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Classical Relations and Fuzzy Relations

A relation is of fundamental importance in all-engineering, science, and mathematically based fields . Relations are involved in logic, approximate reasoning, classification, rule-based systems, pattern recognition, and control. Relations represent the mapping of the sets.

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Classical Relations and Fuzzy Relations

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  1. A relation is of fundamental importance in all-engineering, science, and mathematically based fields. Relations are involved in logic, approximate reasoning, classification, rule-based systems,pattern recognition, and control. Relations represent the mapping of the sets. In the case of crisp relation there are only two degrees of relationship betweenthe elements of sets in a crisp relation, i.e., “completely related” and “notrelated”. A crisp relation represents the presence or absence ofassociation, interaction, or interconnectedness between the elements of two ormore sets. But fuzzy relations have infinite number of relationship between theextremes of completely related and not related between the elements of twoor more sets considered. Classical Relations and Fuzzy Relations

  2. For the crisp sets A1,A2, . . . , An, theset of n-tuples a1, a2, . . ., an, where a1 ∈ A1, a2 ∈ A2, . . . , an ∈ An, is calledthe Cartesian product of A1,A2, . . . , An. The Cartesian product is denotedby A1 × A2 ×· · ·×An. In Cartesian product the first element in each pair isa member of x and the second element is a member of y formally, X x Y Cartesian Product X x Y

  3. The elements in two sets A and B are given as A = {0, 1} and B = {e, f, g} The Cartesian product A × B,B × A,A × A,B × B ?. Cartesian Product

  4. The elements in two sets A and B are given as A = {0, 1} and B = {e, f, g} find the Cartesian product A × B,B × A,A × A,B × B. The Cartesian product for the given sets is as follows: B × A = {(e, 0), (e, 1), (e, 1), (f, 1), (g, 1)}, A × A = A2 = {(0, 0), (0, 1), (1, 0), (1, 1)}, B × B = B2 = {(e, e), (e, f), (e, g), (f, e), (f, f), (f, g), (g, e), (g, f), (g, g)}. Cartesian Product

  5. A relation among classical sets x1, x2, . . . , xn and y1, y2, . . . , yn is a subset ofthe Cartesian product. It is denoted by R X × Y = {(x, y)/x ∈ X, y ∈ Y. The strength of the relationship between ordered pairs of elements in each universe is measured by the characteristic function denoted by χ, where avalue of unity(1) is associated with complete relationship and a value of zero isassociated with no relationship, i.e., Classical Relations

  6. When the universe or the set are finite, a matrix canconveniently represent the relation. The matrix is called as relation matrix A two-dimensional matrix represents thebinary relation. If X = {2, 4, 6} and Y = {p, q, r}, if they both are related toeach other entirely, then the relation between them can be given by:…. Classical Relations

  7. When the universe or the set are finite, a matrix canconveniently represent the relation. The matrix is called as relation matrix A two-dimensional matrix represents thebinary relation. If X = {2, 4, 6} and Y = {p, q, r}, if they both are related toeach other entirely, then the relation between them can be given by: Classical Relations Sagittal diagram

  8. Example: Let R be a relation among the three sets X= {Hindi, English}, Y = {Dollar, Euro, Pound, Rupees}, and Z ={India, Nepal, United States, Canada} R (x, y, z) = {Hindi, Rupees, India} {Hindi, Rupees, Nepal} {English, Dollar, Canada} {English, Dollar, United States}. Relation Matrix ? Classical Relations

  9. Example: Let R be a relation among the three sets X= {Hindi, English}, Y = {Dollar, Euro, Pound, Rupees}, and Z ={India, Nepal, United States, Canada} R (x, y, z) = {Hindi, Rupees, India} {Hindi, Rupees, Nepal} {English, Dollar, Canada} {English, Dollar, United States}. Solution. India Nepal US Canada Dollar 0 0 0 0 Euro 0 0 00 Pound 0 0 0 0 Rupees 1 1 0 0 Hindi India Nepal US Canada Dollar 0 0 1 1 Euro 0 0 0 0 Pound 0 00 0 Rupees 0 0 0 0 English Classical Relations

  10. Example: In many biological models, members of certain species can reproduce only withcertain members of another species. Hence, only some elements in two or more universes havea relationship (nonzero) in the Cartesian product. For example, two-member species, i.e., for X = {1, 2} and for Y = {a, b}. Members 1 - a and 2 – b can reproduce new species. R = ? Relation matrix? Sagittal diagram? Classical Relations

