Crisp sets related to a fuzzy set

For any fuzzy set ${\displaystyle A=(U,m)}$ and ${\displaystyle \alpha \in [0,1]}$ the following crisp sets are defined:

• ${\displaystyle A^{\geq \alpha }=A_{\alpha }=\{x\in U\mid m(x)\geq \alpha \}}$ is called its α-cut (aka α-level set)
• ${\displaystyle A^{>\alpha }=A'_{\alpha }=\{x\in U\mid m(x)>\alpha \}}$ is called its strong α-cut (aka strong α-level set)
• ${\displaystyle S(A)=\operatorname {Supp} (A)=A^{>0}=\{x\in U\mid m(x)>0\}}$ is called its support
• ${\displaystyle C(A)=\operatorname {Core} (A)=A^{=1}=\{x\in U\mid m(x)=1\}}$ is called its core (or sometimes kernel ${\displaystyle \operatorname {Kern} (A)}$).

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

Other definitions

• A fuzzy set ${\displaystyle A=(U,m)}$ is empty (${\displaystyle A=\varnothing }$) iff (if and only if)

${\displaystyle \forall }$${\displaystyle x\in U:\mu _{A}(x)=m(x)=0}$

• Two fuzzy sets ${\displaystyle A}$ and ${\displaystyle B}$ are equal (${\displaystyle A=B}$) iff

${\displaystyle \forall x\in U:\mu _{A}(x)=\mu _{B}(x)}$

• A fuzzy set ${\displaystyle A}$ is included in a fuzzy set ${\displaystyle B}$ (${\displaystyle A\subseteq B}$) iff

${\displaystyle \forall x\in U:\mu _{A}(x)\leq \mu _{B}(x)}$

• For any fuzzy set ${\displaystyle A}$, any element ${\displaystyle x\in U}$ that satisfies

${\displaystyle \mu _{A}(x)=0.5}$

is called a crossover point.

• Given a fuzzy set ${\displaystyle A}$, any ${\displaystyle \alpha \in [0,1]}$, for which ${\displaystyle A^{=\alpha }=\{x\in U\mid \mu _{A}(x)=\alpha \}}$ is not empty, is called a level of A.
• The level set of A is the set of all levels ${\displaystyle \alpha \in [0,1]}$ representing distinct cuts. It is the image of ${\displaystyle \mu _{A}}$:

${\displaystyle \Lambda _{A}=\{\alpha \in [0,1]:A^{=\alpha }\neq \varnothing \}=\{\alpha \in [0,1]:{}}$${\displaystyle \exists }$${\displaystyle x\in U(\mu _{A}(x)=\alpha )\}=\mu _{A}(U)}$

• For a fuzzy set ${\displaystyle A}$, its height is given by

${\displaystyle \operatorname {Hgt} (A)=\sup\{\mu _{A}(x)\mid x\in U\}=\sup(\mu _{A}(U))}$

where ${\displaystyle \sup }$ denotes the supremum, which exists because ${\displaystyle \mu _{A}(U)}$ is non-empty and bounded above by 1. If U is finite, we can simply replace the supremum by the maximum.

• A fuzzy set ${\displaystyle A}$ is said to be normalized iff

${\displaystyle \operatorname {Hgt} (A)=1}$

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 ${\displaystyle A}$ may be normalized with result ${\displaystyle {\tilde {A}}}$ by dividing the membership function of the fuzzy set by its height:

${\displaystyle \forall x\in U:\mu _{\tilde {A}}(x)=\mu _{A}(x)/\operatorname {Hgt} (A)}$

Besides similarities this differs from the usual normalization in that the normalizing constant is not a sum.

