## 3 Conflict Analysis Based on Soft Preference Relation
From Theorem 2: \[\left(\underline{\emptyset}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+} }=\left(\overline{\mathcal{O}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E) ^{+}}=\emptyset\] and: \[\left(\underline{U}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}}= \left(\overline{U}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}}=U.\]
2. We prove that an arbitrary union of soft dominating definable sets is a soft dominating definable set. For this consider: \[\left(\overline{\cup A_{i}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}} = \cup_{i}\left(\overline{A_{i}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}}=\cup_{i}\left(\underline{A_{i}}\right)_{\mathcal{D}_{ 0}\min(\mathcal{F}_{1},E)^{+}}\] \[\subseteq \left(\underline{\cup A_{i}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}}.\] Similarly, \(\left(\underline{\cup A_{i}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}}\subseteq\left(\overline{\cup A_{i}}\right)_{\mathcal{D }_{0}\min(\mathcal{F}_{1},E)^{+}}\). Hence, \(\left(\underline{\cup A_{i}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+ }}=\left(\overline{\cup A_{i}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E) ^{+}}\). 3. Last, we prove a finite intersection of soft dominating definable sets is a soft dominating definable set. For this consider: \[\left(\overline{\cap_{i}^{n}A_{i}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}} \subseteq \cap_{i}^{n}\left(\overline{A_{i}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}}=\cap_{i}^{n}\left(\underline{A_{i}}\right)_{\mathcal{D }_{0}\min(\mathcal{F}_{1},E)^{+}}\] \[= \left(\underline{\cap_{i}^{n}A_{i}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}}.\] Similarly, \(\left(\underline{\cap_{i}^{n}A_{i}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}}\subseteq\left(\overline{\cap_{i}^{n}A_{i}}\right)_{ \mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}}\). Therefore, \(\left(\underline{\cap_{i}^{n}A_{i}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E )^{+}}=\left(\overline{\cap_{i}^{n}A_{i}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}}\). **Theorem 4**: _Let \(\mathcal{IS}=\left(U,V,\vec{f}\right)\) be the Middle East conflict situation. Then the collection of soft dominated definable sets form a topology on \(U\)._Proof. : The proof is similar to the proof of Theorem 3. **Definition 6**. : _Let \(\mathcal{IS}=\left(U,V,\widehat{f}\right)\) be the Middle East conflict situation. For any \(A_{1}\subseteq U\), the approximate precisions \(\rho_{\mathcal{D}\min\left(\mathcal{F}_{1},E\right)^{+}}\left(A_{1}\right)\) and \(\rho_{\mathcal{D}\min\left(\mathcal{F}_{1},E\right)^{-}}\left(A_{1}\right)\) of \(A_{1}\) about \(\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)^{+}\) and \(\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)^{-}\) are defined as:_  
\[\rho_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)^{+}}\left(A_{1}\right)= \frac{\left|\left(A_{1}\right)_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right) ^{+}}}{\left|\left(A_{1}\right)_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right) ^{+}}\right|}\text{ and }\text{ }\rho_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right) ^{-}}\left(A_{1}\right)=\frac{\left|\left(A_{1}\right)_{\mathcal{D}o\min \left(\mathcal{F}_{1},E\right)^{-}}}{\left|\left(A_{1}\right)_{\mathcal{D}o \min\left(\mathcal{F}_{1},E\right)^{-}}\right|},\]  
_where \(A_{1}:\
eq\emptyset\), and \(\left|\cdot\right|\) denotes the cardinality of a set._  
_Let \(\mu_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)^{+}}\left(A_{1}\right)=1- \rho_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)^{+}}\left(A_{1}\right)\) and \(\mu_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)^{-}}\left(A_{1}\right)=1- \rho_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)^{-}}\left(A_{1}\right)\).