## 3 Conflict Analysis Based on Soft Preference Relation
Then there is an associated dominance relation, which can be denoted by \(\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)\) and is defined as follows:  
\[\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)=\bigcap_{a\in E}F\left(a\right).\]  
For any \(u\in U\), define the soft dominance classes by:  
\[[u]^{+}_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}=\left\{u_{i}\in U:u_ {i}\succeq_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}u\right\}\]  
and:  
\[[u]^{-}_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}=\left\{u_{i}\in U:u_ {i}\preceq_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}u\right\},\]  
which represent the \((\mathcal{F},E)\)-dominating set and \((\mathcal{F},E)\)-dominated set with respect to \(u\in U\) of the \(\mathcal{IS}\), respectively. Class \([u]^{+}_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}\) describes the set of all those objects that dominate \(u\) and \([u]^{-}_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}\) describes the set of all those objects that are dominated by \(u\) in terms of \(\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)\), where:  
\[\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)^{+}=\left\{u\in U:[u]^{+}_{ \mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}\right\}\]  
and:  
\[\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)^{-}=\left\{u\in U:[u]^{-}_{ \mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}\right\}.\]  
Theorem 4.1: _Let \(\mathcal{IS}=\left(U,V,\widetilde{f}\right)\) be the Middle East conflict situation and \(\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)\) be a soft dominance relation. Then the following hold:_ \begin{tabular}{l l} _(1)_ & _If_ \(E_{1}\subseteq E\)_, then_ \(\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)\subseteq\mathcal{D}o\min \left(\mathcal{F}_{1},E_{1}\right)\) _;_ \\ _(2)_ & _If_ \(a_{n}\in[a_{m}]^{+}_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}\) _, then_ \([a_{n}]^{-}_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}\subseteq[a_{m}]^{+} _{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}\) _;_ \\ _(3)_ & \([a_{m}]^{+}_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}=\bigcup\left\{[a _{n}]^{-}_{\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}:a_{n}\in[a_{m}]^{+}_ {\mathcal{D}o\min\left(\mathcal{F}_{1},E\right)}\right\}\) _;_ \\ \end{tabular}   
\((5)\)__ \(A_{1}\subseteq A_{2}\) _implies_ \(\left(\overline{A_{1}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}} \subseteq\left(\overline{A_{2}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E) ^{+}}\) _;_  
\((6)\)__ \(\left(\underline{A_{1}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}}\cup \left(\underline{A_{2}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}} \subseteq\left(\underline{A_{1}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E )^{+}}\) _;_  
\((7)\)__ \(\left(\overline{A_{1}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}}\cup \left(\overline{A_{2}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}}= \left(\overline{A_{1}\cup A_{2}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E )^{+}}\) _;_  
\((8)\)__ \(\left(\underline{A_{1}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}} \cap\left(\underline{A_{2}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+ }}\) _._  
\((9)\)__ \(\left(\overline{A_{1}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}} \cap\left(\overline{A_{2}}\right)_{\mathcal{D}_{0}\min(\mathcal{F}_{1},E)^{+}} \supseteq\left(\overline{A_{1}\cap A_{2}}\right)_{\mathcal{D}_{0}\min( \mathcal{F}_{1},E)^{+}}\) _;_  
Proof. : These results follow from the respective definitions. The following theorems give a connection between topology and soft dominating/dominated definable sets. **Theorem 3**: _Let \(\mathcal{IS}=\left(U,V,\vec{f}\right)\) be the Middle East conflict situation. Then the collection of soft dominating definable sets form a topology on \(U\)._  
Proof. :
1.