ABSTRACT

The important role of the concept of type-2 fuzzy point in the formation of type-2 fuzzy open sets such as type-2 fuzzy $\tilde{\tilde{\delta }}$ -closed set this important role make the main objective of this paper is to introduce the concept type-2 fuzzy point of type-2 fuzzy set an important definitions in the composition of this concept as $\tilde{\tilde{\alpha }}$ -plane and the support of type-2 fuzzy set after preliminaries we present the definition of type-1 fuzzy set (fuzzy set) and fuzzy point and the special concepts that helped to configure them as support.

Keywords:

Type-2 Fuzzy Set, $\tilde{\tilde{\alpha }}$ -Plane, Fuzzy Point, Type-1 Fuzzy Set, Type-2 Fuzzy Point

1. Introduction

The main concept of a type-1 fuzzy set (fuzzy set) is introduced by Zadeh in 1965 [1]. In [2], he introduced the concept, namely, type-2 fuzzy set in 1973 to classify a fuzzy set of type-1 fuzzy set when dealing with type-2 fuzzy set. Many of the scientists then introduced the concepts of type-2 fuzzy set, in 1976 introducing Mizumoto and Tanaka some properties of type-2 fuzzy set [3] and how to find the operations of type-2 fuzzy sets using the extension principle, it is the same rule that it has established Zadeh in 1973. Scientists then introduced several important concepts of the type-2 fuzzy set that correspond to the main concepts of the type-1 fuzzy set. In [4], Liu introduced the $\tilde{\tilde{\alpha }}$ -plane concept of type-2 fuzzy set corresponding to the concept $\tilde{\alpha }$ -level of the type-1 fuzzy set and the concept of support type-2 fuzzy set was developed by Mendel [5] corresponding to the concept of support type-1 fuzzy set. Mendel then introduced a very important concept interval type-2 fuzzy sets [5]. The study of type-2 fuzzy sets was expanded to include the topology study of type-1 fuzzy set with the introduction of fuzzy topology by Chang in 1968 [6]. Zhang in 2013 offered the interval type-2 fuzzy topological space [7] and then introduced Mohammad and Munir General type-2 fuzzy topological space [8]. The type-2 fuzzy open sets such as type-2 fuzzy $\tilde{\tilde{\delta }}$ -preopen set and type-2 fuzzy $\tilde{\tilde{e}}$ -open set, we cannot configure its own definition only after introduce type-2 fuzzy point concept that belongs to the general fuzzy type-2 topological space.

2. Preliminaries

This section paves the way to introducing the concept of type-2 fuzzy point by providing definitions of special important concept type-1 fuzzy set and type-2 fuzzy set after submitting the special definition to the clusters.

2.1. Definition [1]

If X is a collection of objects with generic element $\chi$ , then a fuzzy subset $\tilde{A}$ in X is characterized by a membership function; ${\tilde{\aleph }}_{\tilde{A}}:X\to I$ , where I is the closed unit interval [0, 1], then we write a fuzzy set $\tilde{A}$ by the set of points: $\tilde{A}=\left\{\left(\chi ,{\tilde{\aleph }}_{\tilde{A}}\left(\chi \right)\right)|\chi \in X,0\le {\tilde{\aleph }}_{\tilde{A}}\left(\chi \right)\le 1\right\}$ .

2.2. Definition [9]

The support of a type-1 fuzzy set $\tilde{A}$ (denoted by $S\left(\tilde{A}\right)$ ) which is the crisp set of all $\chi \in X$ , $\left\{\chi :{\tilde{\aleph }}_{\tilde{A}}\left(\chi \right)>0\right\}$ .

In order to represent an element of a type-1fuzzy set $\tilde{A}$ , we provide a concept in type-1 fuzzy set theory that is a special case of $\tilde{\alpha }$ -level sets, which is called fuzzy points used for inclusion of elements to fuzzy sets.

