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Implications over intuitionistic fuzzy sets: Difference between revisions

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For the various definitions of implication of over [[intuitionistic fuzzy sets]], the functions sg(''x'') and {{overline|sg}}(''x'') have been used:
For the various definitions of implication of over [[intuitionistic fuzzy sets]], the functions sg(''x'') and {{overline|sg}}(''x'') have been used:


<math>
<math>\text{sg}(x) = \begin{cases} 1 \text{ if } x > 0 \\
\text{sg}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x > 0 \\
0 \text{ if } x \leq 0
0 & \text{if } x \leq 0
\end{cases},</math>&nbsp;&nbsp;&nbsp;<math>
\end{array},</math>&nbsp;&nbsp;&nbsp;<math>
\overline{\text{sg}}(x) = \begin{cases} 0 \text{ if } x > 0 \\
\overline{\text{sg}}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x < 0 \\
1 \text{ if } x \leq 0
0 & \text{if } x \geq 0
\end{cases}.</math>
\end{array}.</math>


ρ== List of intuitionistic fuzzy implications ==  
<!-- == List of intuitionistic fuzzy implications ==  


{| width="100%" class="wikitable sortable" style="font-family:Courier; font-size:120%;"  
{| width="100%" class="wikitable sortable" style="font-family:Courier; font-size:120%;"  
Line 161: Line 160:
|
|
| &#123;&#60;x, <font color=green>{{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>, <font color=red>sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
| &#123;&#60;x, <font color=green>{{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>, <font color=red>sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>25</sub>
|
|
| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x).{{overline|sg}}(ν<sub>B</sub>(x)).{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>26</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>27</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),sg(μ<sub>B</sub>(x)))</font>, <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>28</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>29</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>30</sub>
|
|
| &#123;&#60;x, <font color=green>max(1-μ<sub>A</sub>(x),min(μ<sub>A</sub>(x),1-ν<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>31</sub>
|
|
| &#123;&#60;x, <font color=green>{{overline|sg}}(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>, <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>32</sub>
|
|
| &#123;&#60;x, <font color=green>1-ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>, <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>33</sub>
|
|
| &#123;&#60;x, <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>34</sub>
|
|
| &#123;&#60;x, <font color=green>min(1,2-μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>, <font color=red>max(0,μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>35</sub>
|
|
| &#123;&#60;x, <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>, <font color=red>μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>36</sub>
|
|
| &#123;&#60;x, <font color=green>min(1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)),max(μ<sub>A</sub>(x),1-μ<sub>A</sub>(x)),max(1-ν<sub>B</sub>(x),ν<sub>B</sub>(x)))</font>, <font color=red>max(min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)),min(μ<sub>A</sub>(x),1-μ<sub>A</sub>(x)),min(1-ν<sub>B</sub>(x),ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>37</sub>
|
|
| &#123;&#60;x, <font color=green>1-max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>, <font color=red>max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>38</sub>
|
|
| &#123;&#60;x, <font color=green>1-μ<sub>A</sub>(x)+(μ<sub>A</sub>(x)<sup>2</sup>.(1-ν<sub>B</sub>(x)))</font>, <font color=red>μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x)<sup>2</sup>.ν<sub>B</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>39</sub>
|
|
| &#123;&#60;x, <font color=green>(1-ν<sub>B</sub>(x)).{{overline|sg}}(1-μ<sub>A</sub>(x))+sg(1-μ<sub>A</sub>(x)).({{overline|sg}}(ν<sub>B</sub>(x))+(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)))</font>, <font color=red>ν<sub>B</sub>(x).{{overline|sg}}(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x).sg(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>40</sub>
|
|
| &#123;&#60;x, <font color=green>1-sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>, <font color=red>1-{{overline|sg}}(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>41</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font>, <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>42</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),sg(1-ν<sub>B</sub>(x)))</font>, <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>43</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font>, <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>44</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>45</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),{{overline|sg}}(ν<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>46</sub>
|
|
| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),min(1-ν<sub>A</sub>(x),μ<sub>B</sub>(x)))</font>, <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>47</sub>
|
|
| &#123;&#60;x, <font color=green>{{overline|sg}}(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>, <font color=red>(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>48</sub>
|
|
| &#123;&#60;x, <font color=green>1-(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>, <font color=red>(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>49</sub>
|
|
| &#123;&#60;x, <font color=green>min(1,ν<sub>A</sub>(x)+μ<sub>B</sub>(x))</font>, <font color=red>max(0,1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>50</sub>
|
|
| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>, <font color=red>1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>51</sub>
|
|
| &#123;&#60;x, <font color=green>min(max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),max(1-ν<sub>A</sub>(x),ν<sub>A</sub>(x)),max(μ<sub>B</sub>(x),1-μ<sub>B</sub>(x)))</font>, <font color=red>max(1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),min(1-ν<sub>A</sub>(x),ν<sub>A</sub>(x)),min(μ<sub>B</sub>(x),1-μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>52</sub>
|
|
| &#123;&#60;x, <font color=green>1-(1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x))).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>, <font color=red>1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>53</sub>
|
|
| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+(1-ν<sub>A</sub>(x))<sup>2</sup>.μ<sub>B</sub>(x)</font>, <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+(1-ν<sub>A</sub>(x))<sup>2</sup>.(1-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>54</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x){{overline|sg}}(ν<sub>A</sub>(x))+sg(ν<sub>A</sub>(x)).({{overline|sg}}(1-μ<sub>B</sub>(x))+ν<sub>A</sub>(x).sg(1-μ<sub>B</sub>(x)))</font>, <font color=red>(1-μ<sub>B</sub>(x)).{{overline|sg}}(ν<sub>A</sub>(x))+(1-ν<sub>A</sub>(x)).sg(ν<sub>A</sub>(x)).sg(1-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>55</sub>
|
|
