2. Course : T0423-Current Popular IT III
Year : 2013
Properties of Membership Functions : Fuzzification and Defuzification Session 3

3. Learning Outcome
After taking this course, students should be expected to explain and discuss the fuzzification and defuzzification
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4. Lecture Outline Cartesian product
Classical Relations and Fuzzy Relations
Features of Membership Functions
Fuzzy Set Operations
Crisp Set Operations
Fuzzification and Defuzzification
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5. Cartesian Product
An ordered sequence of r elements, written in the form (a1, a2, a3, , ar ), is called an ordered r-tuple; an unordered r-tuple is simply a collection of r elements without restrictions on order.
In a ubiquitous special case where r = 2, the r-tuple is referred to as an ordered pair. For crisp sets A1,A2, ,Ar, the set of all r-tuples (a1, a2, a3, , ar ),
where a1 ∈ A1, a2 ∈ A2, and ar ∈ Ar , is called the Cartesian product of A1,A2, ,Ar , and is denoted by A1 × A2 × ·· ·×Ar .
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6. Example
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:
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7. Crisp Relations
A subset of the Cartesian product A1 × A2 ×· · ·×Ar is called an r-ary relation over A1,A2, ,Ar . Again, the most common case is for r = 2; in this situation, the relation is a subset of the Cartesian product A1 × A2
The Cartesian product of two universes X and Y is determined as
X × Y = {(x, y) | x ∈ X, y ∈ Y}
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8. Cardinality of Crisp Relations
Suppose n elements of the universe X are related (paired) to m elements of the universe Y. If the cardinality of X is nX and the cardinality of Y is nY, then the cardinality of the relation, R, between these two universes is nX×Y = nX ∗ nY. The cardinality of the power set describing this relation, P(X × Y), is then
nP(X×Y) = 2(nXnY).
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9. example
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10. Fuzzy Relations
Fuzzy relations also map elements of one universe, say X, to those of another universe, say Y, through the Cartesian product of the two universes.
However, the “strength” of the relation between ordered pairs of the two universes is not measured with the characteristic function, but rather with a membership function expressing various “degrees” of strength of the relation on the unit interval [0,1].
A fuzzy relation R∼ is a mapping from the Cartesian space X × Y to the interval [0,1]
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11. Exercise (1)
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. These cities can be enumerated as the elements of a set, that is,
X = {x1, x2, x3, x4, x5} = {Omaha, Chicago, Rome, London, Detroit}.
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12. Exercise (2)
Further, suppose we have a tolerance relation, R1, that expresses relationships among these cities:
This relation is reflexive and symmetric. The graph for this tolerance relation would involve five vertices (five elements in the relation)
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13. Exercise (3)
The property of symmetry might represent proximity: Omaha and Chicago (x1 and x2) are close (in a binary sense) geographically, and Chicago and Detroit (x2 and x5) are close geographically. This relation, R1, does not have properties of transitivity, for example,
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14. Exercise (4) R1 can become an equivalence relation through one
(1 ≤ n, where n = 5) composition.
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15. Features of Membership Functions The core of a membership function :
the region of the universe that is characterized by complete and full membership in the set A∼ That is, the core comprises those elements x of the universe such that μA∼(x) = 1.
The support of a membership function :
The region of the universe that is characterized by nonzero membership in the set A∼. That is, the support comprises those elements x of the universe such that μA∼ (x)>0.
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16. Features of Membership Functions
The boundaries of a membership function for some fuzzy set A∼ are defined as that region of the universe containing elements that have a nonzero membership but not complete membership.
That is, the boundaries comprise those elements x of the universe such that
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17. Features of Membership Functions
A normal fuzzy set is one whose membership function has at least one element x in the universe whose membership value is unity. Figure below illustrates typical normal and subnormal fuzzy sets.
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18. Features of Membership Functions
A convex fuzzy set is described by a membership function whose membership values are strictly monotonically increasing, or whose membership values are strictly monotonically decreasing, or whose membership values are strictly monotonically increasing then strictly monotonically decreasing with increasing values for elements in the universe.
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19. then A∼ is said to be a convex fuzzy set
Said another way, if, for any elements x, y, and z in a fuzzy set A∼, the relation x < y <z implies that
then A∼ is said to be a convex fuzzy set
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20. Type-2 fuzzy set
For a particular element, x = z, the membership in a fuzzy set A∼, that is, μA∼(z), would be expressed by the membership interval [α1, α2]. Interval-valued fuzzy sets can be generalized further by allowing their intervals to become fuzzy. Each membership interval then becomes an ordinary fuzzy set. This type of membership function is referred to in the literature as a type-2 fuzzy set.
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21. Fuzzification (1)
Fuzzification is the process of making a crisp quantity fuzzy. We do this by simply recognizing that many of the quantities that we consider to be crisp and deterministic are actually not deterministic at all; they carry considerable uncertainty.
If the form of uncertainty happens to arise because of imprecision, ambiguity, or vagueness, then the variable is probably fuzzy and can be represented by a membership function.
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22. Fuzzification (con’t)
In the real world, hardware such as a digital voltmeter generates crisp data, but these data are subject to experimental error.
The information shown in Figure below shows one possible range of errors for a typical voltage reading and the associated membership function that might represent such imprecision.
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23. Fuzzification (con’t)
The representation of imprecise data as fuzzy sets is a useful. We consider the data as a crisp reading.
In Figure a below, we might want to compare a crisp voltage reading to a fuzzy set, say “low voltage.” In the figure, we see that the crisp reading intersects the fuzzy set “low voltage” at a membership of 0.3, that is, the fuzzy set and the reading can be said to agree at a membership value of 0.3.
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24. Fuzzification (con’t)
In Figure b, the intersection of the fuzzy set “medium voltage” and a fuzzified voltage reading occurs at a membership of 0.4.
We can see in Figure b that the set intersection of the two fuzzy sets is a small triangle, whose largest membership occurs at the membership value of 0.4.
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25. Defuzzification
Defuzzification is the conversion of a fuzzy quantity to a precise quantity, just as fuzzification is the conversion of a precise quantity to a fuzzy quantity. The output of a fuzzy process can be the logical union of two or more fuzzy membership functions defined on the universe of discourse of the output variable.
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26. Defuzzification to Crisp Sets
We begin by considering a fuzzy set , then define a lambda-cut set, Aλ, where 0 ≤ λ ≤ 1.
The set Aλ is a crisp set called the lambda (λ)-cut (or alpha-cut) set of the fuzzy set , where Aλ = {x|μA∼ (x) ≥ λ}. Note that the λ-cut set Aλ does not have a tilde underscore;
it is a crisp set derived from its parent fuzzy set, . Any particular fuzzy set can be transformed into an infinite number of λ-cut sets, because there are an infinite number of values λ on the interval [0, 1].
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27. Example
Let us consider the discrete fuzzy set, using Zadeh’s notation, defined on universe X = {a, b, c, d, e, f },
This fuzzy set is shown schematically in Figure 4.8. We can reduce this fuzzy set into several λ-cut sets, all of which are crisp. For example, we can define λ-cut sets for the values of λ = 1, 0.9, 0.6, 0.3, 0+, and 0.
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28. Centroid method
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29. Example (cont.)
We can express λ-cut sets using Zadeh’s notation. For example, λ-cut sets for the values λ = 0.9 and 0.25 are given here:
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30. Exercise 4.3(individu)
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33. References Fuzzy Logic with Engineering Applications, Chapter 3 and 4.
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