2.3. Definition [10]

A type-1 fuzzy point $\tilde{\wp }$ in a set X is also a type-1 fuzzy set with membership function:

${\tilde{\aleph }}_{\tilde{\wp }}\left(\chi \right)=\{\begin{array}{l}\kappa \text{for}\chi =\gamma \\ 0\text{for}\chi \ne \gamma \end{array}$ ,

where $\chi \in X$ and $0<\kappa <1$ , $\gamma$ is called the support of $\tilde{\wp }$ and $\kappa$ the value of $\tilde{\wp }$ . We denote this type-1 fuzzy point by $\tilde{\wp }$ . Two fuzzy points ${\tilde{\wp }}_{{\chi }_{\kappa }}$ and ${\tilde{\wp }}_{{\gamma }_{\kappa }}$ are said to be distinct if and only if $\chi \ne \gamma$ . A type-1 fuzzy point $\tilde{\wp }$ is said to belong to a type-1 fuzzy subset $\tilde{A}$ in X, denoted by $\tilde{\wp }\in \tilde{A}$ if and only if $\kappa \le {\tilde{\aleph }}_{\tilde{A}}\left(\chi \right)$ .

2.4. Definition [2] [5]

Let X be a finite and non-empty set , which is referred to as the universe a type-2 fuzzy set, denoted by $\tilde{\tilde{A}}$ is characterized by a type-2 memberships function ${\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi ,\tilde{u}\right)$ , as

${\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}:X\times I\to I^{J_{\chi }}\left(J_{\chi }\subseteq I\right),I=[0,1]$ ,

where $\chi \in X$ and $\tilde{u}\in J_{\chi }$ , that is $\tilde{\tilde{A}}=\left\{\left(\left(\chi ,\tilde{u}\right),{\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi ,\tilde{u}\right)\right):\text{where}\chi \in X\text{and}\tilde{u}\in J_{\chi }\subseteq I,\text{where}0\le {\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi ,\tilde{u}\right)\le 1\right\}$ .

We can give a new wording to $\tilde{\tilde{A}}$

$\begin{array}{l}\tilde{\tilde{A}}={\displaystyle {\sum }_{\chi \in X}{\displaystyle {\sum }_{\tilde{u}\in {\upsilon }_{\chi }}\frac{{\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi ,\tilde{u}\right)}{\left(\chi ,\tilde{u}\right)}}}\\ \tilde{\tilde{A}}={\displaystyle {\sum }_{\chi \in X}{\displaystyle {\sum }_{\tilde{u}\in {\upsilon }_{\chi }}\frac{\frac{{\widetilde{\tilde{\mathcal{F}}}}_{\chi }\left(\tilde{u}\right)}{\tilde{u}}}{\chi }}},{\upsilon }_{\chi }\subseteq \left[0,1\right]\end{array}$ ,

where ${\widetilde{\tilde{\mathcal{F}}}}_{\chi }\left(\tilde{u}\right)={\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi ,\tilde{u}\right)$ and $\Sigma \Sigma$ denote the union in discrete sets and ∑ is replaced by ∫ is continuous universes are set. The class of all type-2 fuzzy set of $\chi \in X\ne \varnothing$ denoted by ${\widetilde{\tilde{\mathbb{F}}}}_{T-2}\left(X\right)$ .

A type-2 fuzzy set universes set [3], denoted by, such that

$\tilde{\tilde{X}}={\displaystyle {\sum }_{\chi \in X}{\displaystyle {\sum }_{\tilde{u}\in \left[l,l\right]}\frac{\frac{1}{\tilde{u}}}{\chi }}}$ .

A type-2 fuzzy empty set [3], denoted by

$\tilde{\tilde{\varnothing }}={\displaystyle {\sum }_{\chi \in X}{\displaystyle {\sum }_{\tilde{u}\in \left[0,0\right]}\frac{\frac{1}{\tilde{u}}}{\chi }}}$ .

Interval type-2 fuzzy set [5], when all the ${\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi ,\tilde{u}\right)=1$ , for all $\chi \in X$ .

