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Classical Relations And Fuzzy Relations

Baran Kaynak

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Relations•This chapter introduce the notion of

relation.•The notion of relation is the basic idea

behind numerous operations on sets such as Cartesian products, composition of relations , difference of relations and intersections of relations and equivalence properties

• In all engineering , science and mathematically based fields, relations is very important

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Relations•Similarities can be described with

relations.•In this sense, relations is a very important

notion to many different technologies like graph theory, data manipulation.

Graph theory

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Data manipulations

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•In classical relations (crisp relations),Relationships between elements of the sets

areonly in two degrees; “completely related”

and“not related”.

•Fuzzy relations take on an infinitive number of degrees of relationships between the extremes of “ completely related” and “ not related”

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Crisp system

-Crisp, exact- Based on models (i.e. differential equations)- Requires complete set of data- Typically linear

Fuzzy system

- Fuzzy, qualitative, vague- Uses knowledge (i.e. rules)- Requires fuzzy data- Nonlinear method

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Crisp system

-Complex systems hard to model-incomplete informationleads to inaccuracy-numerical

Fuzzy logic system

-No traditional modeling, inferences based on knowledge- can handle incomplete information to some degree-linguistic

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•Example 3.1. The elements in two sets A and B are given as A ={0, 1} and B ={a,b, c}.Various Cartesian products of these two sets can be written as shown:A × B ={(0,a),(0,b),(0,c),(1,a),(1,b),(1,c)}B × A ={(a, 0), (a, 1), (b, 0), (b, 1), (c, 0), (c, 1)}A × A = A2={(0, 0), (0, 1), (1, 0), (1, 1)}B × B = B2={(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}

Cartesian Product

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Crisp Relations•Cartesian product is denoted in form A1 x

A2 x…..x Ar•The most common case is for r=2 and

represent with A1 x A2•The Cartesian product of two universes X

and Y is determined asX × Y = {(x, y) | x ∈ X,y ∈ Y}

•This form shows that there is a matching between X and Y , this is a unconstrained matching.

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Crisp Relations•Every element in universe X is related

completely to every element in universe Y•This relationship’s strenght is measured

by the characteristics function χχX×Y(x, y) = 1, (x,y) ∈ X × Y

0, (x,y) ∉ X × Y•Complete relationship is showed with 1

and no relationship is showed with 0

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•When the universes, or sets, are finite the relation can be conveniently represented by amatrix, called a relation matrix.

X ={1, 2, 3} and Y ={a, b, c}

Sagittal diagram of an unconstrained relation

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•Special cases of the constrained Cartesian product for sets where r=2 are called identity relation denoted IA

IA ={(0, 0), (1, 1), (2, 2)}

•Special cases of the unconstrained Cartesian product for sets where r=2 are called universal relation denoted UAUA ={(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)}

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Cardinality Of Crips Relations

•The Cardinality of the relation r between X and Y

is n X x Y = nx * ny

•Power set (P(X x Y)), nP(X×Y) = 2(nXnY)

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Operations On Crips Relations•Define R and S as two separate relations on

the Cartesian universe X × YUnion: R ∪ S → χR∪S(x, y) : χR∪S(x, y) =

max[χR(x, y), χS(x, y)]Intersection: R ∩ S → χR∩S(x, y) : χR∩S(x, y)

= min[χR(x, y), χS(x, y)]Complement: R →χR(x, y) : χR(x, y) = 1 −

χR(x, y) Containment: R ⊂ S →χR(x, y) : χR(x, y) ≤

χS(x, y)

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Properties Of Crips Relations•Commutativity•Associativity•Distributivity•Involution•Idempotency

2 × (1 + 3) = (2 × 1) + (2 × 3).