| &#123;&#60;x, <font color=green>1-sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>, <font color=red>1-{{overline|sg}}(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>56</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>min(sg(1-ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>57</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),sg(μ<sub>B</sub>(x)))</font>, <font color=red>min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>58</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>59</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>60</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>61</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),min(ν<sub>B</sub>(x),ν<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>62</sub>
|
|
| &#123;&#60;x, <font color=green>{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>63</sub>
|
|
| &#123;&#60;x, <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>64</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>B</sub>(x).ν<sub>A</sub>(x)</font>, <font color=red>ν<sub>B</sub>(x).μ<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>65</sub>
|
|
| &#123;&#60;x, <font color=green>1-(1-min(μ<sub>B</sub>(x),ν<sub>A</sub>(x))).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>, <font color=red>max(ν<sub>B</sub>(x),μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>66</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>ν<sub>A</sub>(x)</font>, <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>μ<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>67</sub>
|
|
| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+μ<sub>B</sub>(x).sg(1-ν<sub>A</sub>(x)))</font>, <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>68</sub>
|
|
| &#123;&#60;x, <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>69</sub>
|
|
| &#123;&#60;x, <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))-μ<sub>A</sub>(x).{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>70</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}((ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>71</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>,μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>72</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>, <font color=red>min(1-μ<sub>B</sub>(x),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>73</sub>
|
|
| &#123;&#60;x, <font color=green>max(1-max(sg(ν<sub>B</sub>(x)),sg(1-μ<sub>B</sub>(x))),ν<sub>A</sub>(x))</font>, <font color=red>min(sg(1-μ<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>74</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font>, <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>75</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(ν<sub>A</sub>(x)+μ<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x).(ν<sub>B</sub>(x)+μ<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x)<sup>2</sup>+μ<sub>A</sub>(x))+ν<sub>A</sub>(x).μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>76</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),1-μ<sub>A</sub>(x))</font>, <font color=red>min(1-μ<sub>B</sub>(x),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>77</sub>
|
|
| &#123;&#60;x, <font color=green>1-min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>, <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>78</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>80</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>81</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>82</sub>
|
|
| &#123;&#60;x, <font color=green>max(1-ν<sub>B</sub>(x),min(ν<sub>B</sub>(x),1-μ<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>83</sub>
|
|
| &#123;&#60;x, <font color=green>{{overline|sg}}(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)-1)</font>, <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)-1)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>84</sub>
|
|
| &#123;&#60;x, <font color=green>1-μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)+1)</font>, <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)+1)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>85</sub>
|
|
| &#123;&#60;x, <font color=green>1-ν<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>.(1-μ<sub>A</sub>(x))</font>, <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x)<sup>2</sup></font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>86</sub>
|
|
| &#123;&#60;x, <font color=green>(1-μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)){{overline|sg}}(μ<sub>A</sub>(x)+min(1-ν<sub>B</sub>(x),sg(μ<sub>A</sub>(x))))</font>, <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>87</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font>, <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>88</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(1-μ<sub>A</sub>(x)))</font>, <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>89</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>90</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),{{overline|sg}}(μ<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>91</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),min(1-μ<sub>B</sub>(x),ν<sub>A</sub>(x)))</font>, <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>92</sub>
|
|
| &#123;&#60;x, <font color=green>{{overline|sg}}(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>, <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>93</sub>
|
|
| &#123;&#60;x, <font color=green>1-min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>, <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>94</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.ν<sub>A</sub>(x))</font>, <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.(1-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>95</sub>
|
|
| &#123;&#60;x, <font color=green>min(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+sg(μ<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+min(μ<sub>B</sub>(x),sg(1-ν<sub>A</sub>(x))))</font>, <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+min(min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))),sg(1-ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>96</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>min(sg(1-μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>97</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font>, <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>98</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>99</sub>
|
|
| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min(1-μ<sub>B</sub>(x),{{overline|sg}}(ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>100</sub>
|
|
| &#123;&#60;x, <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),μ<sub>B</sub>(x))</font>, <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>101</sub>
|
|
| &#123;&#60;x, <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>102</sub>
|
|
| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>103</sub>
|
|
| &#123;&#60;x, <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),1-ν<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>104</sub>
|
|
| &#123;&#60;x, <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>105</sub>
|
|
| &#123;&#60;x, <font color=green>max(1-μ<sub>A</sub>(x),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>106</sub>
|
|
| &#123;&#60;x, <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),μ<sub>B</sub>(x))</font>, <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),1-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>107</sub>
|
|
