New Approaches of Generalised Fuzzy Soft sets on fuzzy Codes and Its Properties on Decision-Makings

Definition 1.

Ozkan & Mehmet (2002) The notion $F_q=(a_1,a_2,\dots ,a_q),$ is a q-ary code which is a collection of symbol sequences in which each symbol is chosen from a set of q different components. The set $F_q$ is sometimes referred to as the alphabet and is defined as the set

Definition 2.

(Amudhambigai & Neeraja 2019) $F^n$ is the collection of all ordered $n-$ tuples

$p=p_1,p_2,\dots ,p_k,$ Where each p _i in $F_q$ . The elements of $F^n$ are referred to as words or vectors.

Definition 3.

Ozkan & Mehmet (2002) Let us assume that $\mathit{C}$ has $\mathit{a}$ as a code word. The relative weight of a code word $\mathit{a},$ denoted as

$R_w\left(\mathit{a}\right)$ , is the sum of $p_1,p_2,\dots ,p_k$ if $p_1,p_2,\dots ,p_k$ are defined as the positions of 1_s in a . For instance, $R_w\left(\mathit{a}\right)=1+3+4+5=13$ if $\mathit{a}=10111$ is a code word of $\mathit{C}$ in $F^5$ .

$1+2+3+\dots +n$ is the relative weight of $\mathrm{111\dots 1},$ which is a code word of $\mathit{C}$ in $F_2^n$ . This can be articulated as the sum of the first n positive integers. The largest relative weight of code $\mathit{C}$ in $F_2^n$ is referred to as this matter. Where $n$ is the length of a code word, where $R_{\mathit{wr}}$ is a mapping defined as $R_{\mathit{wr}}:C\to [0,1]$ for C ∈ F₂ ⁿ and $R_{\mathit{wr}}\left(\mathit{a}\right)\in [0,1],$ for any $\mathit{a}\in \mathit{C}$ .

Definition 4.

(Ozkan & Mehmet 2002) Let $\{c_1,c_2,\dots ,c_n\}$ is the code word of $\mathit{C}$ with length n. Let $\mathit{C}$ be a code.

For each $i=1,2,\dots ,n,$ let $R_{\mathit{wr}}\left(c_i\right)$ be a fuzzy code word related to the code words $c_i$ , $c_j$ in $\mathit{C}$ . Then the fuzzy code intersection and union of any two fuzzy codes $R_{\mathit{wr}}\left(c_i\right)$ and $R_{\mathit{wr}}\left(c_j\right)$ , respectively, are given as follows:

Definition 5.

(Amudhambigai and Neeraja 2019) For any fuzzy code $R_{\mathit{wr}}\left(c_i\right)$ , the complement is $c\left(R_{\mathit{wr}}\left(c_i\right)\right)$ is calculated by subtracting from 1 the relative weight of each member of $\mathit{C}$ .

That is,

Definition 6.

(Molodtsov 2011) Consider the idea of a set of parameters represented by $E$ and $U$ be the notion of a starting universe of items. $P\left(U\right)$ is the power set of $U$ and $A\subseteq E$ . A pair $(F,A)$ is a soft set over $U$ if and only if $F$ is a mapping given by

In other words, the soft set is a parameterized family of subjects from the set U.

Consider the set of e ∈ A, or the set of $e-$ approximate components of the soft set, where $F\left(e\right)$ can be arbitrary, some of which can have nonempty intersection, and some of which can be empty. This applies to every F (e) from this family for e in $A$ . where the attributes, traits, or properties of objects are usually the arguments. We might consider Zadeh’s fuzzy set as a subset of the soft set. Given a fuzzy set $A$ , its membership function is ${\mu }_A$ , that is, μ _A is a mapping from $U$ into $[0,1],$ that is

Definition 7.

(Majumdar & Samanta, 2010) The set of all fuzzy subsets within a universal set $U$ is denoted by $I^U$ .

Assume $A\subseteq E$ and that $E$ is a parameter set. Then, a pair $(F,A)$ is a fuzzy soft set over $U$ , where $F$ is a mapping denoted by

Definition 8.

(Ali et al. 2018) Assume that a vector space of size n over the field $K$ is represented by $W=K^n$ . $(F,E)$ is as of t algebraic linear code over $K$ . For every $F\left(e_i\right);1\le i\le t$ , the symbol $F\left(E\right)=F\left(e1\right),\dots ,F\left(\mathit{et}\right)$ in x indicates a linear algebraic code of $W$ . The number of linearly independent elements of $F\left(e_i\right)$ is denoted by $d_i,$ which is the dimension of $F\left(e_i\right)$ . Each $F\left(e_i\right)\in F\left(E\right)$ in this instance represents a linear algebraic code. The soft dimension of $(F,E)$ is $(F,E)=\{d_1,d_2\dots ,d_t\}$ and the number of soft code words of $(F,E)$ is given by:

Where $1\le i\le t$ .

Keep in mind that $\mathit{dim}(F,A)$ represents the soft dimension $(F,E)$ .

Definition 9.