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Composition

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Crisp Binary Relation

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Composition•For these two relations lets make a

composition named T ▫R = {(x1, y1), (x1, y3), (x2, y4)}▫S = {(y1, z2), (y3, z2)}

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There are two common forms of the

composition operation;•max–min composition•T = R.S•χT(x, z)(χR(x, y) ∧ χS(y, z))•max–product composition•T = R.S•χT(x, z)(χR(x, y) . χS(y, z))

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A chain is only as strong as its weakest link

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Example

µT(x1, z1) = max[min(1, 0), min(0, 0), min(1, 0), min(0, 0)] = 0

• Using the max–min composition operation, relation matrices for R and S would be expressed as

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Example• Using the max–min composition operation, relation

matrices for R and S would be expressed as

µT(x1, z1) = max[min(1, 0), min(0, 0), min(1, 0), min(0, 0)] = 0µT(x1, z2) = max[min(1, 1), min(0, 0), min(1, 1), min(0, 0)] = 1

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Fuzzy Relations•A fuzzy relation R is a mapping from the

Cartesian space X x Y to the interval [0,1], where the strength of the mapping is expressed by the membership function of the relation μR(x,y)▫μR : A × B → [0, 1]▫R = {((x, y), μR(x, y))| μR(x, y) ≥ 0 , x ∈ A, y

∈ B}

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Crisp relation vs. Fuzzy relation

Crisp relation Fuzzy relation

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Cardinality of Fuzzy Relations•Since the cardinality of fuzzy sets on any

universe is infinity, the cardinality of a fuzzy relation between two or more universes is also infinity.

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Operations on Fuzzy Relations• Let R and S be fuzzy relations on the Cartesian space X ×

Y. Then the following operations apply for the membership values for various set operations:

Union: µR∪S (x, y) = max(µR (x, y),µS(x, y)) Intersection: µR∩S (x, y) = min(µR (x, y),µS (x, y)) Complement:µR(x, y) = 1 − µR(x, y) Containment:R⊂ S ⇒ µR (x, y) ≤ µS (x, y)

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Fuzzy Cartesian Product and Composition•A fuzzy relation R is a mapping from the

Cartesian space X x Y to the interval [0,1], where the strength of the mapping is expressed by the membership function of the relation μR(x,y)▫μR : A × B → [0, 1]▫R = {((x, y), μR(x, y))| μR(x, y) ≥ 0 , x ∈ A, y

∈ B}

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Max-min Composition•Two fuzzy relations R and S are defined

on sets A, B and C. That is, R ⊆ A × B, S ⊆ B × C. The composition S•R = SR of two relations R and S is expressed by the relation from A to C:▫For (x, y) ∈ A × B, (y, z) ∈ B × C,

µS•R (x, z) = max [min (µR (x, y), µS (y, z))] = ∨ [μR (x, y) ∧ μS (y, z)]

MS•R = MR•MS (matrix notation)

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Max-min Composition

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Max-product Composition•Two fuzzy relations R and S are defined

on sets A, B and C. That is, R ⊆ A × B, S ⊆ B × C. The composition S•R = SR of two relations R and S is expressed by the relation from A to C:▫For (x, y) ∈ A × B, (y, z) ∈ B × C,

μS•R (x, z) = maxy [μR (x, y) • μS (y, z)] = ∨y [μR (x, y) • μS (y, z)MS•R = MR • MS (matrix notation)

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Example• Suppose we have two fuzzy sets, A defined on a universe of

three discrete temperatures, X = {x1, x2, x3}, and B defined on a universe of two discrete pressures, Y = {y1, y2}, and we want to find the fuzzy Cartesian product between them. Fuzzy set A could represent the ‘‘ambient’’ temperature and fuzzy set B the ‘‘near optimum’’ pressure for a certain heat exchanger, and the Cartesian product might represent the conditions (temperature–pressure pairs) of the exchanger that are associated with ‘‘efficient’’ operations.