| &#123;&#60;x, <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font>, <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>108</sub>
|
|
| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font>, <font color=red>min(1-ν<sub>A</sub>(x),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>109</sub>
|
|
| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>110</sub>
|
|
| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>111</sub>
|
|
| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>112</sub>
|
|
| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>, <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)).ν<sub>B</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>113</sub>
|
|
| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+(μ<sub>B</sub>(x).ν<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x))).(ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>114</sub>
|
|
| &#123;&#60;x, <font color=green>1-μ<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>115</sub>
|
|
| &#123;&#60;x, <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>116</sub>
|
|
| &#123;&#60;x, <font color=green>max(1-μ<sub>A</sub>(x),(1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>117</sub>
|
|
| &#123;&#60;x, <font color=green>1-μ<sub>A</sub>(x)-ν<sub>B</sub>(x)+μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>, <font color=red>(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x))).ν<sub>B</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>118</sub>
|
|
| &#123;&#60;x, <font color=green>(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))+μ<sub>A</sub>(x).ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))</font>, <font color=red>(μ<sub>A</sub>(x)-μ<sub>A</sub>(x)<sup>2</sup>+{{overline|sg}}(1-μ<sub>A</sub>(x))).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)<sup>2</sup>+{{overline|sg}}(1-ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x))</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>119</sub>
|
|
| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>120</sub>
|
|
| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>, <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>121</sub>
|
|
| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),(1-μ<sub>B</sub>(x)).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>122</sub>
|
|
| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>, <font color=red>((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x))).(1-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>123</sub>
|
|
| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x))).(((1-μ<sub>B</sub>(x)).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x))))+{{overline|sg}}(μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>124</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>125</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>126</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>127</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font>, <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>128</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>A</sub>(x)-μ<sub>B</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>129</sub>
|
|
| &#123;&#60;x, <font color=green>1-ν<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font>, <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>130</sub>
|
|
| &#123;&#60;x, <font color=green>1-min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>131</sub>
|
|
| &#123;&#60;x, <font color=green>max(1-ν<sub>B</sub>(x),(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>132</sub>
|
|
| &#123;&#60;x, <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>, <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>133</sub>
|
|
| &#123;&#60;x, <font color=green>1-ν<sub>B</sub>(x)+(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)-(1-ν<sub>B</sub>(x)).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font>, <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>134</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>135</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>, <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>136</sub>
|
|
| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),(1-ν<sub>A</sub>(x)).(ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>137</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font>, <font color=red>((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x))).(1-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| →<sub>138</sub>
|
|
| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))-μ<sub>B</sub>(x).</font>, <font color=red></font>&#62;&#124;x &#8712; E&#125;
|}
|}
-->
=== List of implications ===
{| width="100%" class="wikitable" style="font-family:Courier; font-size:120%;"
|- valign="top"
! width="5%" | No.
! width="5%" | Ref.
! width="5%" | Year
! width="85%" | Implication:
&#123;&#60;x, <font color=green>Implication MEMBERSHIP expression</font>, <font color=red>Implication NON-MEMBERSHIP expression</font> &#62;&#124;x &#8712; E&#125;
|}
{| width="100%" class="wikitable sortable" style="font-family:Courier; font-size:120%;"
|- valign="top"
! width="5%" | No.
! width="5%" | Ref.
! width="5%" | Year
! width="40%" | Implication MEMBERSHIP expression<br/>
! width="45%" | Implication NON-MEMBERSHIP expression<br/>
|- valign="top"
| →<sub>1</sub>
|
|
| <font color=green> max(ν<sub>A</sub>(x),min(μ<sub>A</sub>(x),μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>2</sub>
|
|
| <font color=green> {{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red> ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>3</sub>
|
|
| <font color=green> 1-(1-μ<sub></sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>ν<sub>B</sub>.sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)) </font>
|- valign="top"
| →<sub>4</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>5</sub>
|
|
| <font color=green>min(1,ν<sub>A</sub>(x)+μ<sub>B</sub>(x))</font> || <font color=red>max(0,μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
|- valign="top"
| →<sub>6</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+μ<sub>A</sub>(x)μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x)ν<sub>B</sub>(x)</font>
|- valign="top"
| →<sub>7</sub>
|
|
| <font color=green>min(max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),max(μ<sub>A</sub>(x),ν<sub>A</sub>(x)), max(μ<sub>B</sub>(x),ν<sub>B</sub>(x)))</font> || <font color=red>max(min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)), min(μ<sub>A</sub>(x),ν<sub>A</sub>(x)),min(μ<sub>B</sub>(x),ν<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>8</sub>
|
|
| <font color=green>1-(1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x))).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)),sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
|- valign="top"
| →<sub>9</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+μ<sub>A</sub>(x)<sup>2</sup>μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x)ν<sub>A</sub>(x)+μ<sub>A</sub>(x)<sup>2</sup>ν<sub>B</sub>(x)</font>
|- valign="top"
| →<sub>10</sub>
|
|
| <font color=green>μ<sub>A</sub>(x).{{overline|sg}}(1-μ<sub>A</sub>(x))+sg(1-μ<sub>A</sub>(x)).({{overline|sg}}(1-μ<sub>B</sub>(x))+ν<sub>A</sub>(x).sg(1-μ<sub>B</sub>(x)))</font> || <font color=red>ν<sub>B</sub>.{{overline|sg}}(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x).sg(1-μ<sub>A</sub>(x)).sg(1-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>11</sub>
