Fuzzy subset

The notion of fuzzy subset

Consider a vague property as big in a set S, for example a set of apples. Imagine we will define in some way "the set of big apples". Then to do this we can 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 apples 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 say that s is crisp if $s(x)\in \{0,1\}$ for every $x\in S$.

The idea is that such a notion enables us to represent the extension of predicates as "big","slow", "near" "similar", which are vague in nature. Indeed, 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. The notion of fuzzy subset is on the basis of fuzzy logic. 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 $x\in S$.

If we denote by $F(S)$ the class of all the fuzzy subsets of S, in such a way an algebraic structure $(F(S),\cup ,\cap ,-,\emptyset ,S)$ is defined. Such a 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 the logical connectives is conservative in the sense that the restriction to {0,1} coincide with the classical one. This entails that the whole fuzzy logic is conservative, i.e. is an extension of classical logic, in a sense. In particular, we have 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 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

Bibliography

• Cox E., The Fuzzy Systems Handbook (1994), ISBN 0-12-194270-8
• Elkan C.. The Paradoxical Success of Fuzzy Logic. November 1993. Available from Elkan's home page.
• Gerla G., Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer, 2001.
• Hájek P., Metamathematics of fuzzy logic. Kluwer 1998.
• Höppner F., Klawonn F., Kruse R. and Runkler T., Fuzzy Cluster Analysis (1999), ISBN 0-471-98864-2.
• Klir G. and Folger T., Fuzzy Sets, Uncertainty, and Information (1988), ISBN 0-13-345984-5.
• Klir G. , UTE H. St. Clair and Bo Yuan Fuzzy Set Theory Foundations and Applications,1997.
• Klir G. and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995) ISBN 0-13-101171-5
• Bart Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic (1993), Hyperion. ISBN 0-7868-8021-X
• Novák V., Perfilieva I, Mockor J., Mathematical Principles of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, (1999).
• Yager R. and Filev D., Essentials of Fuzzy Modeling and Control (1994), ISBN 0-471-01761-2
• Zimmermann H., Fuzzy Set Theory and its Applications (2001), ISBN 0-7923-7435-5.
• Zadeh L.A., Fuzzy Sets, ‘’Information and Control’’, 8 (1965) 338­353.