The operations of type-2 fuzzy set [3], consider $\tilde{\tilde{A}}$ and $\tilde{\tilde{B}}$ are two type-2 fuzzy sets and the membership grades of $\tilde{\tilde{A}}$ and $\tilde{\tilde{B}}$ respectively, we can represented by

${\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi \right)={\displaystyle {\sum }_{\tilde{u}\in {\upsilon }_{\chi }^{\tilde{u}}}\frac{{\widetilde{\tilde{\mathcal{F}}}}_{\chi }\left(\tilde{u}\right)}{\tilde{u}}}\text{and}{\tilde{\tilde{\aleph }}}_{\tilde{\tilde{B}}}\left(\chi \right)={\displaystyle {\sum }_{\tilde{w}\in {\upsilon }_{\chi }^{\tilde{w}}}\frac{{\widetilde{\tilde{\mathcal{R}}}}_{\chi }\left(\tilde{w}\right)}{\tilde{w}}}$ ,

where ${\widetilde{\tilde{\mathcal{F}}}}_{\chi }\left(\tilde{u}\right),{\widetilde{\tilde{\mathcal{R}}}}_{\chi }\left(\tilde{w}\right)\in \mathfrak{T}=\left[0,1\right]$ and.

The union of two type-2 fuzzy sets is defined as

$\tilde{\tilde{A}}\cup \tilde{\tilde{B}}\iff {\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}\cup \tilde{\tilde{B}}}\left(\chi \right)={\displaystyle {\sum }_{\tilde{u}\in {\upsilon }_{\chi }^{\tilde{u}}}{\displaystyle {\sum }_{\tilde{w}\in {\upsilon }_{\chi }^{\tilde{w}}}\frac{{\widetilde{\tilde{\mathcal{F}}}}_{\chi }\left(\tilde{u}\right)\tilde{\wedge }{\widetilde{\tilde{\mathcal{R}}}}_{\chi }\left(\tilde{w}\right)}{\tilde{u}\tilde{\vee }\tilde{w}}}}$ .

The intersection of two type-2 fuzzy sets is defined as

$\tilde{\tilde{A}}\cap \tilde{\tilde{B}}\iff {\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}\cap \tilde{\tilde{B}}}\left(\chi \right)={\displaystyle {\sum }_{\tilde{u}\in {\upsilon }_{\chi }^{\tilde{u}}}{\displaystyle {\sum }_{\tilde{w}\in {\upsilon }_{\chi }^{\tilde{w}}}\frac{{\widetilde{\tilde{\mathcal{F}}}}_{\chi }\left(\tilde{u}\right)\tilde{\wedge }{\widetilde{\tilde{\mathcal{R}}}}_{\chi }\left(\tilde{w}\right)}{\tilde{u}\tilde{\wedge }\tilde{w}}}}$ .

The containment type-2 fuzzy sets are defined as $\tilde{\tilde{A}}\subseteq \tilde{\tilde{B}}\iff \tilde{\tilde{A}}\cap \tilde{\tilde{B}}=\tilde{\tilde{A}}$ .

The complement of type-2 fuzzy set defined as $\left(\neg \tilde{\tilde{A}}\right)={\displaystyle {\sum }_{\tilde{u}\in {\upsilon }_{\chi }^{\tilde{u}}}\frac{{\widetilde{\tilde{\mathcal{F}}}}_{\chi }\left(\tilde{u}\right)}{1-\tilde{u}}}$ .

A normal type-1 fuzzy set $\tilde{A}$ is one for height equals 1, otherwise it is called subnormal [9].

2.5. Definition [3]

A normal type-2 fuzzy set $\tilde{\tilde{A}}$ is one for which ${\max }_{\chi \in X}{\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi ,\tilde{u}\right)=1$ .

2.6. Definition [5]

The support of a type-2 fuzzy set $\tilde{\tilde{A}}$ denoted $S\left(\tilde{\tilde{A}}\right)$ comprises all $\left(\chi ,\tilde{u}\right)\in X\times I$ such that ${\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi ,\tilde{u}\right)>0$ .

