# Linguistic Interval-Valued Spherical Fuzzy Soft Set and Its Application in Decision Making
## 3 Linguistic Interval-Valued Spherical Fuzzy Soft Set
### Operations on \(Livesfsss\)
\right.\]  
where \(\circ(\tilde{F}(\varepsilon)\cup\tilde{G}(\varepsilon))=([max\{s\sqrt{(h)^{2 }-(\varepsilon 1)^{2}},s\sqrt{(h)^{2}-(\varepsilon 2)^{2}}\},max\{s\sqrt{(h)^{2}-( \varepsilon 1)^{2}},s\sqrt{(h)^{2}-(\varepsilon 2)^{2}}\}],\)\([s_{0},s_{0}],[min\{s_{e1},s_{\alpha 2}\},min\{s_{e1},s_{\alpha 2}\}])\). According to Definitions 12 and 17, suppose \(\circ(\tilde{F},A)\tilde{\cup}\circ(\tilde{G},B)=(\tilde{O},C)\), where \(C=A\cup B\) and \(\forall\varepsilon\in C\),  
\[\tilde{O}(\varepsilon)=\left\{ \begin{array}{l}\circ\tilde{F}(\varepsilon)= (\left[s\sqrt{(h)^{2}-(\varepsilon 1)^{2}},s\sqrt{(h)^{2}-(\varepsilon 1)^{2}}\right],[s_{0},s_{0}],[s_{e1},s_{01}]), \varepsilon\in A-B;\\ \circ\tilde{G}(\varepsilon)=(\left[s\sqrt{(h)^{2}-(\varepsilon 2)^{2}},s\sqrt{(h)^{2}-(\varepsilon 2)^{2}}\right],[s_{0},s_{0}],[s_{e2},s_{02}]), \varepsilon\in B-A;\\ \circ\tilde{F}(\varepsilon)\cup\circ\tilde{G}(\varepsilon),\ \varepsilon\in A\cap B; \end{array} \right.\]  
where \(\circ\tilde{F}(\varepsilon)\cup\circ\tilde{G}(\varepsilon)=([max\{s\sqrt{(h)^{2 }-(\varepsilon 1)^{2}},s\sqrt{(h)^{2}-(\varepsilon 2)^{2}}\},max\{s\sqrt{(h)^{2}-(\varepsilon 1)^{2}},s\sqrt{(h)^{2}-(\varepsilon 2)^{2}}\}],\)\([s_{0},s_{0}],[min\{s_{e1},s_{\alpha 2}\},min\{s_{\alpha 1},s_{\alpha 2}\}])\). Consequently, \(((\tilde{F},A)\tilde{\cup}(\tilde{G},B))=(\tilde{F},A)\ \tilde{\cup}\ \circ(\tilde{G},B)\). (3). The proof is similar to that of (2). (4). Suppose that \((\tilde{F},A)\wedge(\tilde{G},B)=(\tilde{H},\text{A}\times\text{B})\), where \(\forall(\alpha,\beta)\in A\times B\), \(\tilde{H}(\alpha,\beta)=([min\{s_{a1},s_{a2}\},min\{s_{b1},s_{b2}\}],[min\{s_{c1}, s_{c2}\},min\{s_{d1},s_{d2}\}],[max\{s_{e1},s_{c2}\},max\{s_{\alpha 1},s_{\alpha 2}\}])\). Then we have \(\circ((\tilde{F},A)\wedge(\tilde{G},B))=\circ(\tilde{H},\text{A}\times\text{B})\). According to Definition 17, we have \(\circ\tilde{H}(\varepsilon)=(\left[s\sqrt{(h)^{2}-(max\{c1,\alpha 2})},s\sqrt{(h)^{2}-(max\{c1,\alpha 2})}\right]^{2},[s_{0},s_{0}],[max\{s_{e1},s_{c2}\},max\{s_{\alpha 1},s_{\alpha 2}\}])\). According to Definitions 14 and 17, assume that \(\circ(\tilde{F},A)\ \wedge\ \circ(\tilde{G},B)=(\tilde{I},A\times\text{B})\), where \(\forall(\alpha,\beta)\in A\times\text{B}\), \(\tilde{I}(\alpha,\beta)=\circ\tilde{F}(\alpha)\cap\circ\tilde{G}(\beta)=([min\{s \sqrt{(h)^{2}-(\varepsilon 1)^{2}},s\sqrt{(h)^{2}-(\varepsilon 2)^{2}}\},min\{s\sqrt{(h)^{2}-(\varepsilon 1)^{2}},s\sqrt{(h)^{2}-(\varepsilon 2)^{2}}\}],[s_{0},s_{0}],[max\{s_{e1},s_{c2}\},max\{s_{\alpha 1},s_{\alpha 2}\}])=([s\sqrt{(h)^{2}-(max\{c1,\alpha 2})},\)\([s\sqrt{(h)^{2}-(\varepsilon 1,\alpha 2})]^{2},[s_{0},s_{0}],[max\{s_{e1},s_{c2}\},max\{s_{\alpha 1},s_{\alpha 2}\}])=( (\tilde{F},A)\wedge(\tilde{G},B))\). (5). The proof is similar to that of (4). **Theorem 5**: _Let \((\bar{F},A)\) be a linguistic interval-valued spherical fuzzy soft set over \(U\), then we have the following properties._  
\(\Box(\bar{F},A)\tilde{\subset}(\bar{F},A)\)_;_  
\(\circ\Box(\bar{F},A)=\Box(\bar{F},A)\)_;_  
\(\Box\circ(\bar{F},A)=\circ(\bar{F},A)\)_._  
**Proof. ** (1). According to Definitions 16 and 17, we have \(\Box\bar{F}(\alpha)=([s_{a},s_{b}],[s_{0},s_{0}],\)\([\sideset{}{{}_{8}}{{}_{8}}{{}_{8}}\sqrt{(h)^{2}-(b)^{2}},\sideset{}{{}_{8}}{{}_{8}}{{}_{8}} \sqrt{(h)^{2}-(a)^{2}}]\)) and \(\circ\bar{F}(\alpha)=([\sideset{}{{}_{8}}{{}_{8}}\sqrt{(h)^{2}-(\alpha)^{2}}, \sideset{}{{}_{8}}{{}_{8}}{{}_{8}}\sqrt{(h)^{2}-(\alpha)^{2}}]\), \([s_{0},s_{0}],[s_{e},s_{0}])\). Since \(0\leq b^{2}+d^{2}+f^{2}\leq h^{2}\), then we have \(\sideset{}{{}_{8}}{{}_{8}}{{}_{8}}{{}_{8}}\sqrt{(h)^{2}-(a)^{2}}\geq s_{0}\). Thus, \(\sideset{}{{}_{8}}{{}_{8}}{{}_{8}}{{}_{8}}\sqrt{(h)^{2}-(a)^{2}}\geq s_{0}\)\(\geq s_{e}\). Since \(s_{a}\leq s_{a}\), \(s_{b}\leq s_{b}\), \(s_{0}\leq s_{c}\), \(s_{0}\leq s_{d}\), hence \(\Box(\bar{F},A)\tilde{\subset}(\bar{F},A)\). (2).