  11. Let R be a relation that relates, or maps, elements from universe X to universe Y, and let Sbe a relation that relates, or maps, elements from universe Y to universe Z. A useful question we seek to answer is whether we can find a relation, T, that relatesthe same elements in universe X that R contains to the same elements in universe Z that Scontains. we can find such a relation using an operation known as composition Classical Composition

  12. The two methods of the composition operations are: – Max–min composition, – Max–product composition. The max–min composition is defined by the set-theoretic and membership function-theoretic expressions: The max–product composition is defined by the set-theoretic and membership function-theoretic expressions: Classical Composition

  13. We wish to find a relation T that relates the ordered pair (x1, z2), i.e., (x1, z2) ∈ T. In this example, R = {(x1, y1), (x1, y3), (x2, y4)} S = {(y1, z2), (y3, z2)} Sagittal diagram? Relation matrix? Max-min composition operations? Composition

  14. We wish to find a relation T that relates the ordered pair (x1, z2), i.e., (x1, z2) ∈ T. In this example, R = {(x1, y1), (x1, y3), (x2, y4)} S = {(y1, z2), (y3, z2)} Sagittal diagram Classical Composition

  15. We wish to find a relation T that relates the ordered pair (x1, z2), i.e., (x1, z2) ∈ T. In this example, R = {(x1, y1), (x1, y3), (x2, y4)} S = {(y1, z2), (y3, z2)} Relation matrix Classical Composition

  16. We wish to find a relation T that relates the ordered pair (x1, z2), i.e., (x1, z2) ∈ T. In this example, R = {(x1, y1), (x1, y3), (x2, y4)} S = {(y1, z2), (y3, z2)} Max-Min composition operations Classical Composition

  17. Fuzzy relations are fuzzy subsets of X×Y , i.e., mapping from X → Y . Fuzzy relations mapselements of one universe, X to those of another universe, say Y , through theCartesian product of the two universes. A fuzzy relation R∼is mapping fromthe Cartesian space X × Y to the interval [0, 1] where the strength of themapping is expressed by the membership function of the relation for orderedpairs. This can be expressed as is called a fuzzy relation on X × Y . Fuzzy Relation

  18. Let A∼be a fuzzy set on universe X and B∼be a fuzzy set on universe Y , then the Cartesian product between fuzzy sets A∼and B∼ will result in a fuzzy relation R∼ which is contained with the full Cartesian product space or where the fuzzy relation R∼has membership function. Fuzzy Relation Cartesian Product

  19. Example: Suppose we have two fuzzy sets, A∼defined on a universe of three discretetemperatures, 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 setB∼the ‘‘near optimum’’ pressure for a certainheat exchanger, and the Cartesian productmight represent the conditions (temperature–pressurepairs) of the exchanger that are associated with ‘‘efficient’’ operations. For example, let Fuzzy Relation Cartesian Product

  20. Example: Suppose we have two fuzzy sets, A∼defined on a universe of three discretetemperatures, 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 setB∼the ‘‘near optimum’’ pressure for a certainheat exchanger, and the Cartesian productmight represent the conditions (temperature–pressurepairs) of the exchanger that are associated with ‘‘efficient’’ operations. For example, let The fuzzy Cartesian productresults in a fuzzy relation R∼representing ‘‘efficient’’ conditions, Fuzzy Relation Cartesian Product

  21. Fuzzy composition can be defined just as it is for crisp (binary) relations. Suppose R∼is a fuzzy relation on the Cartesian space X × Y, S∼ is a fuzzy relation on Y × Z, and T∼is a fuzzy relation on X × Z; then fuzzy max–min composition is defined in terms of theset-theoretic notation and membership function-theoretic notation in the following manner: fuzzy max–product composition is defined in terms of the membershipfunctiontheoretic notation as Fuzzy Relation Composition

  22. Example: Fuzzy relationships for X × Y (denoted by the fuzzy relationR∼) and Y× Z (denoted by the fuzzy relation S∼). X = {x1, x2}, Y = {y1, y2}, and Z = {z1, z2, z3} Consider the following fuzzy relations: Find the resulting relation,T∼, which relates elements of universe X to elements of universe Z,i.e., defined on Cartesian space X × Z using max–min composition and using max-productcomposition Fuzzy Relation Composition

  23. Solution:Max–Min Composition Fuzzy Relation Composition T

  24. Solution:Max–Product Composition Fuzzy Relation Composition

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