|
|
| <font color=green>1-(1-μ<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
|- valign="top"
| →<sub>12</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>13</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>
|- valign="top"
| →<sub>14</sub>
|
|
| <font color=green>1-(1-μ<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))-ν<sub>B</sub>(x).{{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
|- valign="top"
| →<sub>15</sub>
|
|
| <font color=green>1-sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>sg({{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))+{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>
|- valign="top"
| →<sub>16</sub>
|
|
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>17</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+μ<sub>A</sub>(x)<sup>2</sup>,ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>18</sub>
|
|
| <font color=green> max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(1-ν<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>19</sub>
|
|
| <font color=green>max(1-sg(sg(μ<sub>A</sub>(x))+sg(1-ν<sub>A</sub>(x))),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>20</sub>
|
|
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),sg(μ<sub>A</sub>(x))))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(μ<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>21</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).(μ<sub>B</sub>(x)+ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x).(μ<sub>A</sub>(x)+ν<sub>A</sub>(x)),ν<sub>B</sub>(x).(μ<sub>B</sub>(x)<sup>2</sup>+ν<sub>B</sub>(x)+μ<sub>B</sub>(x).ν<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>22</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),1-ν<sub>B</sub>(x))</font> || <font color=red>min(1-ν<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>23</sub>
|
|
| <font color=green>1-min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>24</sub>
|
|
| <font color=green>{{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
|- valign="top"
| →<sub>25</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x).{{overline|sg}}(ν<sub>B</sub>(x)).{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>26</sub>
|
|
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>27</sub>
|
|
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),sg(μ<sub>B</sub>(x)))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>28</sub>
|
|
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>29</sub>
|
|
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>30</sub>
|
|
| <font color=green>max(1-μ<sub>A</sub>(x),min(μ<sub>A</sub>(x),1-ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>31</sub>
|
|
| <font color=green>{{overline|sg}}(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
|- valign="top"
| →<sub>32</sub>
|
|
| <font color=green>1-ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
|- valign="top"
| →<sub>33</sub>
|
|
| <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>34</sub>
|
|
| <font color=green>min(1,2-μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>max(0,μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
|- valign="top"
| →<sub>35</sub>
|
|
| <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>
|- valign="top"
| →<sub>36</sub>
|
|
| <font color=green>min(1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)),max(μ<sub>A</sub>(x),1-μ<sub>A</sub>(x)),max(1-ν<sub>B</sub>(x),ν<sub>B</sub>(x)))</font> || <font color=red>max(min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)),min(μ<sub>A</sub>(x),1-μ<sub>A</sub>(x)),min(1-ν<sub>B</sub>(x),ν<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>37</sub>
|
|
| <font color=green>1-max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
|- valign="top"
| →<sub>38</sub>
|
|
| <font color=green>1-μ<sub>A</sub>(x)+(μ<sub>A</sub>(x)<sup>2</sup>.(1-ν<sub>B</sub>(x)))</font> || <font color=red>μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x)<sup>2</sup>.ν<sub>B</sub>(x)</font>
|- valign="top"
| →<sub>39</sub>
|
|
| <font color=green>(1-ν<sub>B</sub>(x)).{{overline|sg}}(1-μ<sub>A</sub>(x))+sg(1-μ<sub>A</sub>(x)).({{overline|sg}}(ν<sub>B</sub>(x))+(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)))</font> || <font color=red>ν<sub>B</sub>(x).{{overline|sg}}(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x).sg(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>40</sub>
|
|
| <font color=green>1-sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>1-{{overline|sg}}(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
|- valign="top"
| →<sub>41</sub>
|
|
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>42</sub>
|
|
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),sg(1-ν<sub>B</sub>(x)))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>43</sub>
|
|
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>44</sub>
|
|
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>45</sub>
|
|
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),{{overline|sg}}(ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>46</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),min(1-ν<sub>A</sub>(x),μ<sub>B</sub>(x)))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>47</sub>
|
|
| <font color=green>{{overline|sg}}(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>48</sub>
|
|
| <font color=green>1-(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>49</sub>
|
|
| <font color=green>min(1,ν<sub>A</sub>(x)+μ<sub>B</sub>(x))</font> || <font color=red>max(0,1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>50</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>
|- valign="top"
| →<sub>51</sub>
|
|
| <font color=green>min(max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),max(1-ν<sub>A</sub>(x),ν<sub>A</sub>(x)),max(μ<sub>B</sub>(x),1-μ<sub>B</sub>(x)))</font> || <font color=red>max(1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),min(1-ν<sub>A</sub>(x),ν<sub>A</sub>(x)),min(μ<sub>B</sub>(x),1-μ<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>52</sub>
|
|
| <font color=green>1-(1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x))).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>53</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+(1-ν<sub>A</sub>(x))<sup>2</sup>.μ<sub>B</sub>(x)</font> || <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+(1-ν<sub>A</sub>(x))<sup>2</sup>.(1-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>54</sub>
|
|
| <font color=green>μ<sub>B</sub>(x){{overline|sg}}(ν<sub>A</sub>(x))+sg(ν<sub>A</sub>(x)).({{overline|sg}}(1-μ<sub>B</sub>(x))+ν<sub>A</sub>(x).sg(1-μ<sub>B</sub>(x)))</font> || <font color=red>(1-μ<sub>B</sub>(x)).{{overline|sg}}(ν<sub>A</sub>(x))+(1-ν<sub>A</sub>(x)).sg(ν<sub>A</sub>(x)).sg(1-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>55</sub>
|
|
| <font color=green>1-sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>1-{{overline|sg}}(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>56</sub>
|
|
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>57</sub>
|
|
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),sg(μ<sub>B</sub>(x)))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(μ<sub>B</sub>(x)))</font>
|- valign="top"
| →<sub>58</sub>
|
|
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>
|- valign="top"
| →<sub>59</sub>
|
|
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>