(Majumdar & Samanta 2010). The universal set of elements is $=\{x_1,x_2,\dots ,x_n\}$ , and the universal set of parameters is $E=\{e_1,e_2,\dots ,e_m\}.$ We shall refer to the pair $(U,E)$ as a soft universe. Let $\mu$ be a fuzzy subset of $E$ , that is,

$\mathit{F\mu }:E\to I^U\times I$ have the following definition:

$\mathit{F\mu }\left(e\right)=$ $F\left(e\right),\mu \left(e\right))$ , where $\left(e\right)\in I^U$ .

A generalised fuzzy soft set (in short GFSS) over the soft universe $(U,E)$ is then denoted by $\mathit{F\mu }$ . Here, for every parameter $e_i$ , $\mathit{F\mu }\left(e_i\right)=(F\left(e_i\right),\mu \left(e_i\right))$ showed both the degree of belongingness that $\mu \left(e_i\right)$ represents and the degree of belongingness of the elements of $U$ in $F\left(e_i\right).$

Definition 10.

(Majumdar & Samanta 2010) Let $\mathit{F\mu }$ and $G_{\sigma }$ be two generalised fuzzy soft sets over $(U,E).$ Now $\mathit{F\mu }$ is said to be a generalised fuzzy soft subset of $G_{\sigma }$ if

Definition 11.

(Majumdar and Samanta 2010) Consider a generalised fuzzy soft set over $(U,E)$ , denoted by $\mathit{F\mu }$ . Then, $F^{\complement }$ represents the complement of $\mathit{F\mu }$ , which is defined as

Definition 12.

(Majumdar & Samanta 2010) $\mathit{F\mu }\bigcup G_{\sigma }$ represents the union of two generalised fuzzy soft sets $\mathit{F\mu }$ and $G_{\sigma }$ . This results in a generalised fuzzy soft set $H_{\nu }$ , which is defined as $H_{\nu }:E\to I^U\times I$ such that $H_{\nu }\left(e\right)=(H\left(e\right),\nu \left(e\right))$ where $H\left(e\right)=F\left(e\right)°G\left(e\right),\nu \left(e\right)=\mu \left(e\right)°\sigma \left(e\right),$ and ° denotes a t-co norm.

Definition 13.

(Majumdar & Samanta 2010) $\mathit{F\mu }\bigcup G_{\sigma }$ represents the intersection of two generalised fuzzy soft sets, $\mathit{F\mu }$ and $G_{\sigma }$ . This leads to a generalised fuzzy soft set $H_{\nu }$ , which is defined as

$H_{\nu }:E\to I^U\times I$ such that $H_{\nu }\left(e\right)=(H\left(e\right),\nu \left(e\right))$ , where $\ast$ is a t-co norm and

Definition 14.

(Majumdar & Samanta 2010) ${\varnothing }_{\theta }:E\to I^U\times I$ indicates a generalised null fuzzy soft set, which is referred to as a generalised fuzzy soft set, such that ${\varnothing }_{\theta }\left(e\right)=(F\left(e\right),\theta \left(e\right)),$ where $\varnothing \left(e\right)=\overline{0}$ (null set or no element is included under any parameter within the soft set), and, $\theta \left(e\right)=0$ , for all $e\in E$ .

Definition 15.

(Majumdar & Samanta 2010) A generalised absolute fuzzy soft set, represented by

$A_{\alpha }:E\to I^U\times I,$ is refered to as a generalised fuzzy soft set, where

$A_{\alpha }\left(e\right)=F\left(e\right),\alpha \left(e\right)$ is defined by $A\left(e\right)=\overline{1}$ (maximal soft set or where every parameter is associated with the whole set under consideration) and $\alpha \left(e\right)=1$ , for all $e\in E$ .

Definition 24.

Let $U=R_{\mathit{wr}}\left(a_i\right)$ for which $C$ is the universal element, consider the set of parameters be $E,$ and $a_i$ is an element of a code vector. The fuzzy soft code universe is a pair $(F,E).$

Let $F:E\to P\left(R_{\mathit{wr}}\left(a_i\right)\right),$ where all fuzzy code subsets of $R_{\mathit{wr}}\left(a_i\right)\in [0,1]$ are collected as $P\left(R_{\mathit{wr}}\left(a_i\right)\right)$ . And consider a mapping $\mu$ and $F_{\mu }$ such that

is a fuzzy subset of $E$ and

$(R_{\mathit{wr}}\left(a_i\right),E).\mu \left(e_i\right)$ is a representation of the possibility of membership, and the degree of belongingness of $R_{\mathit{wr}}\left(a_i\right)\in F\left(e_i\right)$ is indicated for each parameter $e_i$ by $F_{\mu }\left(e_i\right)=(F\left(e_i\right),\mu \left(e_i\right)).$

Example 3.

Consider $C\subseteq F_2^4$ parameter set E, and define a fuzzy set on this parameter set μ(e) ∈ [0,1] for any set of R _wr(a). And let

$=\{0.2,0.5,0.8,0.6,0.4,0.9,0.3,1\}$ be a fuzzy code of the eight computers under consideration.

