Fuzzy set

A diffuse or blurry set (in English, fuzzy set) is a set that may contain elements in part, that is, that the property of an element x{displaystyle x} belong to the whole A{displaystyle A} (x한 한 A{displaystyle xin A}) can be true with a partial degree of truth. This degree of belonging is a proposition in the context of diffuse logic, and not of the usual binary logic, which only admits two values: true or false.

The degree of belonging x{displaystyle x} a A{displaystyle A}or the degree of truth of belonging to the whole, is measured with a real number μ μ A(x){displaystyle mu _{A}(x)} from 0 and 1Both inclusive. rigorously, the value corresponding to each element defines an indicative function μ μ A(x):X→ → [chuckles]0,1]{displaystyle mu _{A}(x):Xrightarrow [0.1]}Where X{displaystyle X} represents the universal set of which the whole A{displaystyle A} take your elements. This is why diffuse subsets and not diffuse assemblies are usually spoken.

If the value of this function is 0, x{displaystyle x} does not belong to A{displaystyle A}. If it's 1, then x한 한 A{displaystyle xin A} totally, and if <math alttext="{displaystyle 0<mu _{A}(x)0.μ μ A(x).1{displaystyle 0 ingredientmu _{A}(x) tax1}<img alt="{displaystyle 0<mu _{A}(x) then. x{displaystyle x} belongs to A{displaystyle A} in a partial way.

History

The theory of fuzzy subsets was developed by Lofti A. Zadeh in 1965 in order to mathematically represent the intrinsic imprecision of certain categories of objects.

Fuzzy subsets (or fuzzy parts of a set) make it possible to model the human representation of knowledge (for example to measure our ignorance or an objective imprecision) and thus improve decision systems, decision support, and artificial intelligence.

Operations

With diffuse assemblies you can perform the same actions as with classic sets. Being two diffuse sets A{displaystyle A} and B{displaystyle B} The usual operations are defined:

Sets A{displaystyle A} and B{displaystyle B}	Defined by μ μ A(x){displaystyle mu _{A}(x)} and μ μ B(x){displaystyle mu _{B}(x)}
Usual intersection	μ μ A B(x)=min{μ μ A(x),μ μ B(x)!{displaystyle mu _{Acap B}(x)=min{mu _{A}(x),mu _{B}(x)}
Usual Union	μ μ A B(x)=max{μ μ A(x),μ μ B(x)!{displaystyle mu _{Acup B}(x)=max{mu _{A}(x),mu _{B}(x)}
Usual supplement	μ μ A! ! (x)=1− − μ μ A(x){displaystyle mu _{bar {A}(x)=1-mu _{A}(x)}}

Other concepts

• The core of a diffuse subset A{displaystyle ~ A} is the set of elements x{displaystyle ~x} who belong entirely to A{displaystyle A}I mean, they verify μ μ A(x)=1{displaystyle mu _{A}(x)=1}.
• The support of diffuse subset A{displaystyle A} is the whole of the x{displaystyle x} that belong, to a certain extent, to A{displaystyle ~ A}. I mean, they check. 0}" xmlns="http://www.w3.org/1998/Math/MathML">μ μ A(x)▪0{displaystyle mu _{A}(x) tax0}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5176b96865c7e4f359841158a58641c5fc1565b" style="vertical-align: -0.838ex; width:10.266ex; height:2.843ex;"/>.
• Sean. A{displaystyle ~ A} and B{displaystyle ~B} two diffuse subsets of the whole C{displaystyle ~ C}. It's said that A{displaystyle ~ A} There. Included in B{displaystyle ~B} Yes, for everything. x한 한 C{displaystyle xin ~ C}We have μ μ A(x)≤ ≤ μ μ B(x){displaystyle mu _{A}(x)leq mu _{B}(x)}, i.e. the elements of A{displaystyle A} always belong to a greater extent B{displaystyle ~B} to A{displaystyle ~ A}.
• Starting from the diffuse subset A{displaystyle A}, you can define the family of classic sets At{displaystyle}A_{t}, with t{displaystyle t} varying in [chuckles]0,1]{displaystyle [0.1]}For At={x한 한 X/μ μ A(x)≥ ≥ t!{displaystyle A_{t}={xin ~X/mu _{A}(x)geq t}}}. The knowledge of this family fully defines A{displaystyle A}.

Therefore, a fuzzy set is equivalent, in terms of information, to an uncountable infinite family of classical sets. The theory of fuzzy subsets is therefore very different and much more complex than the theory of usual sets. For example, a classical finite set has a finite number of classical subsets, but an infinite number of fuzzy subsets.