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3.7. Set theory operations and their properties

Before getting into the operations on fuzzy sets themselves, it is highly instructive to review the basic operations encountered in set theory and highlight relationships between them and fuzzy set relatives. The main properties of these operations form a suitable reference one can apply when analyzing various models of fuzzy set operations. These, in particular, concern the properties of commutativity, idempotency, associativity, ditributivity, and transitivity. As characteristic functions are equivalent representations of sets, the basic intersection, union and complement operations are conveniently represented by taking minimum, maximum, and complement of the corresponding characteristic functions for all x ∈ X,

where A and B are sets defined in an universe (universe of discourse) X, and (A∩B)(x), (A∪B)(x) denote the values of the membership functions of the set resulting from the intersection and union of A and B, respectively.

While the main features to be expressed below use standard set notation, the same formulas can be immediately re-expressed in the language of characteristic functions,

Commutativity

Associativity

Idempotency

Distributivity

Boundary conditions

Involution

Transitivity

The following are the relationships expressing properties of containment and cardinality. As a matter of fact, they are directly implied by the properties outlined above,

Historically, while relaxing the constraints on membership values, fuzzy sets have embarked on the minimum and maximum operations as the basic models of logic operations. An important breakthrough has been made by adapting t-norms and s-norm (t-conorms) as models of fuzzy set connectives.

3.8. Triangular norms

The concept of triangular norms comes from the ideas of so-called probabilistic metric spaces originally proposed by Menger (1942), Schweizer and Sklar (1983) and actively pursued by Weber (1983) and Butnariu and Klement (1993), to name a few pursuits in this area. The underlying crux of these spaces was to put a geometric concept of a triangular inequality in the setting of probability theory.

In fuzzy sets, triangular norms play the key role by providing generic models for intersection and union operations on fuzzy sets. To be qualified as such, these operations must possess the properties of commutativity, associativity and monotonicity. Boundary conditions to assure that they behave as set operations must also be satisfied. Therefore, triangular norms form general classes of intersection and union operators. Formally they are introduced as follows.

t-norm A t-norm is a binary operation t : [0,1]2 →[0, 1] satisfying the requirements of

• commutativity x t y = y t x
• associativity x t (y t z) = (x t y) t z
• monotonicity if x ≤ y and w ≤ z then x t w ≤ y t z
• boundary conditions 0 t x = 0     1 t x = x.

As a formal construct, s-norms are dual to the t-norms.

s-norm An s-norm (known also as a triangular co-norm) is a binary operation s : [0,1]2 →[0, 1] satisfying the requirements of

• commutativity x s y = y s x
• associativity x s(y s z) = (x z y) s z
• monotonicity if x ≤ y and w ≤ z then x t w ≤ y’ t z’
• boundary conditions x s 0 = x     x s 1 = 1

Clearly, the min operator (⊥) is a t-norm whereas the max operator (⊦) is a s-norm. They correspond to set intersection and union operators, respectively, when membership degrees are constrained to the two-element set of grades of belongingness, {0,1}. Thus, they may be regarded as natural extensions of set intersection and union operations to fuzzy sets. Some of the most frequently triangular norms encountered in the literature are listed below

Examples of t-norms:

Examples of s-norms:

For each t-norm there exists a dual s-norm; this means

and, alternatively,

Once rewritten in the form

it is seen immediately that these two relationships are just De Morgan laws being commonly encountered in set theory

The complement of a fuzzy set A, , is defined analogously as in set theory, namely:

According to this definition, the complement is involutive, meaning that

so it behaves analogously as its well-known counterpart found in set theory.

An interesting generalization of t-norms comes in the form of a so-called ordinal sum (Butnariu and Klement, 1993). This construct is developed as a combination of a countable number of t-norms each of them is defined in some region of the unit square, see Fig.3.5. More formally, the ordinal sum of tj with respect to {(aj, bj)}jJ is defined as


Figure 3.5  Construction of the ordinal sum

3.9. Triangular norms as the models of operations on fuzzy sets

In general, the triangular norms do not satisfy the laws of contradiction and excluded middle, and , respectively. This can be easily checked for the min and max operations. More specifically, we obtain a so-called overlap and underlap effect where the intersection and union of the fuzzy set and its complement do not produce A. These two important departures from the classic laws of set theory arise because of operating on membership values situated between 0 and 1. Observe that the most profound deviation occurs for the membership value equal 1/2. Consider A with the constant membership value of 1/2 over X. In this situation the minimum and maximum operations return the same value for intersection and union of A and its complement,

for each x in X. One can say that A is the most fuzzy set among all fuzzy sets defined in X. In other words, a level of failing the requirements of excluded middle and contradiction can serve as an useful measure of expressing how fuzzy a certain fuzzy set is. We will return to this way of evaluating fuzziness. An exception is a bounded sum (t2 with p = 0) and bounded product (s2 with p = 0); they always produce


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