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•Fuzzy Cartesian product, using µS•R (x, z) = max [min (µR (x, y), µS (y, z))]

results in a fuzzy relation R (of size 3 × 2) representing ‘‘efficient’’ conditions,

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Example•X = {x1, x2}, Y = {y1, y2}, and Z = {z1,

z2, z3} Consider the following fuzzy relations:

Then the resulting relation, T, which relates elements of universe X to elements of universe Z,μT(x1, z1) = max[min(0.7, 0.9), min(0.5, 0.1)] = 0.7

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and by max–product composition,

μT(x2, z2) = max[(0.8 . 0.6), (0.4 . 0.7)] = 0.48

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Example• A simple fuzzy system is given, which models the brake

behaviour of a car driver depending on the car speed. The inference machine should determine the brake force for a given car speed. The speed is specified by the two linguistic terms "low" and "medium", and the brake force by "moderate" and "strong". The rule base includes the two rules

• (1) IF the car speed is low THEN the brake force is moderate• (2) IF the car speed is medium THEN the brake force is strong

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Crisp Equivalence Relation•A relation R on a universe X can also be

thought of as a relation from X to X. The relation R is

an equivalence relation and it has the following three properties:

•Reflexivity•Symmetry•Transitivity

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Reflexivity•(xi ,xi ) ∈ R or χR(xi ,xi ) = 1•When a relation is reflexive every vertex

in the graph originates a single loop, as shown in

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Symmetry• (xi, xj ) ∈ R → (xj, xi) ∈ R

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Transitivity•(xi ,xj ) ∈ R and (xj ,xk) ∈ R → (xi ,xk) ∈ R

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Crisp Tolerance Relation•A tolerance relation R (also called a

proximity relation) on a universe X is a relation that

exhibits only the properties of reflexivity and symmetry.

•A tolerance relation, R, can be reformed into an equivalence relation by

at most (n − 1) compositions with itself, where n

is the cardinal number of the set defining R, in this case X

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Example

•Suppose in an airline transportation system we have a universe composed of five elements: the cities Omaha, Chicago, Rome, London, and Detroit. The airline is studying locations of potential hubs in various countries and must consider air mileage between cities and takeoff and landing policies in the various countries.

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Example•These cities can be enumerated as the elements of a set, i.e., X ={x1,x2,x3,x4,x5}={Omaha, Chicago,

Rome, London, Detroit}• Suppose we have a tolerance relation, R1,

that expresses relationships among these cities:

This relation is reflexive and symmetric.

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Example•The graph for this tolerance relation

•If (x1,x5) ∈ R1 can become an equivalence relation

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Example:•This matrix is equivalence relation

because it has (x1,x5)

Five-vertex graph of equivalence relation (reflexive, symmetric,

transitive)

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FUZZY TOLERANCE AND EQUIVALENCE RELATIONS

•Reflexivity μR(xi, xi) = 1

•Symmetry μR(xi, xj ) = μR(xj, xi)

•Transitivity μR(xi, xj ) = λ1 and μR(xj, xk) = λ2 μR(xi, xk) = λ where λ ≥ min[λ1, λ2].

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Example• Suppose, in a biotechnology experiment, five potentially new

strains of bacteria have been detected in the area around an anaerobic corrosion pit on a new aluminum-lithium alloy used in the fuel tanks of a new experimental aircraft. In order to propose methods to eliminate the biocorrosion caused by these bacteria, the five strains must first be categorized. One way to categorize them is to compare them to one another. In a pairwise comparison, the following " similarity" relation,R1, is developed. For example, the first strain (column 1) has a strength of similarity to the second strain of 0.8, to the third strain a strength of 0 (i.e., no relation), to the fourth strain a strength of 0.1, and so on. Because the relation is for pairwise similarity it will be reflexive and symmetric. Hence,

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is reflexive and symmetric. However, it is not transitive

μR(x1, x2) = 0.8, μR(x2, x5) = 0.9 ≥ 0.8butμR(x1, x5) = 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,μR

2(x1, x2) = 0.8 ≥ 0.5 and μR2(x2, x4) = 0.5

butμR

2(x1, x4) = 0.2 ≤ min(0.8, 0.5)

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Finally, after one or two more compositions, transitivity results: