A fuzzy set is a pair (U, m) where U is a set (often required to be
non-empty) and m\colon U \rightarrow [0,1] 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=\{x_1,\dots,x_n\}, the fuzzy set (U, m) is often denoted by \{m(x_1)/x_1,\dots,m(x_n)/x_n\}. 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 [0,1] the following crisp sets are defined: • A^{\ge\alpha} = A_\alpha = \{x \in U \mid m(x)\ge\alpha\} is called its
α-cut (aka
α-level set) • A^{>\alpha} = A'_\alpha = \{x \in U \mid m(x)>\alpha\} is called its
strong α-cut (aka
strong α-level set) • S(A) = \operatorname{Supp}(A) = A^{>0} = \{x \in U \mid m(x)>0\} is called its
support • C(A) = \operatorname{Core}(A) = A^{=1} = \{x \in U \mid m(x)=1\} is called its
core (or sometimes
kernel \operatorname{Kern}(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 [0,1], for which A^{=\alpha} = \{x \in U \mid \mu_A(x) = \alpha\} is not empty, is called a
level of A. • The
level set of A is the set of all levels \alpha\in[0,1] representing distinct cuts. It is the
image of \mu_A: ::\Lambda_A = \{\alpha \in [0,1] : A^{=\alpha} \ne \varnothing\} = \{\alpha \in [0, 1] : {}
\existx \in U(\mu_A(x) = \alpha)\} = \mu_A(U) • For a fuzzy set A, its
height is given by ::\operatorname{Hgt}(A) = \sup \{\mu_A(x) \mid x \in U\} = \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{Hgt}(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{A} by dividing the membership function of the fuzzy set by its height: ::\forall x \in U: \mu_{\tilde{A}}(x) = \mu_A(x)/\operatorname{Hgt}(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 \subseteq \mathbb{R}) with
bounded support, the
width is defined as ::\operatorname{Width}(A) = \sup(\operatorname{Supp}(A)) - \inf(\operatorname{Supp}(A)) :In the case when \operatorname{Supp}(A) is a finite set, or more generally a
closed set, the width is just ::\operatorname{Width}(A) = \max(\operatorname{Supp}(A)) - \min(\operatorname{Supp}(A)) :In the
n-dimensional case (U \subseteq \mathbb{R}^n) the above can be replaced by the
n-dimensional volume of \operatorname{Supp}(A). :In general, this can be defined given any
measure on
U, for instance by integration (e.g.
Lebesgue integration) of \operatorname{Supp}(A). • A real fuzzy set A (U \subseteq \mathbb{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[0,1]: \mu_A(\lambda{x} + (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 [x,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) = \{f(x) \mid x \in 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{A} (sometimes denoted as A^c or cA) is defined by the following membership function: ::\forall x \in U: \mu_{\neg{A}}(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{B} is defined by: ::\forall x \in U: \mu_{A\cap{B}}(x) = t(\mu_A(x),\mu_B(x)), :and their
union A\cup{B} is defined by: ::\forall x \in U: \mu_{A\cup{B}}(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. It is noteworthy that the generally accepted standard operators for the union and intersection of fuzzy sets are the max and min operators: • \forall x \in U: \mu_{A\cup{B}}(x) = \max(\mu_A(x),\mu_B(x)) and \mu_{A\cap{B}}(x) = \min(\mu_A(x),\mu_B(x)). • If the standard negator n(\alpha) = 1 - \alpha, \alpha \in [0, 1] is replaced by another
strong negator, the fuzzy set difference (defined below) may be generalized by ::\forall x \in U: \mu_{\neg{A}}(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_{A^{\nu}}(x) = \mu_{A}(x)^{\nu}. 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_{CON(A)}(x) = \mu_{A^2}(x) = \mu_{A}(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_{A\setminus{B}}(x) = t(\mu_A(x),n(\mu_B(x))), :which means A \setminus B = A \cap \neg{B}, e. g.: ::\forall x \in U: \mu_{A\setminus{B}}(x) = \min(\mu_A(x),1 - \mu_B(x)). :Another proposal for a set difference could be: ::\forall x \in U: \mu_{A-{B}}(x) = \mu_A(x) - t(\mu_A(x),\mu_B(x)). A classical corollary may be indicating truth and membership values by {f, t} instead of {0, 1}. An extension of fuzzy sets has been provided by
Atanassov. An
intuitionistic fuzzy set (IFS) A is characterized by two functions: :1. \mu_A(x) – degree of membership of
x :2. \nu_A(x) – degree of non-membership of
x with functions \mu_A, \nu_A: U \to [0,1] with \forall x \in U: \mu_A(x) + \nu_A(x) \le 1. This resembles a situation like some person denoted by x voting • for a proposal A: (\mu_A(x)=1, \nu_A(x)=0), • against it: (\mu_A(x)=0, \nu_A(x)=1), • or abstain from voting: (\mu_A(x)=\nu_A(x)=0). After all, we have a percentage of approvals, a percentage of denials, and a percentage of abstentions. For this situation, special "intuitive fuzzy" negators, t- and s-norms can be defined. With D^* = \{(\alpha,\beta) \in [0, 1]^2 : \alpha + \beta = 1 \} and by combining both functions to (\mu_A,\nu_A): U \to D^* this situation resembles a special kind of
L-fuzzy sets. Once more, this has been expanded by defining
picture fuzzy sets (PFS) as follows: A PFS A is characterized by three functions mapping
U to [0, 1]: \mu_A, \eta_A, \nu_A, "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition \forall x \in U: \mu_A(x) + \eta_A(x) + \nu_A(x) \le 1 This expands the voting sample above by an additional possibility of "refusal of voting". With D^* = \{(\alpha,\beta,\gamma) \in [0, 1]^3 : \alpha + \beta + \gamma = 1 \} and special "picture fuzzy" negators, t- and s-norms this resembles just another type of
L-fuzzy sets.
Pythagorean fuzzy sets One extension of IFS is what is known as Pythagorean fuzzy sets. Such sets satisfy the constraint \mu_A(x)^2 + \nu_A(x)^2 \le 1, which is reminiscent of the Pythagorean theorem. Pythagorean fuzzy sets can be applicable to real life applications in which the previous condition of \mu_A(x) + \nu_A(x) \le 1 is not valid. However, the less restrictive condition of \mu_A(x)^2 + \nu_A(x)^2 \le 1 may be suitable in more domains. == Fuzzy logic ==