# Linguistic Interval-Valued Spherical Fuzzy Soft Set and Its Application in Decision Making
## 3 Linguistic Interval-Valued Spherical Fuzzy Soft Set
### Operations on \(Livesfsss\)
Suppose \((\tilde{G},B)\wedge(\tilde{M},C)=(\tilde{H},B\times C)\), where \(\forall(\beta,\gamma)\in B\times C\), \(\tilde{H}(\beta,\gamma)=\tilde{G}(\beta)\cap\tilde{M}(\gamma)\). Then we have \((\tilde{F},A)\vee((\tilde{G},B)\wedge(\tilde{M},C))=(\tilde{F},A)\vee(\tilde {H},B\times C)\). And suppose \((\tilde{F},A)\vee(\tilde{H},B\times C)=(\tilde{I},A\times(B\times C))\), \(\forall(\alpha,(\beta,\gamma))\in A\times(B\times C)\), \(\tilde{I}(\alpha,(\beta,\gamma))=\tilde{F}(\alpha)\cup(\tilde{G}(\beta)\cap \tilde{M}(\gamma))\). Assume that \((\tilde{F},A)\vee(\tilde{G},B)=(\tilde{J},A\times B)\) and \((\tilde{F},A)\vee(\tilde{M},C)=(\tilde{K},A\times C)\), where \(\forall(\alpha,\beta)\in A\times B\), \(\tilde{J}(\alpha,\beta)=\tilde{F}(\alpha)\cup\tilde{G}(\beta)\) and \(\forall(\alpha,\gamma)\in A\times C\), \(\tilde{K}(\alpha,\gamma)=\tilde{F}(\alpha)\cup\tilde{M}(\gamma)\). Then we have \(((\tilde{F},A)\vee(\tilde{G},B))\wedge((\tilde{F},A)\vee(\tilde{M},C))=( \tilde{J},A\times B)\wedge(\tilde{K},A\times C)\). Suppose \((\tilde{J},A\times B)\wedge(\tilde{K},A\times C)=(\tilde{L},(A\times B)\times (A\times C))\), where \(\forall(\alpha,\gamma)\in A\times C\times B\), then, \(\tilde{L}((\alpha,\beta),(\alpha,\gamma))=(\tilde{F}(\alpha)\cup\tilde{G}(\beta) )\cap(\tilde{F}(\alpha)\cup\tilde{M}(\gamma))\). According to Theorem 1(5), \((\tilde{F},A)\tilde{\cup}((\tilde{G},B)\wedge(\tilde{M},C))=((\tilde{F},A) \tilde{\cup}(\tilde{G},B))\wedge((\tilde{F},A)\tilde{\cup}(\tilde{M},C))\). Consequently, \((\tilde{F},A)\vee(((\tilde{G},B)\wedge(\tilde{M},C))=((\tilde{F},A)\vee( \tilde{G},B))\wedge((\tilde{F},A)\vee(\tilde{M},C))\). Similarly, \((\tilde{F},A)\wedge((\tilde{G},B)\vee(\tilde{M},C))=((\tilde{F},A)\wedge( \tilde{G},B))\vee((\tilde{F},A)\wedge(\tilde{M},C))\). (3). Suppose \((\tilde{F},A)\vee(\tilde{G},B)=(\tilde{H},A\times B)\), where \(\forall(\alpha,\beta)\in A\times B\), \(\tilde{H}(\alpha,\beta)=\tilde{F}(\alpha)\cup\tilde{G}(\beta)\). Then we have \(((\tilde{F},A)\vee(\tilde{G},B))^{C}=(\tilde{H},A\times B)^{C}\). According to Definition 15, \(\forall(\alpha,\beta)\in A\times B\), \((\tilde{H}(\alpha,\beta))^{C}=(\tilde{F}(\alpha)\cup\tilde{G}(\beta))^{C}\). Assume that \((\tilde{F},A)\wedge(\tilde{G},B)=(\tilde{I},A\times B)\), where \(\forall(\alpha,\beta)\in A\times B\), \(\tilde{I}(\alpha,\beta)=\tilde{F}(\alpha)\cap\tilde{G}(\beta)\). According to Definition 9, \((\tilde{F},A)^{C}=(\tilde{F}^{C},\
eg A)\), \((\tilde{G},B)^{C}=(\tilde{G}^{C},\
eg B)\). Then we have \((\tilde{F},A)^{C}\wedge(\tilde{G},B)^{C}=(\tilde{I}^{C},\
eg(A\times B))\), where \(\forall(\alpha,\beta)\in C\
eg(A\times B)\), \(\tilde{F}^{C}(\alpha,\beta)=(\tilde{F}(\alpha))^{C}\cap(\tilde{G}(\beta))^{C}\). According to Theorem 1(7), \((\tilde{F}(\alpha)\cup\tilde{G}(\beta))^{C}=(\tilde{F}(\alpha))^{C}\cap(\tilde{ G}(\beta))^{C}\). Consequently, \(((\tilde{F},A)\vee(\tilde{G},B))^{C}=(\tilde{F},A)^{C}\wedge(\tilde{G},B)^{C}\). Similarly, we can obtain \(((\tilde{F},A)\wedge(\tilde{G},B))^{C}=(\tilde{F},A)^{C}\vee(\tilde{G},B)^{C}\). \(\Box\)  
**Definition 16**. : _The necessity operation on a linguistic interval-valued spherical fuzzy soft set \((\tilde{F},E)\) is denoted as \(\Box(\tilde{F},E)\), \(\forall\varepsilon\in E\),_  
\[\Box\tilde{F}(\varepsilon)=\Big{\{}\Big{\langle}x,\Big{(}\tilde{s}_{\Box\tilde {F}(\varepsilon)}(x),\tilde{s}_{\Box\tilde{F}(\varepsilon)}(x),\tilde{s}_{\Box \tilde{F}(\varepsilon)}(x)\Big{)}\Big{\rangle}\Big{|}x\in U.\Big{\}} \tag{17}\]  
Here, \(\tilde{s}_{\Box\tilde{F}(\varepsilon)}(x)=[\tilde{s}_{uF(\varepsilon)}^{L}(x), \tilde{s}_{uF(\varepsilon)}^{U}(x)]\) is the necessary membership degree that object \(x\) holds on parameter \(\varepsilon\), \(\tilde{s}_{\Box\tilde{F}(\varepsilon)}(x)=[\tilde{s}_{\sqrt{(h)^{2}-((u\tilde{F}( \varepsilon))^{U})^{2}}}(x),\tilde{s}_{\sqrt{(h)^{2}-((u\tilde{F}(\varepsilon))^{ U})^{2}}}(x)]\) is the possible non-membership degree that object \(x\) does not hold on parameter \(\varepsilon\). **Theorem 3**.