|- valign="top"
| →<sub>60</sub>
|
|
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))</font>


|- valign="top"
| →<sub>61</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),min(ν<sub>B</sub>(x),ν<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>


=== Alternative separated view ===
|- valign="top"
| →<sub>62</sub>
|
|
| <font color=green>{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>63</sub>
|
|
| <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>64</sub>
|
|
| <font color=green>μ<sub>B</sub>(x)+ν<sub>B</sub>(x).ν<sub>A</sub>(x)</font> || <font color=red>ν<sub>B</sub>(x).μ<sub>A</sub>(x)</font>
 
|- valign="top"
| →<sub>65</sub>
|
|
| <font color=green>1-(1-min(μ<sub>B</sub>(x),ν<sub>A</sub>(x))).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>max(ν<sub>B</sub>(x),μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>66</sub>
|
|
| <font color=green>μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>ν<sub>A</sub>(x)</font> || <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>μ<sub>A</sub>(x)</font>
 
|- valign="top"
| →<sub>67</sub>
|
|
| <font color=green>ν<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+μ<sub>B</sub>(x).sg(1-ν<sub>A</sub>(x)))</font> || <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>68</sub>
|
|
| <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>69</sub>
|
|
| <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))-μ<sub>A</sub>(x).{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>70</sub>
|
|
| <font color=green>max({{overline|sg}}((ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>71</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>,μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>72</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min(1-μ<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>73</sub>
|
|
| <font color=green>max(1-max(sg(ν<sub>B</sub>(x)),sg(1-μ<sub>B</sub>(x))),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>74</sub>
|
|
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>75</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(ν<sub>A</sub>(x)+μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x).(ν<sub>B</sub>(x)+μ<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x)<sup>2</sup>+μ<sub>A</sub>(x))+ν<sub>A</sub>(x).μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>76</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),1-μ<sub>A</sub>(x))</font> || <font color=red>min(1-μ<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>77</sub>
|
|
| <font color=green>1-min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>78</sub>
|
|
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>80</sub>
|
|
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>81</sub>
|
|
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>82</sub>
|
|
| <font color=green>max(1-ν<sub>B</sub>(x),min(ν<sub>B</sub>(x),1-μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>83</sub>
|
|
| <font color=green>{{overline|sg}}(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)-1)</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)-1)</font>
 
|- valign="top"
| →<sub>84</sub>
|
|
| <font color=green>1-μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)+1)</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)+1)</font>
 
|- valign="top"
| →<sub>85</sub>
|
|
| <font color=green>1-ν<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>.(1-μ<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x)<sup>2</sup></font>
 
|- valign="top"
| →<sub>86</sub>
|
|
| <font color=green>(1-μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)){{overline|sg}}(μ<sub>A</sub>(x)+min(1-ν<sub>B</sub>(x),sg(μ<sub>A</sub>(x))))</font> || <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>87</sub>
|
|
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>88</sub>
|
|
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(1-μ<sub>A</sub>(x)))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>89</sub>
|
|
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>90</sub>
|
|
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),{{overline|sg}}(μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>91</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),min(1-μ<sub>B</sub>(x),ν<sub>A</sub>(x)))</font> || <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>92</sub>
|
|
| <font color=green>{{overline|sg}}(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>93</sub>
|
|
| <font color=green>1-min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font> || <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>94</sub>
|
|
| <font color=green>μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.ν<sub>A</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.(1-ν<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>95</sub>
|
|
| <font color=green>min(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+sg(μ<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+min(μ<sub>B</sub>(x),sg(1-ν<sub>A</sub>(x))))</font> || <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+min(min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))),sg(1-ν<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>96</sub>
|
|
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x)</font>
 
|- valign="top"
| →<sub>97</sub>
|
|
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>98</sub>
|
|
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>99</sub>
|
|
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min(1-μ<sub>B</sub>(x),{{overline|sg}}(ν<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>100</sub>
|
|
| <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),μ<sub>B</sub>(x))</font> || <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),ν<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>101</sub>
|
|
| <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>
 
|- valign="top"
| →<sub>102</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>
 
|- valign="top"
| →<sub>103</sub>
|
|
| <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),1-ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>104</sub>
|
|
| <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>
 
|- valign="top"
| →<sub>105</sub>
|
|
| <font color=green>max(1-μ<sub>A</sub>(x),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>
 
|- valign="top"
| →<sub>106</sub>
|
|
| <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),μ<sub>B</sub>(x))</font> || <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),1-μ<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>107</sub>
|
|
| <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font> || <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>
 
|- valign="top"
| →<sub>108</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font> || <font color=red>min(1-ν<sub>A</sub>(x),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>
 
|- valign="top"
| →<sub>109</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>110</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>111</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
 
|- valign="top"
| →<sub>112</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)).ν<sub>B</sub>(x)</font>
 
|- valign="top"
| →<sub>113</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+(μ<sub>B</sub>(x).ν<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x))).(ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
 
|- valign="top"
| →<sub>114</sub>
|
|
| <font color=green>1-μ<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>115</sub>
|
|
| <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>116</sub>
|
|
| <font color=green>max(1-μ<sub>A</sub>(x),(1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
 
|- valign="top"
| →<sub>117</sub>
|
|
| <font color=green>1-μ<sub>A</sub>(x)-ν<sub>B</sub>(x)+μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> || <font color=red>(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x))).ν<sub>B</sub>(x)</font>
 
|- valign="top"
| →<sub>118</sub>
|
|
| <font color=green>(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))+μ<sub>A</sub>(x).ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))</font> || <font color=red>(μ<sub>A</sub>(x)-μ<sub>A</sub>(x)<sup>2</sup>+{{overline|sg}}(1-μ<sub>A</sub>(x))).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)<sup>2</sup>+{{overline|sg}}(1-ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x))</sub>(x))</font>
 
|- valign="top"
| →<sub>119</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>120</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>121</sub>
|
|
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),(1-μ<sub>B</sub>(x)).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(μ<sub>B</sub>(x)))</font>
 
|- valign="top"
| →<sub>122</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x))).(1-μ<sub>B</sub>(x))</font>
 
|- valign="top"
| →<sub>123</sub>
|
|
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x))).(((1-μ<sub>B</sub>(x)).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x))))+{{overline|sg}}(μ<sub>B</sub>(x)))</font>
 
|- valign="top"
| →<sub>124</sub>
|
|
| <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>125</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>126</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>127</sub>
|
|
| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font> || <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>
 
|- valign="top"
| →<sub>128</sub>
|
|
| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>A</sub>(x)-μ<sub>B</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>129</sub>
|
|
| <font color=green>1-ν<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>130</sub>
|
|
| <font color=green>1-min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>131</sub>
|
|
| <font color=green>max(1-ν<sub>B</sub>(x),(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>132</sub>
|
|
| <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> || <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>
 
|- valign="top"
| →<sub>133</sub>
|
|
| <font color=green>1-ν<sub>B</sub>(x)+(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)-(1-ν<sub>B</sub>(x)).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font> || <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>134</sub>
|
|
| <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>135</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>136</sub>
|
|
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),(1-ν<sub>A</sub>(x)).(ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(ν<sub>A</sub>(x)))</font>
 
|- valign="top"
| →<sub>137</sub>
|
|
| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font> || <font color=red>((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x))).(1-ν<sub>A</sub>(x))</font>
 
|- valign="top"
| →<sub>138</sub>
|
|
| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))-μ<sub>B</sub>(x).</font> || <font color=red>
((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x))).(1-ν<sub>A</sub>(x).(ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x)) + {{overline|sg}}(1-ν<sub>A</sub>(x)) + {{overline|sg}}(ν<sub>A</sub>(x))
 
|- valign="top"
| →<sub>139</sub>
|
|
| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x))/2</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x))/2</font>
 
|- valign="top"
| →<sub>140</sub>
|
|
| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x) + min(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + max(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font>
 
|- valign="top"
| →<sub>141</sub>
|
|
| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x) + max(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + min(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font>
 
|- valign="top"
| →<sub>142</sub>
|
|
| <font color=green>(3 - μ<sub>A</sub>(x) - ν<sub>B</sub>(x) - max(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + max(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font>
 
|- valign="top"
| →<sub>143</sub>
|
|
| <font color=green>(1 - μ<sub>A</sub>(x) + μ<sub>b</sub>(x) + min(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)) )/3</font> || <font color=red>(2 + μ<sub>A</sub>(x) - μ<sub>B</sub>(x) - min(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font>
 
|- valign="top"
| →<sub>144</sub>
|
|
| <font color=green>(1 + ν<sub>A</sub>(x) - ν<sub>b</sub>(x) + min(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x)) )/3</font> || <font color=red>(2 - ν<sub>A</sub>(x) + ν<sub>B</sub>(x) + min(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x)))/3</font>
 
|- valign="top"
| →<sub>145</sub>
|
|
| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x) + min(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> || <font color=red>(3 - ν<sub>A</sub>(x) - μ<sub>B</sub>(x) - min(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font>
 
|- valign="top"
| →<sub>146</sub>
|
|
| <font color=green>(3 - μ<sub>A</sub>(x) - ν<sub>B</sub>(x) - min(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + min(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font>
 
|- valign="top"
| →<sub>147</sub>
|
|
| <font color=green>(1 - μ<sub>A</sub>(x) + μ<sub>b</sub>(x) + max(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)) )/3</font> || <font color=red>(2 + μ<sub>A</sub>(x) - μ<sub>b</sub>(x) - max(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)) )/3</font>
 
 
</font>
|}


== References ==
== References ==
Line 182: Line 1,759:
* [[Issue:On some two-parametric intuitionistic fuzzy implications|On some two-parametric intuitionistic fuzzy implications]], Piotr Dworniczak, 2011
* [[Issue:On some two-parametric intuitionistic fuzzy implications|On some two-parametric intuitionistic fuzzy implications]], Piotr Dworniczak, 2011
* [[Issue:Second Zadeh's intuitionistic fuzzy implication|Second Zadeh's intuitionistic fuzzy implication]], Krassimir Atanassov, 2011
* [[Issue:Second Zadeh's intuitionistic fuzzy implication|Second Zadeh's intuitionistic fuzzy implication]], Krassimir Atanassov, 2011
; "What Links Here" References
{{Special:WhatLinksHere/{{PAGENAME}}|namespace=102|hidetrans=1|hideredirs=1}}


== See also ==
== See also ==

Latest revision as of 09:41, 29 April 2022

For the various definitions of implication of over intuitionistic fuzzy sets, the functions sg(x) and sg(x) have been used:

[math]\displaystyle{ \text{sg}(x) = \begin{cases} 1 \text{ if } x \gt 0 \\ 0 \text{ if } x \leq 0 \end{cases}, }[/math]   [math]\displaystyle{ \overline{\text{sg}}(x) = \begin{cases} 0 \text{ if } x \gt 0 \\ 1 \text{ if } x \leq 0 \end{cases}. }[/math]


List of implications

No. Ref. Year Implication:

{<x, Implication MEMBERSHIP expression, Implication NON-MEMBERSHIP expression >|x ∈ E}