• For fuzzy sets ${\displaystyle A}$ of real numbers (U ⊆ ℝ) with bounded support, the width is defined as

${\displaystyle \operatorname {Width} (A)=\sup(\operatorname {Supp} (A))-\inf(\operatorname {Supp} (A))}$

In the case when ${\displaystyle \operatorname {Supp} (A)}$ is a finite set, or more generally a closed set, the width is just

${\displaystyle \operatorname {Width} (A)=\max(\operatorname {Supp} (A))-\min(\operatorname {Supp} (A))}$

In the n-dimensional case (U ⊆ ℝⁿ) the above can be replaced by the n-dimensional volume of ${\displaystyle \operatorname {Supp} (A)}$.

In general, this can be defined given any measure on U, for instance by integration (e.g. Lebesgue integration) of ${\displaystyle \operatorname {Supp} (A)}$.

• A real fuzzy set ${\displaystyle A}$ (U ⊆ ℝ) is said to be convex (in the fuzzy sense, not to be confused with a crisp convex set), iff

${\displaystyle \forall x,y\in U,\forall \lambda \in [0,1]:\mu _{A}(\lambda {x}+(1-\lambda )y)\geq \min(\mu _{A}(x),\mu _{A}(y))}$.

Without loss of generality, we may take x ≤ y, which gives the equivalent formulation

${\displaystyle \forall z\in [x,y]:\mu _{A}(z)\geq \min(\mu _{A}(x),\mu _{A}(y))}$.

This definition can be extended to one for a general topological space U: we say the fuzzy set ${\displaystyle A}$ is convex when, for any subset Z of U, the condition

${\displaystyle \forall z\in Z:\mu _{A}(z)\geq \inf(\mu _{A}(\partial Z))}$

holds, where ${\displaystyle \partial Z}$ denotes the boundary of Z and ${\displaystyle f(X)=\{f(x)\mid x\in X\}}$ denotes the image of a set X (here ${\displaystyle \partial Z}$) under a function f (here ${\displaystyle \mu _{A}}$).

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 ${\displaystyle A}$, its complement ${\displaystyle \neg {A}}$ (sometimes denoted as ${\displaystyle A^{c}}$ or ${\displaystyle cA}$) is defined by the following membership function:

${\displaystyle \forall x\in U:\mu _{\neg {A}}(x)=1-\mu _{A}(x)}$.

• Let t be a t-norm, and s the corresponding s-norm (aka t-conorm). Given a pair of fuzzy sets ${\displaystyle A,B}$, their intersection ${\displaystyle A\cap {B}}$ is defined by:

${\displaystyle \forall x\in U:\mu _{A\cap {B}}(x)=t(\mu _{A}(x),\mu _{B}(x))}$,

and their union ${\displaystyle A\cup {B}}$ is defined by:

${\displaystyle \forall x\in U:\mu _{A\cup {B}}(x)=s(\mu _{A}(x),\mu _{B}(x))}$.

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.

• If the standard negator ${\displaystyle n(\alpha )=1-\alpha ,\alpha \in [0,1]}$ is replaced by another strong negator, the fuzzy set difference may be generalized by

${\displaystyle \forall x\in U:\mu _{\neg {A}}(x)=n(\mu _{A}(x)).}$

• 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 min 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 ${\displaystyle A}$ and ${\displaystyle \nu \in \mathbb {R} ^{+}}$ the ν-th power of ${\displaystyle A}$ is defined by the membership function:

${\displaystyle \forall x\in U:\mu _{A^{\nu }}(x)=\mu _{A}(x)^{\nu }.}$

The case of exponent two is special enough to be given a name.

• For any fuzzy set ${\displaystyle A}$ the concentration ${\displaystyle CON(A)=A^{2}}$ is defined

${\displaystyle \forall x\in U:\mu _{CON(A)}(x)=\mu _{A^{2}}(x)=\mu _{A}(x)^{2}.}$

Taking ${\displaystyle 0^{0}=1}$, we have ${\displaystyle A^{0}=U}$ and ${\displaystyle A^{1}=A.}$

• Given fuzzy sets ${\displaystyle A,B}$, the fuzzy set difference ${\displaystyle A\setminus B}$, also denoted ${\displaystyle A-B}$, may be defined straightforwardly via the membership function:

${\displaystyle \forall x\in U:\mu _{A\setminus {B}}(x)=t(\mu _{A}(x),n(\mu _{B}(x))),}$

which means ${\displaystyle A\setminus B=A\cap \neg {B}}$, e. g.:

${\displaystyle \forall x\in U:\mu _{A\setminus {B}}(x)=\min(\mu _{A}(x),1-\mu _{B}(x)).}$^[7]

Another proposal for a set difference could be:

${\displaystyle \forall x\in U:\mu _{A-{B}}(x)=\mu _{A}(x)-t(\mu _{A}(x),\mu _{B}(x)).}$^[7]

• Proposals for symmetric fuzzy set differences have been made by Dubois and Prade (1980), either by taking the absolute value, giving

${\displaystyle \forall x\in U:\mu _{A\triangle B}(x)=|\mu _{A}(x)-\mu _{B}(x)|,}$

or by using a combination of just max, min, and standard negation, giving

${\displaystyle \forall x\in U:\mu _{A\triangle B}(x)=\max(\min(\mu _{A}(x),1-\mu _{B}(x)),\min(\mu _{B}(x),1-\mu _{A}(x))).}$^[7]

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).^[7]

• 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 ${\displaystyle A,B}$ are disjoint iff

${\displaystyle \forall x\in U:\mu _{A}(x)=0\lor \mu _{B}(x)=0}$

which is equivalent to

${\displaystyle \nexists }$ ${\displaystyle x\in U:\mu _{A}(x)>0\land \mu _{B}(x)>0}$

and also equivalent to

${\displaystyle \forall x\in U:\min(\mu _{A}(x),\mu _{B}(x))=0}$

We keep in mind that min/max 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 according to the standard definition for crisp sets.

For disjoint fuzzy sets ${\displaystyle A,B}$ any intersection will give ∅, and any union will give the same result, which is denoted as

${\displaystyle A\,{\dot {\cup }}\,B=A\cup B}$

with its membership function given by

${\displaystyle \forall x\in U:\mu _{A{\dot {\cup }}B}(x)=\mu _{A}(x)+\mu _{B}(x)}$

Note that only one of both summands is greater than zero.

For disjoint fuzzy sets ${\displaystyle A,B}$ the following holds true:

${\displaystyle \operatorname {Supp} (A\,{\dot {\cup }}\,B)=\operatorname {Supp} (A)\cup \operatorname {Supp} (B)}$

This can be generalized to finite families of fuzzy sets as follows: Given a family ${\displaystyle A=(A_{i})_{i\in I}}$ of fuzzy sets with index set I (e.g. I = {1,2,3,...,n}). This family is (pairwise) disjoint iff

${\displaystyle {\text{for all }}x\in U{\text{ there exists at most one }}i\in I{\text{ such that }}\mu _{A_{i}}(x)>0.}$

A family of fuzzy sets ${\displaystyle A=(A_{i})_{i\in I}}$ is disjoint, iff the family of underlying supports ${\displaystyle \operatorname {Supp} \circ A=(\operatorname {Supp} (A_{i}))_{i\in I}}$ is disjoint in the standard sense for families of crisp sets.

Independent of the t/s-norm pair, intersection of a disjoint family of fuzzy sets will give ∅ again, while the union has no ambiguity:

${\displaystyle {\dot {\bigcup \limits _{i\in I}}}\,A_{i}=\bigcup _{i\in I}A_{i}}$

with its membership function given by

${\displaystyle \forall x\in U:\mu _{{\dot {\bigcup \limits _{i\in I}}}A_{i}}(x)=\sum _{i\in I}\mu _{A_{i}}(x)}$

Again only one of the summands is greater than zero.

For disjoint families of fuzzy sets ${\displaystyle A=(A_{i})_{i\in I}}$ the following holds true:

${\displaystyle \operatorname {Supp} \left({\dot {\bigcup \limits _{i\in I}}}\,A_{i}\right)=\bigcup \limits _{i\in I}\operatorname {Supp} (A_{i})}$

Scalar cardinality

For a fuzzy set ${\displaystyle A}$ with finite support ${\displaystyle \operatorname {Supp} (A)}$ (i.e. a "finite fuzzy set"), its cardinality (aka scalar cardinality or sigma-count) is given by

${\displaystyle \operatorname {Card} (A)=\operatorname {sc} (A)=|A|=\sum _{x\in U}\mu _{A}(x)}$.

In the case that U itself is a finite set, the relative cardinality is given by

${\displaystyle \operatorname {RelCard} (A)=\|A\|=\operatorname {sc} (A)/|U|=|A|/|U|}$.

This can be generalized for the divisor to be a non-empty fuzzy set: For fuzzy sets ${\displaystyle A,G}$ with G ≠ ∅, we can define the relative cardinality by:

${\displaystyle \operatorname {RelCard} (A,G)=\operatorname {sc} (A|G)=\operatorname {sc} (A\cap {G})/\operatorname {sc} (G)}$,

which looks very similar to the expression for conditional probability. Note:

• ${\displaystyle \operatorname {sc} (G)>0}$ here.
• The result may depend on the specific intersection (t-norm) chosen.
• For ${\displaystyle G=U}$ the result is unambiguous and resembles the prior definition.

Distance and similarity

For any fuzzy set ${\displaystyle A}$ the membership function ${\displaystyle \mu _{A}:U\to [0,1]}$ can be regarded as a family ${\displaystyle \mu _{A}=(\mu _{A}(x))_{x\in U}\in [0,1]^{U}}$. The latter is a metric space with several metrics ${\displaystyle d}$ known. A metric can be derived from a norm (vector norm) ${\displaystyle \|\,\|}$ via

${\displaystyle d(\alpha ,\beta )=\|\alpha -\beta \|}$.

For instance, if ${\displaystyle U}$ is finite, i.e. ${\displaystyle U=\{x_{1},x_{2},...x_{n}\}}$, such a metric may be defined by:

${\displaystyle d(\alpha ,\beta ):=\max\{|\alpha (x_{i})-\beta (x_{i})|:i=1,...,n\}}$ where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are sequences of real numbers between 0 and 1.

For infinite ${\displaystyle U}$, the maximum can be replaced by a supremum. Because fuzzy sets are unambiguously defined by their membership function, this metric can be used to measure distances between fuzzy sets on the same universe:

${\displaystyle d(A,B):=d(\mu _{A},\mu _{B})}$,

which becomes in the above sample:

${\displaystyle d(A,B)=\max\{|\mu _{A}(x_{i})-\mu _{B}(x_{i})|:i=1,...,n\}}$.

Again for infinite ${\displaystyle U}$ the maximum must be replaced by a supremum. Other distances (like the canonical 2-norm) may diverge, if infinite fuzzy sets are too different, e.g., ${\displaystyle \varnothing }$ and ${\displaystyle U}$.

Similarity measures (here denoted by ${\displaystyle S}$) may then be derived from the distance, e.g. after a proposal by Koczy:

${\displaystyle S=1/(1+d(A,B))}$ if ${\displaystyle d(A,B)}$ is finite, ${\displaystyle 0}$ else,

or after Williams and Steele:

${\displaystyle S=\exp(-\alpha {d(A,B)})}$ if ${\displaystyle d(A,B)}$ is finite, ${\displaystyle 0}$ else

where ${\displaystyle \alpha >0}$ is a steepness parameter and ${\displaystyle \exp(x)=e^{x}}$.^[6]

Another definition for interval valued (rather 'fuzzy') similarity measures ${\displaystyle \zeta }$ is provided by Beg and Ashraf as well.^[6]

L-fuzzy sets

Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure ${\displaystyle L}$ of a given kind; usually it is required that ${\displaystyle L}$ be at least a poset or lattice. These are usually called L-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. These kinds of generalizations were first considered in 1967 by Joseph Goguen, who was a student of Zadeh.^[8] 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) ${\displaystyle A}$ is characterized by two functions:

1. ${\displaystyle \mu _{A}(x)}$ – degree of membership of x

2. ${\displaystyle \nu _{A}(x)}$ – degree of non-membership of x

with functions ${\displaystyle \mu _{A},\nu _{A}:U\to [0,1]}$ with ${\displaystyle \forall x\in U:\mu _{A}(x)+\nu _{A}(x)\leq 1}$.

This resembles a situation like some person denoted by ${\displaystyle x}$ voting

• for a proposal ${\displaystyle A}$: (${\displaystyle \mu _{A}(x)=1,\nu _{A}(x)=0}$),
• against it: (${\displaystyle \mu _{A}(x)=0,\nu _{A}(x)=1}$),
• or abstain from voting: (${\displaystyle \mu _{A}(x)=\nu _{A}(x)=0}$).

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 ${\displaystyle D^{*}=\{(\alpha ,\beta )\in [0,1]^{2}:\alpha +\beta =1\}}$ and by combining both functions to ${\displaystyle (\mu _{A},\nu _{A}):U\to D^{*}}$ 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]: ${\displaystyle \mu _{A},\eta _{A},\nu _{A}}$, "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition ${\displaystyle \forall x\in U:\mu _{A}(x)+\eta _{A}(x)+\nu _{A}(x)\leq 1}$ This expands the voting sample above by an additional possibility of "refusal of voting".

With ${\displaystyle D^{*}=\{(\alpha ,\beta ,\gamma )\in [0,1]^{3}:\alpha +\beta +\gamma =1\}}$ and special "picture fuzzy" negators, t- and s-norms this resembles just another type of L-fuzzy sets.^[9][10]

Neutrosophic fuzzy sets

Some Key Developments in the Introduction of Fuzzy Set Concepts.^[11]

The concept of IFS has been extended into two major models. The two extensions of IFS are neutrosophic fuzzy sets and Pythagorean fuzzy sets.^[11]

Neutrosophic fuzzy sets were introduced by Smarandache in 1998.^[12] Like IFS, neutrosophic fuzzy sets have the previous two functions: one for membership ${\displaystyle \mu _{A}(x)}$ and another for non-membership ${\displaystyle \nu _{A}(x)}$. The major difference is that neutrosophic fuzzy sets have one more function: for indeterminate ${\displaystyle i_{A}(x)}$. This value indicates that the degree of undecidedness that the entity x belongs to the set. This concept of having indeterminate ${\displaystyle i_{A}(x)}$ value can be particularly useful when one cannot be very confident on the membership or non-membership values for item x.^[13] In summary, neutrosophic fuzzy sets are associated with the following functions:

1. ${\displaystyle \mu _{A}(x)}$—degree of membership of x

2. ${\displaystyle \nu _{A}(x)}$—degree of non-membership of x

3. ${\displaystyle i_{A}(x)}$—degree of indeterminate value of x

Pythagorean fuzzy sets

The other extension of IFS is what is known as Pythagorean fuzzy sets. Pythagorean fuzzy sets are more flexible than IFSs. IFSs are based on the constraint ${\displaystyle \mu _{A}(x)+\nu _{A}(x)\leq 1}$, which can be considered as too restrictive in some occasions. This is why Yager proposed the concept of Pythagorean fuzzy sets. Such sets satisfy the constraint ${\displaystyle \mu _{A}(x)^{2}+\nu _{A}(x)^{2}\leq 1}$, which is reminiscent of the Pythagorean theorem.^[14][15][16] Pythagorean fuzzy sets can be applicable to real life applications in which the previous condition of ${\displaystyle \mu _{A}(x)+\nu _{A}(x)\leq 1}$ is not valid. However, the less restrictive condition of ${\displaystyle \mu _{A}(x)^{2}+\nu _{A}(x)^{2}\leq 1}$ may be suitable in more domains.^[11][13]