Let $E=\{e_1,e_2,e_3,e_4\}$ be the set of qualities of the given computers, where the symbols $e_1$ denote a super computer, $e_2$ a micro computer, $e_3$ a workstation computer, and $e_4$ a personal computer.

Let $\mu :E\to [0,1]$ be a fuzzy code subset of R _wr : C → [0,1] defined as follows:

And we define a function

$\mathit{F\mu }:E\to P\left(R_{\mathit{wr}}\left(a_i\right)\right)\times R_{\mathit{wr}}\left(a_i\right)$ , be defined as follows:

A Generalised fuzzy soft code over $(R_{\mathit{wr}}\left(a\right),E)$ is then $F_{\mu }$ .

This can be represented in matrix form as follows:

The GFSC is then can be defined as

$F_{\mu }\left(e_i\right),\left(e_i\right)=(F\left(e_i\right),\mu \left(e_i\right)),\forall e\in E,F\left(e\right)\in P\left(R_{\mathit{wr}}\left(a_i\right)\right)$ and $\forall i$ .

Definition 25.

Given two GFSCs over $(R_{\mathit{wr}}\left(a\right),E),$ let $F_{\mu }$ and $G_{\sigma }$ . Specifically, $F_{\mu }$ is a subset of $G_{\sigma }$ that is considered a generalised fuzzy soft code if

Here, we compose $F_{\mu }\subseteq G_{\sigma }.$

Example 4.

The GFSC $F_{\mu }$ over $(R_{\mathit{wr}}\left(a\right),E)$ is the one presented in example 1. Let $G_{\sigma }$ be an additional GFSC defined as follows over $(R_{\mathit{wr}}\left(a\right),E).$

Here $\sigma$ and $R_{\mathit{wr}}\left(a\right)$ are the respective mapping given by

Then, a generalised fuzzy soft code subset of $G_{\sigma }$ is $F_{\mu }$ .

Matrix representation of $G_{\sigma }$ is given by

Definition 26.

(Intersection of GFSC). $F_{\mu }\bigcap G_{\sigma }$ represents the GFSC intersection of two GFSCs of $F_{\mu }$ and $G_{\sigma }$ over $(R_{\mathit{wr}}\left(a\right),E)$ , and is defined as

Definition 27.

(Union of GFSC) $F_{\mu }\bigcup G_{\sigma }$ represents the GFSC union of two GFSCs of $F_{\mu }$ and $G_{\sigma }$ over a soft universe $(R_{\mathit{wr}}\left(a\right),E)$ , and is defined as

Definition 28.

(Complement of GFSC). The following notation represents the complement of the GFSC of $F_{\mu }:{\left(F_{\mu }\right)}^{\complement }$ or Subtracting the GFSC of each member of $(F_{\mu }\left(e\right),\mu \left(e\right))$ from 1 yield ${F^{\complement }}_{{\mu }^{\complement }}$ . That is,

Theorem 1.

If $F_{\mu }$ and $G_{\sigma }$ represent any GFSC over a shared soft universe $(R_{\mathit{wr}}\left(a\right),E)$ , then the following result is hold:

Proof:

The proof follows directly from the definition.

Theorem 2.

The following properties are hold, if $(F,A)$ and $(G,B)$ are two generalised fuzzy soft codes over the common universe $R_{\mathit{wr}}\left(a\right).$

Proof

where $H(\neg e,\neg e^{\prime })=F^{\complement }\left(e\right)\bigcup G^{\complement }\left(e^{\prime }\right)=(H,\neg (A\times B)))$

Assume that, $((F,A)\bigcap (G,B))=(H,A\times B))$

Thus, from our discussion we get the result,

The proof of (ii) is similar to the above.

Definition 29.

A generalised null fuzzy soft code, represented by $F_{\mu },$ is considered to be a generalised null fuzzy soft code if

$F_{\mu }:E\to P\left(R_{\mathit{wr}}\left(a\right)\right)\times R_{\mathit{wr}}\left(a\right)$ and $F_{\mu }\left(e\right)=(F\left(e\right),\mu \left(e\right)),$ where $F\left(e\right)=\overline{0}$ and

Definition 30.

A GFSC is said to be a generalised absolute fuzzy soft code, denoted by $G_{\sigma }$ if $G_{\sigma }:E\to P\left(R_{\mathit{wr}}\left(a\right)\right)\times R_{\mathit{wr}}\left(a\right),$ where $G_{\sigma }\left(e\right)=(G\left(e\right),\sigma \left(e\right))$ is defined by $G\left(e\right)=\overline{1},\forall e\in E,$ and $\sigma \left(e\right)=1\forall e\in E.$

Theorem 3.

Given any GFSC over $(R_{\mathit{wr}}\left(a\right),E)$ , let’s say $F_{\mu }$ . The generalised null fuzzy soft code and generalised absolute fuzzy soft code over $(R_{\mathit{wr}}\left(a\right),E)$ denoted as $G_{\sigma }$ and $H_{\theta }$ , respectively. Then, the following result is hold: