29 Mar 2017

Kaufmann (1.3) Introduction to the Theory of Fuzzy Subsets, “Le concept de sous-ensemble flou” / “The Concept of a Fuzzy Subset”


by Corry Shores


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[The following is summary. Unless otherwise noted, boldface is my own. Page citations refer to the French edition first / then the English. I apologize in advance for my distracting typos or other mistakes, because proofreading is incomplete.]




Summary of


Arnold Kaufmann


Introduction à la théorie des sous-ensembles flous

à l’usage des ingénieurs

(Fuzzy sets theory)

1. Eléments théoriques de base


Introduction to the Theory of Fuzzy Subsets.

Vol.1 Fundamental Theoretical Elements



Notions de base

Fundamental Notions



Le concept de sous-ensemble flou

The Concept of a Fuzzy Subset



Brief summary:

A fuzzy subset is one where the members admit of a variety of degrees of membership, ranging from 0 for not at all a member to 1 for fully a member, with all the decimal values between for the varying degrees of membership. We use a wavy line under the set name to designate it as a fuzzy subset, or we may use the wavy line under the set inclusion symbol.


We can define the members by assigning to them the set of values, partial or full, from a set M.


Fuzzy membership could also be written using the wavy line under the membership symbol.


We can also designate the degree of membership by writing it under the membership symbol.


Fuzzy subsets allow us to define imprecise concepts, like the fuzzy subset of integers very near 0. As we move away from 0, the membership values will decrease.






[In the prior section 1.2, Kaufman had an example of two sets:

E = {x1, x2, x3, x4, x5}

A = {x2, x3, x5}

(Here the E seems to operate like a domain of discourse). We can notate this using a valuation function μ which assigns 1 when a member is in a set and 0 when it is not, so:

μA(x) = 1 if xA

            = 0 if xA

We can then designate the memberships for A as:

A = μA(x1) = 0,   μA(x2) = 1, μA(x3) = 1,  μA(x4) = 0, μA(x5) = 1

We can also write these as pairs in this way:

A = {(x1, 0), (x2, 1), (x3, 1), (x4, 0), (x5, 1)}

(page 2).] In the example [see above] from the prior section, we designated members of E as either belonging to A or not belonging to A. The valuation function μ (or “characteristic function” as it is called here) only takes one of two values, 0 and 1. (p.4/4)


Kaufmann then has us think that the valuation function μ can take any value between 0 and 1. This allows for partial membership in the subset.

Imagine now that this characteristic function may take any value whatsoever in the interval [0, 1]. Thus, an element xi of E may not be a member of A (μA = 0), could be a member of A a little (μA near 0), may more or less be a member of A (μA neither too near 0 nor too near 1), could be strongly a member of A (μA near 1), or finally might be a member of A (μA = 1). In this manner the notion of membership takes on an interesting extension and leads, as we shall see, to very useful developments.



Such subsets are called fuzzy subsets, and they are notated by either placing a wavy line under the subset name or under the subset symbol. The degree of membership is then notated by writing the value under the membership symbol. [Perhaps we might say that we deal with fuzzy subsets rather than fuzzy sets, because we always need a domain of discourse whose members are established, and then these fully established members in the domain can then be said to be partial members of fuzzy subsets.]




So consider the fuzzy subset 3.1 above. We see that some members are only partially so. [His next point seems to be that we can use fuzzy subsets when dealing with vague predicates where there is still a scale of fittingness for that predicate. Let me quote.]

Thus, the fuzzy subset defined by (3.1) contains a little x1, does not contain x2, contains a little more x3, contains x4 completely, and a large part of x5. This will allow us to construct a mathematical structure with which one may be able to manipulate concepts that are rather poorly defined but for which membership in a subset is somewhat hierarchical. Thus, one may consider: in the set of men, the fuzzy subset of very tall men; in the set of basic colors, the fuzzy subset of deep green colors; in the set of decisions, a fuzzy subset of good decisions; and so forth. We shall go on to see how to manipulate these concepts that seem particularly well adapted to the imprecision prevalent in the social sciences.



Kaufmann then gives Zadah’s rigorous definition, Z1. [Let us look first at the example, as it can help understand the structures.



Here we see that E is like the domain of discourse, because it has all the possible members. M is the set of membership degrees or values that any member can take. We then define a fuzzy subset A by giving ordered pairs, where the first of the pair is the member name, and the second is the assigned partial value. For some reason, each pairing is enclosed in parentheses and separated by a vertical line. As you will see, there is a special symbol for the function that assigns the partial value to the fuzzy subset member. But I think I am misreading the notation for (3.4). The second member of the pair is ∀x∈E. But I do not know exactly what that would be, as something you could designate as a second member of an ordered pair. It does not seem to show in the examples. Please consult the text below to interpret it for yourself.]




[Kaufmann then restructures the above definition for Boolean functions. I am not sure what the important differences are.]




Of course:




Kaufmann then notes that “Thus, the notion of fuzzy subset is linked with the notion of a set and allows one to study, using mathematical structures, imprecise concepts” (5/5).


He next gives some examples for such imprecise concepts:

the fuzzy subset of numbers x approximately equal to a given real number n, where nR (R being the set of reals);

the fuzzy subset of integers very near 0;

let a be a real number and let x be a small positive increment given to a; then the numbers a + x form a fuzzy subset in the set of reals;

let H be an element of a lattice; the elements most near to H in the order relation form a fuzzy subset in the set of elements of the lattice.

(5/5, boldface his)


Kaufmann will use boldface to designate sets or subsets, and the wavy line under them to designate fuzzy subsets.






Kaufmann then uses these symbols to notate fuzzy membership.




We may also write the degree of membership under the membership symbol.




[Kaufmann then illustrates with three examples. The first one we already examined above. The second one is interesting, because it might remind us of Nolt’s example of a vague predicate, where each iteration of a statement in which we increase a figure by 1 we also decrease the truth value by a small amount. See Nolt Logics section 16.1.]




[Example three introduces some alternate notation. See p.7/7.]





Kaufmann, Arnold. 1975 [1973]. Introduction à la théorie des sous-ensembles flous à l’usage des ingénieurs (Fuzzy sets theory). 1: Eléments théoriques de base. Foreword by L.A. Zadeh. 2nd Edn. Paris: Masson.


Kaufmann, Arnold. 1975. Introduction to the Theory of Fuzzy Subsets. Vol.1: Fundamental Theoretical Elements. Foreword by L.A. Zadeh. English translation by D.L. Swanson. New York / San Francisco / London: Academic Press.




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