2.7. Definition [4]

The two damnation $\tilde{\tilde{\alpha }}$ -plane , denoted ${\tilde{\tilde{A}}}_{\tilde{\tilde{\alpha }}}$ is the union of all primary membership whose secondary grades are greater than or equal special value $\tilde{\tilde{\alpha }}$ that

is: ${\tilde{\tilde{A}}}_{\tilde{\tilde{\alpha }}}={\displaystyle \underset{\chi \in X}{\cup }\left\{\left(\chi ,\tilde{u}\right):{\tilde{\tilde{\aleph }}}_{\tilde{\tilde{A}}}\left(\chi ,\tilde{u}\right)\ge \tilde{\tilde{\alpha }},\tilde{\tilde{\alpha }}\in \left[0,1\right]\right\}}$ .

2.8. Definition [8]

Let $\tilde{\tilde{T}}$ be the collection of type-2 fuzzy sets over $X\ne \varnothing$ then $\tilde{\tilde{T}}$ is called to be general type-2 fuzzy topology on $X\ne \varnothing$ .

If

1) $\tilde{\tilde{X}},\tilde{\tilde{\varnothing }}\in \tilde{\tilde{T}}$ ;

2) $\tilde{\tilde{A}}\cap \tilde{\tilde{B}}\in \tilde{\tilde{T}}$ ;

3) ${\displaystyle {\cup }_{i\in N}\tilde{\tilde{A}}}\in \tilde{\tilde{T}}$ for ${\tilde{\tilde{A}}}_i\in \tilde{\tilde{T}}$ .

The pair $\left(\tilde{\tilde{X}},\tilde{\tilde{T}}\right)$ is said to general type-2 fuzzy topological space over X and

the member of $\tilde{\tilde{T}}$ is said to be type-2 fuzzy $\tilde{\tilde{T}}\sim$ open sets in X and type-2 fuzzy sets are said type-2 fuzzy $\tilde{\tilde{T}}\sim$ closed sets in X, if its complement $\neg \tilde{\tilde{A}}\in \tilde{\tilde{T}}$ .

We must note that all the type-2 fuzzy sets are normal type-2 fuzzy sets so as to complete the topological construction and especially check identity law ( $\tilde{\tilde{A}}\cap \tilde{\tilde{\varnothing }}=\tilde{\tilde{\varnothing }}$ ).

3. Type-2 Fuzzy Point

The introduction of this section is the end of which the paper developed by the special case of $\tilde{\tilde{\alpha }}$ -plane, which claims type-2 fuzzy point.

3.1. Definition

A type-2 fuzzy point $\tilde{\tilde{\wp }}$ in a set X is also a type-2 fuzzy set with secondary membership function:

${\tilde{\tilde{\aleph }}}_{\tilde{\tilde{\wp }}}\left(\chi ,\tilde{u}\right)=\{\begin{array}{l}\kappa \text{for}\chi =\gamma \\ 0\text{for}\chi \ne \gamma \end{array}$ ,

where $\chi \in X$ and $0<\kappa <1$ , $\gamma$ is called the support of $\tilde{\tilde{\wp }}$ and $\kappa$ the value of $\tilde{\tilde{\wp }}$ .

3.2. Example

Let $X=\left\{a,b\right\}$ , $\tilde{\tilde{A}}=\left\{\left(\left(a,0.5\right),0.4\right),\left(\left(b,0.8\right),0\right)\right\}$ is a type-2 fuzzy point

$S\left(\tilde{\tilde{A}}\right)=\left\{\left(a,0.5\right)\right\}$ and we have ${\tilde{\tilde{\aleph }}}_{\tilde{\tilde{\wp }}}\left(\chi ,\tilde{u}\right)=\{\begin{array}{l}\kappa =0.4\chi =\gamma \\ 0\chi \ne \gamma \end{array}$

$\begin{array}{l}\Rightarrow \text{that}\text{is}{\tilde{\tilde{\aleph }}}_{\tilde{\tilde{\wp }}}\left(\chi ,\tilde{u}\right)={\tilde{\tilde{\aleph }}}_{\tilde{\tilde{\wp }}}\left(a,0.5\right)=\kappa =0.4\\ {\tilde{\tilde{\aleph }}}_{\tilde{\tilde{\wp }}}\left(\chi ,\tilde{u}\right)={\tilde{\tilde{\aleph }}}_{\tilde{\tilde{\wp }}}\left(b,0.8\right)=0\end{array}$ .

