Fuzzy Sets

In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
At the same time, defined a more general kind of structure called an ''L''-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of ''L''-relations when ''L'' is the unit interval , 1

In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval , 1 Fuzzy sets generalize classical sets, since the indicator functions (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes values 0 or 1.D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York. In fuzzy set theory, classical bivalent sets are usually called ''crisp sets''. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.

Definition

A fuzzy set is a pair $(U, m)$ where $U$ is a set (often required to be non-empty) and m\colon U \rightarrow ,1/math> a membership function.
The reference set $U$ (sometimes denoted by $\Omega$ or $X$) is called universe of discourse, and for each $x\in U,$ the value $m(x)$ is called the grade of membership of $x$ in $(U,m)$.
The function $m = \mu_A$ is called the membership function of the fuzzy set $A = (U, m)$.

For a finite set $U=\,$ the fuzzy set $(U, m)$ is often denoted by $\.$

Let $x \in U$. Then $x$ is called
* not included in the fuzzy set $(U,m)$ if (no member),
* fully included if (full member),
* partially included if
The (crisp) set of all fuzzy sets on a universe $U$ is denoted with $SF(U)$ (or sometimes just $F(U)$).

Crisp sets related to a fuzzy set

For any fuzzy set $A = (U,m)$ and \alpha \in ,1/math> the following crisp sets are defined:
* $A^ = A_\alpha = \$ is called its α-cut (aka α-level set)
* $A^ = A'_\alpha = \$ is called its strong α-cut (aka strong α-level set)
* $S(A) = \operatorname(A) = A^ = \$ is called its support
* $C(A) = \operatorname(A) = A^ = \$ is called its core (or sometimes kernel $\operatorname(A)$).

Note that some authors understand "kernel" in a different way; see below.

Other definitions

* A fuzzy set $A = (U,m)$ is empty ($A = \varnothing$) iff (if and only if)
:: $\forall$$x \in U: \mu_A(x) = m(x) = 0$

* Two fuzzy sets $A$ and $B$ are equal ($A = B$) iff
::$\forall x \in U: \mu_A(x) = \mu_B(x)$

* A fuzzy set $A$ is included in a fuzzy set $B$ ($A \subseteq B$) iff
::$\forall x \in U: \mu_A(x) \le \mu_B(x)$

* For any fuzzy set $A$, any element $x \in U$ that satisfies
::$\mu_A(x) = 0.5$
:is called a crossover point.

* Given a fuzzy set $A$, any \alpha \in ,1/math>, for which $A^ = \$ is not empty, is called a level of A.
* The level set of A is the set of all levels \alpha\in ,1/math> representing distinct cuts. It is the image of $\mu_A$:
::$\Lambda_A = \ = \ = \mu_A(U)$

* For a fuzzy set $A$, its height is given by
::$\operatorname(A) = \sup \ = \sup(\mu_A(U))$
:where $\sup$ denotes the supremum, which exists because $\mu_A(U)$ is non-empty and bounded above by 1. If ''U'' is finite, we can simply replace the supremum by the maximum.

* A fuzzy set $A$ is said to be normalized iff
::$\operatorname(A) = 1$
:In the finite case, where the supremum is a maximum, this means that at least one element of the fuzzy set has full membership. A non-empty fuzzy set $A$ may be normalized with result $\tilde$ by dividing the membership function of the fuzzy set by its height:
::$\forall x \in U: \mu_(x) = \mu_A(x)/\operatorname(A)$
:Besides similarities this differs from the usual normalization in that the normalizing constant is not a sum.

* For fuzzy sets $A$ of real numbers (''U'' ⊆ ℝ) with bounded support, the width is defined as
::$\operatorname(A) = \sup(\operatorname(A)) - \inf(\operatorname(A))$
:In the case when $\operatorname(A)$ is a finite set, or more generally a closed set, the width is just
::$\operatorname(A) = \max(\operatorname(A)) - \min(\operatorname(A))$
:In the ''n''-dimensional case (''U'' ⊆ ℝ^''n'') the above can be replaced by the ''n''-dimensional volume of $\operatorname(A)$.
:In general, this can be defined given any measure on ''U'', for instance by integration (e.g. Lebesgue integration) of $\operatorname(A)$.

* A real fuzzy set $A$ (''U'' ⊆ ℝ) is said to be convex (in the fuzzy sense, not to be confused with a crisp convex set), iff
::\forall x,y \in U, \forall\lambda\in ,1 \mu_A(\lambda + (1-\lambda)y) \ge \min(\mu_A(x),\mu_A(y)).
: Without loss of generality, we may take ''x'' ≤ ''y'', which gives the equivalent formulation
::\forall z \in ,y \mu_A(z) \ge \min(\mu_A(x),\mu_A(y)).
: This definition can be extended to one for a general topological space ''U'': we say the fuzzy set $A$ is convex when, for any subset ''Z'' of ''U'', the condition
::$\forall z \in Z: \mu_A(z) \ge \inf(\mu_A(\partial Z))$
: holds, where $\partial Z$ denotes the boundary of ''Z'' and $f(X) = \$ denotes the image of a set ''X'' (here $\partial Z$) under a function ''f'' (here $\mu_A$).

Fuzzy set operations

Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity.
* For a given fuzzy set $A$, its complement $\neg$ (sometimes denoted as $A^c$ or $cA$) is defined by the following membership function:
::$\forall x \in U: \mu_(x) = 1 - \mu_A(x)$.
* Let t be a t-norm, and s the corresponding s-norm (aka t-conorm). Given a pair of fuzzy sets $A, B$, their intersection $A\cap$ is defined by:
::$\forall x \in U: \mu_(x) = t(\mu_A(x),\mu_B(x))$,
:and their union $A\cup$ is defined by:
::$\forall x \in U: \mu_(x) = s(\mu_A(x),\mu_B(x))$.

By the definition of the t-norm, we see that the union and intersection are commutative, monotonic, associative, and have both a null and an identity element. For the intersection, these are ∅ and ''U'', respectively, while for the union, these are reversed. However, the union of a fuzzy set and its complement may not result in the full universe ''U'', and the intersection of them may not give the empty set ∅. Since the intersection and union are associative, it is natural to define the intersection and union of a finite family of fuzzy sets recursively.

* If the standard negator $n(\alpha) = 1 - \alpha, \alpha \in$, 1/math> is replaced by another strong negator, the fuzzy set difference may be generalized by
::$\forall x \in U: \mu_(x) = n(\mu_A(x)).$
* The triple of fuzzy intersection, union and complement form a De Morgan Triplet. That is, De Morgan's laws extend to this triple.
:Examples for fuzzy intersection/union pairs with standard negator can be derived from samples provided in the article about t-norms.

:The fuzzy intersection is not idempotent in general, because the standard t-norm is the only one which has this property. Indeed, if the arithmetic multiplication is used as the t-norm, the resulting fuzzy intersection operation is not idempotent. That is, iteratively taking the intersection of a fuzzy set with itself is not trivial. It instead defines the ''m''-th power of a fuzzy set, which can be canonically generalized for non-integer exponents in the following way:

* For any fuzzy set $A$ and $\nu \in \R^+$ the ν-th power of $A$ is defined by the membership function:
::$\forall x \in U: \mu_(x) = \mu_(x)^.$
The case of exponent two is special enough to be given a name.
* For any fuzzy set $A$ the concentration $CON(A) = A^2$ is defined
::$\forall x \in U: \mu_(x) = \mu_(x) = \mu_(x)^2.$
Taking $0^0 = 1$, we have $A^0 = U$ and $A^1 = A.$

* Given fuzzy sets $A, B$, the fuzzy set difference $A \setminus B$, also denoted $A - B$, may be defined straightforwardly via the membership function:
::$\forall x \in U: \mu_(x) = t(\mu_A(x),n(\mu_B(x))),$
:which means $A \setminus B = A \cap \neg$, e. g.:
::$\forall x \in U: \mu_(x) = \min(\mu_A(x),1 - \mu_B(x)).$N.R. Vemuri, A.S. Hareesh, M.S. Srinath
Set Difference and Symmetric Difference of Fuzzy Sets
in: Fuzzy Sets Theory and Applications 2014, Liptovský Ján, Slovak Republic

:Another proposal for a set difference could be:
::$\forall x \in U: \mu_(x) = \mu_A(x) - t(\mu_A(x),\mu_B(x)).$

* Proposals for symmetric fuzzy set differences have been made by Dubois and Prade (1980), either by taking the absolute value, giving
::$\forall x \in U: \mu_(x) = , \mu_A(x) - \mu_B(x), ,$
:or by using a combination of just , , and standard negation, giving
::$\forall x \in U: \mu_(x) = \max(\min(\mu_A(x), 1 - \mu_B(x)), \min(\mu_B(x), 1 - \mu_A(x))).$

:Axioms for definition of generalized symmetric differences analogous to those for t-norms, t-conorms, and negators have been proposed by Vemur et al. (2014) with predecessors by Alsina et al. (2005) and Bedregal et al. (2009).

* In contrast to crisp sets, averaging operations can also be defined for fuzzy sets.

Disjoint fuzzy sets

In contrast to the general ambiguity of intersection and union operations, there is clearness for disjoint fuzzy sets:
Two fuzzy sets $A, B$ are disjoint iff
:$\forall x \in U: \mu_A(x) = 0 \lor \mu_B(x) = 0$
which is equivalent to
: $\nexists$ $x \in U: \mu_A(x) > 0 \land \mu_B(x) > 0$
and also equivalent to
:$\forall x \in U: \min(\mu_A(x),\mu_B(x)) = 0$
We keep in mind that / is a t/s-norm pair, and any other will work here as well.

Fuzzy sets are disjoint if and only if their supports are disjoint

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