# Linguistic Interval-Valued Spherical Fuzzy Soft Set and Its Application in Decision Making
## 3 Linguistic Interval-Valued Spherical Fuzzy Soft Set
### Operations on \(Livesfsss\)
The intersection of two linguistic interval-valued spherical fuzzy soft sets \((\tilde{F},A)\) and \((\tilde{G},B)\) over \(U\) is a linguistic interval-valued spherical fuzzy soft set \((\tilde{H},C)\), where \(C=A\cap B\), \(\forall\varepsilon\in C\), \(\tilde{H}(\varepsilon)=\tilde{F}(\varepsilon)\cap\tilde{G}(\varepsilon)=\{< x,([min\{s_{uF(\varepsilon)}^{L}(x),s_{uG(\varepsilon)}^{L}(x)\},min\{s_{uF( \varepsilon)}^{U}(x),s_{uG(\varepsilon)}^{U}(x)\}],\\ [min\{s_{uF(\varepsilon)}^{L}(x),\tilde{s}_{\pi F(\varepsilon)}^{L}(x)\},min \{\tilde{s}_{\pi F(\varepsilon)}^{U}(x),\tilde{s}_{\pi G(\varepsilon)}^{U}( x)\}],\\ max\{s_{vF(\varepsilon)}^{U}(x),\tilde{s}_{vG(\varepsilon)}^{U}(x)\}]\}>|x \in U}.\) We denote it by \((\tilde{F},A)\cap(\tilde{G},B)=(\tilde{H},C)\). Let \((\tilde{F},A)\), \((\tilde{G},B)\) and \((\tilde{M},C)\) be three linguistic interval-valued spherical fuzzy soft sets over \(U\), then we have the following properties. 1. _Transitive law:_ If \((\tilde{F},A)\subset(\tilde{G},B)\) and \((\tilde{G},B)\subset(\tilde{M},C)\), then \((\tilde{F},A)\subset(\tilde{M},C)\);
2. _Commutative law:_\((\tilde{F},A)\cup(\tilde{G},B)=(\tilde{G},B)\cup(\tilde{F},A)\) and \((\tilde{F},A)\cap(\tilde{G},B)=(\tilde{G},B)\cap(\tilde{F},A)\);
3. _Idempotent law:_\((\tilde{F},A)\cap(\tilde{F},A)=(\tilde{F},A)\) and \((\tilde{F},A)\cap(\tilde{F},A)=(\tilde{F},A)\);
4. _Associative law:_\((\tilde{F},A)\cup((\tilde{G},B)\cap(\tilde{M},C))=((\tilde{F},A)\cup( \tilde{G},B))\cup(\tilde{M},C)\) and \((\tilde{F},A)\cap((\tilde{G},B)\cap(\tilde{M},C))=((\tilde{F},A)\cap(\tilde{ G},B))\cap(\tilde{M},C)\);
5. _Distributive law:_\((\tilde{F},A)\cap((\tilde{G},B)\cap(\tilde{M},C))=((\tilde{F},A)\cap(\tilde{G},B))\cap(( \tilde{F},A)\cap(\tilde{M},C))\) and \((\tilde{F},A)\cap((\tilde{G},B)\cap(\tilde{M},C))=((\tilde{F},A)\cap(\tilde{ G},B))\cap((\tilde{F},A)\cap(\tilde{M},C))\);
6. _Absorption law:_\(((\tilde{F},A)\cup(\tilde{G},B))\cap(\tilde{F},A)=(\tilde{F},A)\) and \(((\tilde{F},A)\cap(\tilde{G},B))\cap(\tilde{F},A)=(\tilde{F},A)\);
7. _[PERSON]'s law:_\(((F,A)\cup(\tilde{G},B))^{C}=(\tilde{F},A)^{C}\cap(\tilde{G},B)^{C}\) and \(((\tilde{F},A)\cap(\tilde{G},B))^{C}=(\tilde{F},A)^{C}\cap(\tilde{G},B)^{C}\). We can obtain (1)-(4) easily according to Definitions 7, 12 and 13. And we prove the other properties as follows. Suppose that, for \((\tilde{F},A)\), \(\forall\alpha\in A\), \(\tilde{F}(\alpha)=\{\langle x,(\tilde{s}_{uF(\alpha)}(x),\tilde{s}_{\pi F( \alpha)}(x),\tilde{s}_{vF(\alpha)}(x))\rangle|x\in U\}\), for \((\tilde{G}  
where \(\tilde{F}(\rho)\cup\tilde{G}(\rho)=([max\{s_{a1},s_{a2}\},max\{s_{b1},s_{b2}\}], \quad[min\{s_{c1},s_{c2}\},min\{s_{d1},s_{d2}\}],\quad[min\{s_{e1},s_{e2}\},min\{s_{ o1},s_{o2}\}])\). For \((\tilde{I},E)\), we have \(E=A\cup C\) and \(\forall\sigma\in E\),  
\[\tilde{I}(\sigma)= \begin{cases}\tilde{F}(\sigma),\sigma\in A-C;\\ \tilde{M}(\sigma),\sigma\in C-A;\\ \tilde{F}(\sigma)\cup\tilde{M}(\sigma),\sigma\in A\cap C;\end{cases} \]  
where \(\tilde{F}(\sigma)\cup\tilde{M}(\sigma)=([max\{s_{a1},s_{a3}\},max\{s_{b1},s_{ b3}\}],\quad[min\{s_{c1},s_{c3}\},min\{s_{d1},s_{d3}\}],\quad[min\{s_{e1},s_{e3}\},min\{s_{ o1},s_{o3}\}])\). Hence \(((\tilde{F},A)\tilde{\cup}(\tilde{G},B))\tilde{\cap}((\tilde{F},A)\tilde{ \cup}(\tilde{M},C))=(\tilde{N},D)\tilde{\cap}(\tilde{I},E)\), suppose \((\tilde{N},D)\cap(\tilde{I},E)\)\(=(\tilde{J},F)\), where \(F=D\cap E\) and \(F\
eq\varnothing\), then  
\[\tilde{J}(z)= \begin{cases}\tilde{F}(z),\forall z\in D\cap E=\rho\cap\sigma=(A- C)\cap(A-B)=A-B\cap C;\\ \tilde{G}(z),\forall z\in B\cap C-A;\\ \tilde{N}(z)\cap\tilde{I}(z),\forall z\in D\cap E=\rho\cap\sigma=(A\cap B) \cap(A\cap C)=A\cap B\cap C;\end{cases} \]  
where \(\tilde{N}(z)\cap\tilde{I}(z)=([min\{max\{s_{a1},s_{a2}\},max\{s_{a1},s_{a3}\}\}, \ min\{max\{s_{b1},s_{b2}\},max\{s_{b1},s_{b3}\}]],\quad[min\{min\{s_{c1},s_{ c2}\},min\{s_{c1},s_{b3}\}\},min\{min\{s_{d1},s_{d2}\},min\{s_{d1},s_{d3}\}\}],\quad[max\{ min\{s_{e1},s_{e2}\},min\{s_{e1},s_{o3}\}\},\ max\{min\{s_{o1},s_{o2}\},min\{s_{o1},s_{o3}\}\}])\).