We denote this a type-2 fuzzy point by $\tilde{\tilde{\wp }}$ . Two a type-2 points ${\tilde{\tilde{\wp }}}_{{\chi }_{\kappa }}$ and ${\tilde{\tilde{\wp }}}_{{\gamma }_{\lambda }}$ are said to be distinct if and only if $\chi \ne \gamma$ .

3.3. Serious Results

The means of determining belonging set or fuzzy point in the subject of the fuzzy set is to compare with the membership function, but in the second type-2 fuzzy set we use the containment property given by Mizumotoand Tanaka $\tilde{\tilde{A}}\subseteq \tilde{\tilde{B}}\iff \tilde{\tilde{A}}\cap \tilde{\tilde{B}}=\tilde{\tilde{A}}$ [3].

3.4. Example

Let

$X=\left\{{\chi }_1,{\chi }_2,{\chi }_3\right\}$

$\tilde{\tilde{A}}=\left\{\left(\left({\chi }_1,0.7\right),0.8\right),\left(\left({\chi }_1,0.6\right),1\right),\left(\left({\chi }_2,0.4\right),1\right),\left(\left({\chi }_3,0.8\right),0.9\right),\left(\left({\chi }_3,0.5\right),1\right)\right\}$

$\tilde{\tilde{\wp }}=\left\{\left(\left({\chi }_1,0.6\right),0.8\right),\left(\left({\chi }_2,0.3\right),0.8\right),\left(\left({\chi }_3,0.5\right),0\right)\right\}$

$\tilde{\tilde{\wp }}\cap \tilde{\tilde{A}}\left({\chi }_1\right)=\frac{0.8\wedge 0.8}{0.7\wedge 0.6}+\frac{1\wedge 0.8}{0.6\wedge 0.6}$

$\tilde{\tilde{\wp }}\cap \tilde{\tilde{A}}\left({\chi }_1\right)=\left(0.6,\max \left(0.8,0.8\right)\right)=\left(\left({\chi }_1,0.6\right),0.8\right)$

$\tilde{\tilde{\wp }}\cap \tilde{\tilde{A}}\left({\chi }_2\right)=\frac{1\wedge 0.8}{0.4\wedge 0.3}\Rightarrow \frac{0.8}{0.3}=\left(\left({\chi }_2,0.3\right),0.8\right)$

$\tilde{\tilde{\wp }}\cap \tilde{\tilde{A}}\left({\chi }_3\right)=\frac{0.9\wedge 0}{0.8\wedge 0.5}+\frac{1\wedge 0}{0.5\wedge 0.5}$

$\tilde{\tilde{\wp }}\cap \tilde{\tilde{A}}\left({\chi }_3\right)=\left(0.5,\max \left(0,0\right)\right)=\left(\left({\chi }_3,0.5\right),0\right)$

$\Rightarrow therefore\tilde{\tilde{\wp }}\cap \tilde{\tilde{A}}=\tilde{\tilde{\wp }}\iff \tilde{\tilde{\wp }}\subseteq \tilde{\tilde{A}}$ .

4. Conclusion

The important role played by the type-2 fuzzy point in configuring continuous function after building new concepts from open type-2 fuzzy set made us offer this new concept. This concept allows for future dealings with the concept of neighborhood in general type-2 fuzzy topological space. In order for us to be able to configure such as type-2 fuzzy $\tilde{\tilde{\delta }}$ -preopenset and type-2 fuzzy $\tilde{\tilde{e}}$ -open set only and to examine the relationships among them.

Acknowledgements

Thanks and appreciation to all who contributed to the publication and composition of the main idea of the research and especially Prof. Jerry Mendel and Prof. Mohammad Reza Rajati.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Hussan, M.S.M. and Al-Khafaji, M.A.K. (2019) Type-2 Fuzzy Point. Journal of Applied Mathematics and Physics, 7, 3067-3072. https://doi.org/10.4236/jamp.2019.712215

References