Lecturer: A. Zadeh Shirazi

Fuzzy logic with engineering applications -- 3rd EDA. Zadeh Shirazi, et al

Cartesian Product Crisp Relations Operations on Crisp Relations Properties of Crisp Relations Composition

Fuzzy Relations Operations on Fuzzy Relations Properties of Fuzzy Relations Fuzzy Cartesian Product and Composition

Tolerance and Equivalence Relations Value Assignments Other Forms of the Composition Operation 2

Relations can also be used to represent similarity Relations are intimately involved in logic, approximate reasoning, rule-based systems,

nonlinear simulation, synthetic evaluation, classification,pattern recognition, and control.

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ordered r-tuple An ordered sequence of r elements (a1, a2, a3, . . . , ar )

Cartesian product of crisp sets A1,A2, . . . ,Ar the set of all r-tuples (a1, a2, a3, . . . , ar ) where a1 ∈ A1, a2 ∈ A2, and ar ∈ Ar denoted by A1 × A2 × ·· ·×Ar

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Example: A = {0, 1} and B = {a, b, c}

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r-ary relation over A1,A2, . . . ,Ar A subset of the Cartesian product A1 × A2 ×· · ·×Ar the most common case is for r = 2

▪ a subset of the Cartesian product A1 × A2▪ binary relation from A1 into A2

Example: Cartesian product of two universes X and Y X × Y = {(x, y) | x ∈ X, y ∈ Y}, Unconstrained matches between X and Y every element in universe X is related completely to every

element in universe Y

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Characteristic function strength of this relationship between ordered pairs of

elements denoted χ

▪ complete relationship▪ no relationship

strength of relation▪ a mapping from ordered pairs to the characteristic function

r-dimensional relation matrix7

Example: an unconstrained relation

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Example: a constrained relation X = {1, 2} , Y = {a, b}

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Example: A = {0, 1, 2} identity relation

▪ IA = {(0, 0), (1, 1), (2, 2)}

universal relation ▪ UA = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)}

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Example: continuous universe

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R and S as two separate relations on the Cartesian universe X × Y

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commutativity, associativity, distributivity, involution, idempotency De Morgan’s principles the excluded middle axioms

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R: relates elements from universe X to universe Y S: relates elements from universe Y to universe Z T: relates the same elements in universe X that R contains

to the same elements in universe Z that S contains

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max–min composition Physical analogy for the max–min composition

max–product (max–dot) composition

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Example:

Five other forms of the composition operator described at the end of this chapter.

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Fuzzy relations Mapping from the Cartesian space X × Y to the interval [0,1]

Strength of the mapping Membership function of the relation for ordered pairs from

the two universes

Cardinality 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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Commutativity Associativity Distributivity Involution Idempotency De Morgan’s principles

excluded middle axioms

null relation, O the complete relation, E

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fuzzy relations in general are fuzzy sets Cartesian product is a relation between some fuzzy sets

A tild : a fuzzy set on universe X B tild : a fuzzy set on universe Y

membership function

Each of the fuzzy sets could be thought of as a vector of

membership values; 20

Example: X = {x1, x2, x3}, Y = {y1, y2} A∼ “ambient” temperature ,B∼ “near-optimum” pressure

the Cartesian product might represent the conditions(temperature–pressure pairs) that are associated with“efficient” operations

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fuzzy max–min composition

fuzzy max–product composition

Caution: neither crisp nor fuzzy compositions are commutative

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Example: X = {x1, x2}, Y = {y1, y2}, Z = {z1, z2, z3}

max–min composition

max–product composition

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P∼ :number of black pixels S∼ : shape of the black pixel clusters (e.g., S1 is an ellipse

and S2 is a circle)

relationship between quantity and the shape

For another P’, what will be S’?

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reflexivity, symmetry, and transitivity

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An equivalence relation R has three properties: (1) reflexivity, (2)symmetry, and (3) transitivity.

Some familiar equivalence relation ▪ parallelism among lines in plane geometry, the relation of similarity

among triangles, the relation “works in the same building as” among workers of a given city, and others

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A tolerance (proximity ) relation R 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 withitself, where n is the cardinal number of the set defining R

Equivalent relation is useful in classification

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Example: ▪ X = {x1, x2, x3, x4, x5} = {Omaha, Chicago, Rome, London, Detroit}

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Example (Cont’) R1 can become an equivalence relation through one (1 ≤ n,

where n = 5) composition.

