Fuzzy subset

The notion of fuzzy subset

Fuzzy set theory is an attempt to represent the extension of vague properties. Given a well defined property P and a set S, the axiom of comprehension reads that there exists a set B whose members are precisely those objects in S that satisfy P. Such a set is called the extension of P. For example if S is the set of natural numbers and P is the property "to be prime", then the set B of prime numbers is defined. Assume that P is a vague property as "to be big", "to be youngh": there is a way to define the extension of P ? For example: is there a precise definition of the notion of set of big objects ? In order to give an aswer to this question recall that the caracteristic function of a classical subset X of a set S is the map $c_{X}:S\rightarrow \{0,1\}$ such that $c_{X}(x)=1$ if x is an element in X and $c_{X}(x)=0$ otherwise. Obviously, it is possible to identify X with its characteristic function $c_{X}$. This suggests that we can define the subset of big elements by a generalized caracteristic function in which instead of the Boolean algebra {0,1} we can consider the complete lattice [0,1]. The following is a precise definition.

Definition Given a nonempty set S, a fuzzy subset of S is a map s from S into the interval [0,1]. We denote by $F(S)$ the class of all the fuzzy subsets of S.

The elements in [0,1] are interpreted as truth values and, in accordance, for every x in S, the value s(x) is interpreted as the membership degree of x to s.

We say that a fuzzy subset s is crisp if $s(x)\in \{0,1\}$ for every $x\in S$. By associating every classical subsets of S with its caracteristic function, we can identify the subsets of S with the crisp fuzzy subsets. In particular we identify $\emptyset$ with the fuzzy subset constantly equal to 0 and $S$ with the fuzzy subset constantly equal to 1.

Some set-theoretical notions for fuzzy subsets

In classical mathematics the definitions of union, intersection and complement are related with the interpretation of the basic logical connectives $\vee ,\wedge ,\neg$. In order to define the same operations for fuzzy subsets, we have to fix a multi-valued logic and therefore suitable operations $\oplus ,\otimes$ and ~ to interpret these connectives. Once this was done, we can set

$(s\cup t)(x)=s(x)\oplus t(x)$,

$(s\cap t)(x)=s(x)\otimes t(x)$,

$(-s)(x)=~s(x)$.

Also, the inclusion relation is defined by setting

$s\subseteq t\Leftrightarrow s(x)\leq t(x)$ for every $x\in S$.

In such a way an algebraic structure $(F(S),\cup ,\cap ,-,\emptyset ,S)$ is defined and this structure is the direct power of the structure $([0,1],\oplus ,\otimes ,$~,0,1) with index set S. In Zadeh's original papers the operations $\oplus ,\otimes$, ~ are defined by setting for every x and y in [0,1]:

$x\otimes y$ = min(x, y)

$x\oplus y$ = max(x,y)

$~x$ = 1-x.

In such a case $(F(S),\cup ,\cap ,-,\emptyset ,S)$ is a complete, completely distributive lattice with an involution. Several authors prefer to consider different operations, as an example to assume that $\otimes$ is a triangular norm and that $\oplus$ is the corresponding triangular co-norm.

In all the cases the interpretation of a logical connectives is conservative in the sense that its restriction to {0,1} coincides with the classical one. This entails that the map associating any subset X of a set S with the related caracteristic function is an embedding of the Boolean algebra $(P(S),\cup ,\cap ,-,\emptyset ,S)$ into the algebra $(F(S),\cup ,\cap ,-,\emptyset ,S)$.

An extension of these definitions to the general case in which instead of [0,1] we consider different algebraic structures L is obvious. In such a case one uses the expression L-subset.

Is fuzzy logic a proper extension of classical logic ?

In all the cases the interpretation of a logical connectives is conservative in the sense that its restriction to {0,1} coincides with the classical one. This entails that the map associating any subset X of a set S with the related caracteristic function is an embedding of the Boolean algebra $(P(S),\cup ,\cap ,-,\emptyset ,S)$ into the algebra $(F(S),\cup ,\cap ,-,\emptyset ,S)$. This fact can be interpreted by saying that fuzzy logic is conservative, i.e. it is a proper extension of classical logic. On the other hand it is evident also that fuzzy logic is defined inside classical mathematics and therefore inside classical logic. As a matther of fact fuzzy logic is a (small) chapter of classical mathematics. This means that, differently from intuitionistic logic, fuzzy logic cannot be considered as an alternative philosophy in a trict sense. An extension of these definitions to the general case in which instead of [0,1] we consider different algebraic structures is obvious.

Fuzzy logic and probability

Many peoples compare fuzzy logic with probability theory since both refer to the interval [0,1]. However, they are conceptually distinct since we have not confuse a degree of truth with a probability measure. To illustrate the difference, consider the following example: Let $\alpha$ be the claim "the rose on the table is red" and imagine we can freely examine the rose (complete knowledge) but, as a matter of fact, the color looks not exactly red. Then $\alpha$ is neither fully true nor fully false and we can express that by assigning to $\alpha$ a truth value, as an example 0.8, different from 0 and 1 (fuzziness). This truth value does not depend on the information we have since this information is complete.

Now, imagine a world in which all the roses are either clearly red or clearly yellow. In such a world $\alpha$ is either true or false but, inasmuch as we cannot examine the rose on the table, we are not able to know what is the case. Nevertheless, we have an opinion about the possible color of that rose and we could assign to $\alpha$ a number, as an example 0.8, as a subjective measure of our degree of belief in $\alpha$ (probability). In such a case this number depends strongly from the information we have and, for example, it can vary if we have some new information on the taste of the possessor of the rose.

See also

• Neuro-fuzzy
• Fuzzy subalgebra
• Fuzzy associative matrix
• FuzzyCLIPS expert system
• Fuzzy control system
• Fuzzy set
• Paradox of the heap
• Pattern recognition
• Rough set

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