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. In the case of crisp relation there are only two degrees of relationship between the elements of sets in a crisp relation, i.e., “completely related” and “not related”. A crisp relation represents the presence or absence of association, interaction, or interconnectedness between the elements of two or more sets. But fuzzy relations have infinite number of relationship between the extremes of completely related and not related between the elements of two or more sets considered.

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

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

Cartesian Product The elements in two sets A and B are given as 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)}.

Classical Relations A relation among classical sets x1, x2, . . . , xn and y1, y2, . . . , yn is a subset of the 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 a value of unity (1) is associated with complete relationship and a value of zero is associated with no relationship, i.e.,

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

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

Classical Relations 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 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 0 0 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 0 0 0 Rupees 0 0 0 0 English

Classical Relations Example: In many biological models, members of certain species can reproduce only with certain members of another species. Hence, only some elements in two or more universes have a 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 Composition Let R be a relation that relates, or maps, elements from universe X to universe Y, and let S be 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 relates the same elements in universe X that R contains to the same elements in universe Z that S contains. we can find such a relation using an operation known as composition

Classical Composition 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:

Composition 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?

Classical Composition 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 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 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

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

Fuzzy Relation Cartesian Product 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 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. For example, let

Fuzzy Relation Cartesian Product 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. For example, let The fuzzy Cartesian product results in a fuzzy relation R∼ representing ‘‘efficient’’ conditions,

Fuzzy Relation Composition 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 the set-theoretic notation and membership function-theoretic notation in the following manner: fuzzy max–product composition is defined in terms of the membership function theoretic notation as

Fuzzy Relation Composition Example: Fuzzy relationships for X × Y (denoted by the fuzzy relation R∼) 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-product composition

Fuzzy Relation Composition Solution:Max–Min Composition T

Fuzzy Relation Composition Solution:Max–Product Composition