Inspection of the matrix shows that the first, second,andfifth columns are identical, that is, Omaha, Chicago, andDetroit are in the same class

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the strength of the link between two elements must be at

least equal to the strength of the weakest indirect chain involving other elements

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Any fuzzy tolerance relation can be reformed into a fuzzy equivalence relation by at most (n-1) compositions

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Example: “similarity” relation between five parameters

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Where do the membership values that are contained in a relation come from? 1. Cartesian product▪ to calculate relations from the Cartesian product of two or more fuzzy

sets

2. Closed-form expression▪ simple observation of a physical process▪ For a given set of inputs, we observe a process yielding a set of

outputs.▪ to express the relation as a closed-form algorithm of the form Y = f

(X), where X is a vector of inputs and Y is a vector of outputs.

3. Lookup table▪ If some variability exists, membership values on the interval [0, 1]

may lead us to develop a fuzzy relation from a lookup table.

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4. Linguistic rules of knowledge▪ linguistic knowledge, expressed as if–then rules. Such knowledge

may come from experts (Chapter 5 to 8)

5. Classification▪ issues associated with similarity are central to determining

relationships (Chapter 10)

6. Automated methods from input/output data▪ procedures used on input and output data, which could be

observed and measured from some complex process (chapter 7)

7. Similarity methods in data manipulation.

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Attempt to determine some sort of similar pattern or structure in data through various metrics The more robust a data set, the more accurate the

relational entities are in establishing relationships amongelements of two or more data sets.

1. Cosine Amplitude2. Max–Min Method

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With n data samples form a data array, X

xi is itself a vector of length m (features)

rij , a pairwise comparison of two data samples rij = μR(xi , yj )

Relation matrix size n × n reflexive and symmetric – hence a tolerance relation

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Cosine Amplitude relation

is related to the dot product When two vectors are colinear (most similar), their dot

product is unity; when the two vectors are at right angles to one another

(most dissimilar), their dot product is zero. 37

Example: Five separate regions with three damage states

Use the cosine amplitude method to express these data asa fuzzy relation?

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Example (Cont’):

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Relation is computationally simpler is different from max–min composition

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Example:

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f (·) is a logistic function (such as a sigmoid or a stepfunction) that limits the value of the function within theinterval [0, 1].

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Classical Relations and Fuzzy RelationsOutlineIntroductionCartesian ProductCartesian ProductCrisp RelationsCrisp RelationsCrisp RelationsCrisp RelationsCrisp RelationsCrisp RelationsCrisp Relations�Operations on Crisp RelationsCrisp Relations�Properties of Crisp RelationsCrisp Relations�CompositionCrisp Relations�CompositionCrisp Relations�CompositionFuzzy RelationsFuzzy Relations�Operations on Fuzzy RelationsFuzzy Relations�Properties of Fuzzy RelationsFuzzy Relations�Fuzzy Cartesian ProductFuzzy Relations�Fuzzy Cartesian ProductFuzzy Relations�Fuzzy CompositionFuzzy Relations�Fuzzy CompositionFuzzy Relations�Fuzzy CompositionTolerance and Equivalence RelationsTolerance and Equivalence Relations�Crisp Equivalence RelationTolerance and Equivalence Relations�Crisp Tolerance RelationTolerance and Equivalence Relations�Crisp Tolerance RelationTolerance and Equivalence Relations�Crisp Tolerance RelationTolerance and Equivalence Relations�Fuzzy Equivalence RelationTolerance and Equivalence Relations�Fuzzy Tolerance RelationTolerance and Equivalence Relations�Fuzzy Tolerance and Equivalence RelationValue AssignmentsValue AssignmentsValue Assignments�Similarity MethodsSimilarity Methods�Cosine AmplitudeSimilarity Methods�Cosine AmplitudeSimilarity Methods�Cosine AmplitudeSimilarity Methods�Cosine AmplitudeSimilarity Methods�Max–Min MethodSimilarity Methods�Max–Min MethodOther Forms of the Composition OperationExercises