# Linguistic Interval-Valued Spherical Fuzzy Soft Set and Its Application in Decision Making
## 3 Linguistic Interval-Valued Spherical Fuzzy Soft Set
### Operations on \(Livesfsss\)
According to Definition 16, \(\forall\epsilon\in A\times B\), \(\square\tilde{H}(\epsilon)=([min\{s_{a1},s_{a2}\},min\{s_{b1},s_{b2}\}],[s_{0},s_{0} ],[s_{\sqrt{(h)^{2}-(min\{s_{b1},s_{b2}\})}},s_{\sqrt{(h)^{2}-(min\{s_{a1},s_{ a2}\})}}])\). According to Definition 14 and Definition 16, assume that \(\square(\tilde{F},A)\wedge\square(\tilde{G},B)=(\tilde{I},A\times B)\), where \(\forall(\alpha,\beta)\in A\times B\), \(\tilde{I}(\alpha,\beta)=\square\tilde{F}(\alpha)\cap\square\tilde{G}(\beta)=([min\{s_{ a1},s_{a2}\},min\{s_{b1},s_{b2}\}],[s_{0},s_{0}],\)\([max\{s_{\sqrt{(h)^{2}-(b1)^{2}}},s_{\sqrt{(h)^{2}-(b2)^{2}}}\},max\{s_{\sqrt{(h)^{2}-(a1)^{2}}},s_{ \sqrt{(h)^{2}-(a2)^{2}}}\}])=([min\{s_{a1},s_{a2}\},min\{s_{b1},s_{b2}\}],[s_{0},s_{0} ],[s_{\sqrt{(h)^{2}-(min\{s_{b1},s_{b2}\})}}]^{2},s_{\sqrt{(h)^{2}-(min\{s_{a1},s_{ a2}\})}}])=\square((\tilde{F},A)\wedge(\tilde{G},B))\). (5). The proof is similar to that of (4). \(\square\)  
**Definition 17**. : _The possibility operation on a linguistic interval-valued spherical fuzzy soft set \((\tilde{F},E)\) is denoted as \(\circ(\tilde{F},E)\), \(\forall\epsilon\in E\),_  
\[\circ\tilde{F}(\epsilon)=\Big{\{}\Big{\langle}x_{s}\big{(}\tilde{s}_{\circ u \tilde{F}(\epsilon)}(x),\tilde{s}_{\circ u\tilde{F}(\epsilon)}(x),\tilde{s}_{ \circ v\tilde{F}(\epsilon)}(x)\big{)}\Big{\rangle}\Big{|}x\in U\Big{\}} \tag{18}\]Here, \(\hat{s}_{\omega F(\varepsilon)}(x)=[\hat{s}_{\sqrt{(h)^{2}-((v\tilde{F}( \varepsilon))^{U})^{2}}}(x),\hat{s}_{\sqrt{(h)^{2}-((v\tilde{F}(\varepsilon))^{ U})^{2}}}(x)]\) is the possible membership degree that object \(x\) holds on parameter \(\varepsilon\), \(\hat{s}_{\omega F(\varepsilon)}(x)=[s_{0},s_{0}]\) is the empty hesitancy degree that object \(x\) holds on parameter \(\varepsilon\), \(\hat{s}_{\omega F(\varepsilon)}(x)=[s_{0}^{L}(x),\hat{s}_{\omega F(\varepsilon )}^{U}(x)]\) is the necessary non-membership degree that object \(x\) does not hold on parameter \(\varepsilon\). Let \((\tilde{F},A)\) and \((\tilde{G},B)\) be two linguistic interval-valued spherical fuzzy soft sets over \(U\), then we have the following properties. (1) \(\circ\circ(\tilde{F},E)=\circ(\tilde{F},E)\);  
(2) \(\circ((F,A)\tilde{\cup}(\tilde{G},B))=\circ(\tilde{F},A)\ \tilde{\cup}\ \circ( \tilde{G},B)\);  
(3) \(\circ((\tilde{F},A)\tilde{\cap}(\tilde{G},B))=\circ(\tilde{F},A)\ \cap\ \circ( \tilde{G},B)\);  
(4) \(\circ((\tilde{F},A)\wedge(\tilde{G},B))=\circ(\tilde{F},A)\ \wedge\ \circ( \tilde{G},B)\);  
(5) \(\circ((\tilde{F},A)\vee(\tilde{G},B))=\circ(\tilde{F},A)\ \vee\ \circ( \tilde{G},B)\). Proof. : We can obtain (1) easily according to Definition 17. And we prove the other properties as follows. (2). Suppose that \((\tilde{F},A)\tilde{\cup}(\tilde{G},B)=(\tilde{H},C)\), where \(C=A\cup B\) and \(\forall\varepsilon\in C\),  
\[\tilde{H}(\varepsilon)=\left\{ \begin{array}{l}\tilde{F}(\varepsilon)=([s_{ \alpha 1},s_{b1}],[s_{c1},s_{d1}],[s_{e1},s_{\alpha 1}]),\varepsilon\in A-B;\\ \tilde{G}(\varepsilon)=([s_{\alpha 2},s_{b2}],[s_{\alpha 2},s_{\alpha 2}],[s_{ \alpha 2},s_{\alpha 2}]),\varepsilon\in B-A;\\ \tilde{F}(\varepsilon)\cup\tilde{G}(\varepsilon),\ \varepsilon\in A\cap B;\end{array} \right.\]  
where \(\tilde{F}(\varepsilon)\cup\tilde{G}(\varepsilon)=([max\{s_{\alpha 1},s_{\alpha 2}\},max\{s_{b1},s_{b2}\}],[min\{s_{c1},s_{ \alpha 2}\},min\{s_{d1},s_{d2}\}],\ [min\{s_{c1},s_{\alpha 2}\},\)\(min\{s_{\alpha 1},s_{\alpha 2}\}])\). Then we have \(\circ((\tilde{F},A)\tilde{\cup}(\tilde{G},B))=\circ(\tilde{H},C)\). According to Definition 17, we have  
\[\circ\tilde{H}(\varepsilon)=\left\{ \begin{array}{l}\circ\tilde{F}(\varepsilon )=(\left[s\sqrt{(h)^{2}-(\varepsilon 1)^{2}},s\sqrt{(h)^{2}-(\varepsilon 1)^{2}}\right],[s_{0},s_{0}],[s_{e1},s_{01}]), \varepsilon\in A-B;\\ \circ\tilde{G}(\varepsilon)=(\left[s\sqrt{(h)^{2}-(\varepsilon 2)^{2}},s\sqrt{(h)^{2}-( \varepsilon 2)^{2}}\right],[s_{0},s_{0}],[s_{\alpha 2},s_{\alpha 2}]), \varepsilon\in B-A;\\ \circ(\tilde{F}(\varepsilon)\cup\tilde{G}(\varepsilon)),\ \varepsilon\in A\cap B; \end{array}