuzzy relations. But since there is overlap between a relation andits complement:

Where O= the null relation, and E=the complete relation e.g.

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3.5. Fuzzy Tolerance and Equivalence Relations:Let R be the fuzzy relation defined on the set of cities andrepresenting the concept very near. We may assume that a city iscertainly (i.e., to a degree of 1) very near to itself. The relation istherefore reflexive. Furthermore, if city A is very near to city B,then B is certainly very near to A. Therefore, the relation is alsosymmetric. Finally, if city A is very near to city B to somedegree, say .7, and city B is very near to city C to some degree,say .8, it is possible (although not necessary) that city A is verynear to city C to a smaller degree, say 0.5. Therefore, the relationis nontransitive.

A fuzzy relation is a fuzzy equivalence relation if all three of thefollowing properties for matrix relations define it:Reflexivity μR(xi, yi) = 1Symmetry μR(xi, yj ) = μR(xj, yi)Transitivity μR(xi, yj ) = λ1 and μR(xj, yk) = λ2

→ μR(xi, yk) =λ where λ ≥ min[λ1, λ2].2012

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Fuzzy Tolerance and Equivalence Relations (Contd.)

Tolerance relation (Aehnlichkeitsrelation), has only the properties ofreflexivity and symmetry.A tolerance relation, R, can be reformed into an equivalence relation by atmost (n − 1) compositions with itself, where n is is the number of rowsor columns of R.

Example:Consider the relation

is reflexive and symmetric. However, it is not transitive, e.g.,μR(x1, y2) = 0.8, and μR(x2, y5) = 0.9 ≥ 0.8

but μR(x1, y5) = 0.2 ≤ min(0.8, 0.9)

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One composition results in the following relation:

where transitivity still does not result; for example,μR2(x1, y2) = 0.8 ≥ 0.5 , and μR2(x2, y4) = 0.5But μR2(x1, y4) = 0.2 ≤ min(0.8, 0.5)Finally, after one or two more compositions, transitivity results:

R3(x1, y2) = 0.8 , and R3(x2, y4) = 0.5 , and R3(x1, y4) = 0.5 ≥ min(0.8, 0.5)2012

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