# Linguistic Interval-Valued Spherical Fuzzy Soft Set and Its Application in Decision Making
## 3 Linguistic Interval-Valued Spherical Fuzzy Soft Set
### Operations on \(Livesfsss\)
Consequently, we can obtain \((\tilde{L},Z)=(\tilde{J},F)\). Thus \((\tilde{F},A)\tilde{\cup}((\tilde{G},B)\tilde{\cap}(\tilde{M},C))=((\tilde{F },A)\tilde{\cup}(\tilde{G},B))\tilde{\cap}((\tilde{F},A)\tilde{\cup}(\tilde{ M},C))\). Similarly, \((\tilde{F},A)\tilde{\cap}((\tilde{G},B)\tilde{\cup}(\tilde{M},C))=((\tilde{F },A)\tilde{\cap}(\tilde{G},B))\tilde{\cup}((\tilde{F},A)\tilde{\cap}(\tilde{ M},C))\). (6). The proof is similar to that of (5). (7). Suppose that \((\tilde{F},A)\tilde{\cup}(\tilde{G},B)=(\tilde{H},C)\), where \(C=A\cup B\) and \(\forall\varepsilon\in C\),  
\[\tilde{H}(\varepsilon)= \begin{cases}&\tilde{F}(\varepsilon),\varepsilon\in A-B; \\ &\tilde{G}(\varepsilon),\varepsilon\in B-A;\\ &\tilde{F}(\varepsilon)\cup\tilde{G}(\varepsilon),\,\varepsilon\in A\cap B; \end{cases} \]  
where \(\tilde{F}(\varepsilon)\cup\tilde{G}(\varepsilon)=([max\{s_{a1},s_{a2}\},\ max\{s_{b1},s_{b2}\}],[min\{s_{c1},s_{c2}\},min\{s_{d1},s_{d2}\}],\quad[min\{s_{e1},s_{e2}\},min\{s_{ o1},s_{o2}\}])\). Then we have \(((\tilde{F},A)\tilde{\cup}(\tilde{G},B))^{C}=(\tilde{H},C)^{C}\), where \(\forall\varepsilon\in C\),  
\[(\tilde{H}(\varepsilon))^{C}= \begin{cases}(\tilde{F}(\varepsilon))^{C}=([s_{e1},s_{o1}],[s_{c1},s_{d1}],[s_{a1},s_{b1}]),\varepsilon\in A-B;\\ ((\tilde{G}(\varepsilon))^{C}=([s_{e2},s_{o2}],[s_{c2},s_{d2}],[s_{a2},s_{b2}]), \varepsilon\in B-A;\\ &(\tilde{F}(\varepsilon)\cup\tilde{G}(\varepsilon))^{C},\varepsilon\in A\cap B; \end{cases} \]  
where \((\tilde{F}(\varepsilon)\cup\tilde{G}(\varepsilon))^{C}=([min\{s_{c1},s_{e2}\}, min\{s_{o1},s_{o2}\}],[min\{s_{c1},s_{c2}\},min\{s_{d1},s_{d2}\}],\ [max\{s_{b1},s_{b2}\}])\). Assume that \((\tilde{F},A)^{C}\tilde{\cap}(\tilde{G},B)^{C}=(\tilde{I},D)\), where \(D=A\cap B\), \(D\
eq\varnothing\) and \(\forall\varepsilon\in D\), then \(\tilde{I}(\varepsilon)=(\tilde{F},A)^{C}\tilde{\cap}(\tilde{G},B)^{C}=([min\{s_{ e1},s_{e2}\},\ min\{s_{o1},s_{o2}\}],\ [min\{s_{c1},s_{c2}\},min\{s_{d1},s_{d2}\}],\)\([max\{s_{a1},s_{a2}\},max\{s_{b1},s_{b2}\}])\). Consequently, \((\tilde{H},C)=(\tilde{I},D)\). Thus \(((\tilde{F},A)\tilde{\cup}(\tilde{G},B))^{C}=(\tilde{F},A)^{C}\tilde{\cap}( \tilde{G},B)^{C}\). Similarly, we can obtain \(((\tilde{F},A)\tilde{\cap}(\tilde{G},B))^{C}=(\tilde{F},A)^{C}\tilde{\cup}( \tilde{G},B)^{C}\). \(\square\)  
**Definition 14**. : _The \"AND\" operation on the two linguistic interval-valued spherical fuzzy soft sets \((\tilde{F},A)\) and \((\tilde{G},B)\) is defined by_  
\[(\tilde{F},A)\wedge(\tilde{G},B)=(\tilde{H},A\times B) \tag{15}\]  
_where \(\tilde{H}(\alpha,\beta)=\tilde{F}(\alpha)\cap\tilde{G}(\beta)\), \(\forall(\alpha,\beta)\in A\times B\)._  
**Definition 15**. : _The \"OR\" operation on the two linguistic interval-valued spherical fuzzy soft sets \((\tilde{F},A)\) and \((\tilde{G},B)\) is defined by_  
\[(\tilde{F},A)\vee(\tilde{G},B)=(\tilde{H},A\times B) \tag{16}\]  
_where \(\tilde{H}(\alpha,\beta)=\tilde{F}(\alpha)\cup\tilde{G}(\beta)\), \(\forall(\alpha,\beta)\in A\times B\)._  
**Theorem 2**. : _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. _Associative law:_ \((\tilde{F},A)\vee((\tilde{G},B)\vee(\tilde{M},C))=((\tilde{F},A)\vee(\tilde{ G},B))\vee(\tilde{M},C)\) _and_ \((\tilde{F},A)\wedge((\tilde{G},B)\wedge(\tilde{M},C))=((\tilde{F},A)\wedge( \tilde{G},B))\wedge(\tilde{M},C)\)_;_
2. _Distributive law:_ \((\tilde{F},A)\vee((\tilde{G},B)\wedge(\tilde{M},C))=((\tilde{F},A)\vee(\tilde {G},B))\wedge((\tilde{F},A)\vee(\tilde{M},C))\) _and_ \((\tilde{F},A)\wedge((\tilde{G},B)\vee(\tilde{M},C))=((\tilde{F},A)\wedge( \tilde{G},B))\vee((\tilde{F},A,A)\wedge(\tilde{M},C))\)_;_
3. _[PERSON]'s law:_ \(((\tilde{F},A)\vee(\tilde{G},B))^{C}=(\tilde{F},A)^{C}\wedge(\tilde{G},B)^{C}\) _and_ \(((\tilde{F},A)\wedge(\tilde{G},B))^{C}=(\tilde{F},A)^{C}\vee(\tilde{G},B)^{C}\)_._  
Proof. : We can obtain (1) easily according to Definitions 14 and 15. And we prove the other properties as follows. (2).