Kaufmann (1.2) Introduction to the Theory of Fuzzy Subsets, “Rappel sur la notion d’appartenance” / “Review of the Notion of Membership”

 

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

 

Ch.1

Notions de base

Fundamental Notions

 

1.2

Rappel sur la notion d’appartenance

Review of the Notion of Membership

 

 

Brief summary:

Set A being a subset of E is written:

A ⊂ E

x being a member of A is written:

x ∈ A

We can also describe membership using a valuation function μ:

μ_A(x) = 1 if x ∈ A

            = 0 if x ∉ A

The complement of a set is notated by writing a line above its symbol. We can calculate whether or not some element is a member of a union or intersection of sets by using Boolean operations.

μ_{A ∩B}(x) = μ_A (x) .  μ_B(x)

 

μ_{A ∪B}(x) = μ_A (x) + μ_B(x)

 

 

 

 

 

Summary

 

[Kaufmann will define membership and subset inclusion. See especially Suppes’ Introduction to Logic sections 9.1-9.2 and section 9.3 for another great introduction to the concepts and notation.]

We suppose that E is a set and A is a subset of E. We then would write:

A ⊂ E.

(Kaufmann p.1/1). [This symbol is used to mean a proper subset, in Suppes’ text; (if all the members of one set are included in a second, but not all of the second are included in the first, then we call the first a proper subset of the second)]

Suppose E has an element x. If it is a member of A, we write it:

x ∈ A.

[The next idea seems to be that we can describe membership using a valuation function.]

In order to indicate this membership one may also use another concept, a characteristic function μ_A(x), whose value indicates (yes or no) whether x is a member of A:

μ_A(x) = 1 if x ∈ A

            = 0 if x ∉ A.

(p.1/1)

 

Kaufmann then gives an example. Here we have two sets with members:

E = {x₁, x₂, x₃, x₄, x₅}

A = {x₂, x₃, x₅}

(As you can see, the E seems to function like a domain of discourse).

We can then designate the memberships for A as:

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

We can also write these as pairs in this way:

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

(2/2)

 

Kaufmann next has us consider Boolean binary algebra. [Recall the following from Suppes’ Introduction section 9.5:

Certain operations can be performed on sets. If we find all the members shared in common between two sets, we are finding their intersection (∩):

(x)(x ∈ A ∩ B ↔ x ∈ A & x ∈ B)

When two intersecting sets share no members in common, that is, when they are mutually exclusive sets, their intersection is the empty set. The set containing all the members in total from two sets is their union (∪):

(x)(x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B)

All the members in set A that are not in set B is the difference (∼) of A and B.

(x)(x ∈ A ∼ B ↔ x ∈ A & x ∉ B)

And recall the following from section 9.6. A domain of individuals (also called a domain of discourse) is a specific set.  We use the symbol “V” to denote a domain. Suppose we have a domain V and a set A. The complement of A relative to the domain are all those items in the domain that are not in A. We symbolize it either as V∼A or just ∼A. In section 9.9 he noted a number of identities (here Λ is the empty set and ~ is the complement):

] [In the following, I cannot duplicate the notation using typographical symbols, so I will paste images for certain parts. A set’s complement has all the members in the domain that the first set lacks. The intersection of sets are all those members shared by both sets. Thus the intersection of a set and its complement will be empty. The union of sets are all those found in either set. So the union of a set and its complement will be all those in the domain.]

 

(2/2)

 

Kaufmann next will examine intersections in terms of Boolean products. We first consider two subsets, A and B, and their intersection A ∩ B. We thus can make the following determinations:

μ_A(x) = 1 if x ∈ A

            = 0 if x ∉ A ,

μ_B(x) = 1 if x ∈ B

            = 0 if x ∉ B ,

μ_{A ∩ B}(x) = 1 if x ∈ A ∩ B

            = 0 if x ∉ A ∩ B.

(2/2)

[The next piece of notation is a ‘.’, which seems aligned not above the base-line but right at it, like a period. I am not sure what it means at this point. I first I thought it was like arithmetical multiplication, which would hold in this case, but the ‘+’ operation to follow does not add the 1 values to get 2. It instead seems to be an operator that simply should be understood as conjunction, and + as disjunction.]

(2/2)

(3/3)

We then define the union of two subsets A and B by using the ‘+’ or Boolean sum operator [which seems to correspond to disjunction.]

μ_{A ∪B}(x) = 1 if x ∈ A ∪ B

            = 0 if x ∉ A ∪ B

μ_{A ∪B}(x) = μ_A (x) + μ_B(x)

or

(3/3)

 

[Kaufmann then gives more examples of these operations and also adds to them complementarity. See page 3/3.]

 

 

 

 

From:

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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