No. Ref. Year Implication MEMBERSHIP expression
 Implication NON-MEMBERSHIP expression
→1 max(νA(x),min(μA(x),μB(x))) min(μA(x),νB(x))
→2 sg(μA(x)-μB(x)) νB(x).sg(μA(x)-μB(x))
→3 1-(1-μ(x)).sg(μA(x)-μB(x)) νB.sg(μA(x)-μB(x))
→4 max(νA(x),μB(x)) min(μA(x),νB(x))
→5 min(1,νA(x)+μB(x)) max(0,μA(x)+νB(x)-1)
→6 νA(x)+μA(x)μB(x) μA(x)νB(x)
→7 min(max(νA(x),μB(x)),max(μA(x),νA(x)), max(μB(x),νB(x))) max(min(μA(x),νB(x)), min(μA(x),νA(x)),min(μB(x),νB(x)))
→8 1-(1-min(νA(x),μB(x))).sg(μA(x)-μB(x)) max(μA(x),νB(x)).sg(μA(x)-μB(x)),sg(νB(x)-νA(x))
→9 νA(x)+μA(x)2μB(x) μA(x)νA(x)+μA(x)2νB(x)
→10 μA(x).sg(1-μA(x))+sg(1-μA(x)).(sg(1-μB(x))+νA(x).sg(1-μB(x))) νB.sg(1-μA(x))+μA(x).sg(1-μA(x)).sg(1-μB(x))
→11 1-(1-μB(x)).sg(μA(x)-μB(x)) νB(x).sg(μA(x)-μB(x)).sg(νB(x)-νA(x))
→12 max(νA(x),μB(x)) 1-max(νA(x),μB(x))
→13 νA(x)+μB(x)-νA(x).μB(x) μA(x).νB(x)
→14 1-(1-μB(x)).sg(μA(x)-μB(x))-νB(x).sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) νB(x).sg(νB(x)-νA(x))
→15 1-sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) sg(sg(μA(x)-μB(x))+sg(νB(x)-νA(x)))
→16 max(sg(μA(x)),μB(x)) min(sg(μA(x)),νB(x))
→17 max(νA(x),μB(x)) min(μA(x).νA(x)+μA(x)2,νB(x))
→18 max(νA(x),μB(x)) min(1-νA(x),νB(x))
→19 max(1-sg(sg(μA(x))+sg(1-νA(x))),μB(x)) min(sg(1-νA(x)),νB(x))
→20 max(sg(μA(x)),sg(μA(x)))) min(sg(μA(x)),sg(μB(x)))
→21 max(νA(x),μB(x).(μB(x)+νB(x))) min(μA(x).(μA(x)+νA(x)),νB(x).(μB(x)2+νB(x)+μB(x).νB(x)))
→22 max(νA(x),1-νB(x)) min(1-νA(x),νB(x))
→23 1-min(sg(1-νA(x)),sg(1-νB(x))) min(sg(1-νA(x)),sg(1-νB(x)))
→24 sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) sg(μA(x)-μB(x)).sg(νB(x)-νA(x))
→25 max(νA(x),sg(μA(x)).sg(1-νA(x)),μB(x).sg(νB(x)).sg(1-μB(x))) min(μA(x),νB(x))
→26 max(sg(1-νA(x)),μB(x)) min(sg(μA(x)),νB(x))
→27 max(sg(1-νA(x)),sg(μB(x))) min(sg(μA(x)),sg(1-νB(x)))
→28 max(sg(1-νA(x)),μB(x)) min(μA(x),νB(x))
→29 max(sg(1-νA(x)),sg(1-μB(x))) min(μA(x),sg(1-νB(x)))
→30 max(1-μA(x),min(μA(x),1-νB(x))) min(μA(x),νB(x))
→31 sg(μA(x)+νB(x)-1) νB(x).sg(μA(x)+νB(x)-1)
→32 1-νB(x).sg(μA(x)+νB(x)-1) νB(x).sg(μA(x)+νB(x)-1)
→33 1-min(μA(x),νB(x)) min(μA(x),νB(x))
→34 min(1,2-μA(x)-μB(x)) max(0,μA(x)+νB(x)-1)
→35 1-μA(x).νB(x) μA(x).νB(x)
→36 min(1-min(μA(x),νB(x)),max(μA(x),1-μA(x)),max(1-νB(x),νB(x))) max(min(μA(x),νB(x)),min(μA(x),1-μA(x)),min(1-νB(x),νB(x)))
→37 1-max(μA(x),νB(x)).sg(μA(x)+νB(x)-1) max(μA(x),νB(x)).sg(μA(x)+νB(x)-1)
→38 1-μA(x)+(μA(x)2.(1-νB(x))) μA(x)(1-μA(x))+μA(x)2.νB(x)
→39 (1-νB(x)).sg(1-μA(x))+sg(1-μA(x)).(sg(νB(x))+(1-μA(x)).sg(νB(x))) νB(x).sg(1-μA(x))+μA(x).sg(1-μA(x)).sg(νB(x))
→40 1-sg(μA(x)+νB(x)-1) 1-sg(μA(x)+νB(x)-1)
→41 max(sg(μA(x)),1-νB(x)) min(sg(μA(x)),νB(x))
→42 max(sg(μA(x)),sg(1-νB(x))) min(sg(μA(x)),sg(1-νB(x)))
→43 max(sg(μA(x)),1-νB(x)) min(sg(μA(x)),νB(x))
→44 max(sg(μA(x)),1-νB(x)) min(μA(x),νB(x))
→45 max(sg(μA(x)),sg(νB(x))) min(μA(x),sg(1-νB(x)))
→46 max(νA(x),min(1-νA(x),μB(x))) 1-max(νA(x),μB(x))
→47 sg(1-νA(x)-μB(x)) (1-μB(x)).sg(1-νA(x)-μB(x))
→48 1-(1-μB(x)).sg(1-νA(x)-μB(x)) (1-μB(x)).sg(1-νA(x)-μB(x))
→49 min(1,νA(x)+μB(x)) max(0,1-νA(x)-μB(x))
→50 νA(x)+μB(x)-νA(x).μB(x) 1-νA(x)-μB(x)+νA(x).μB(x)
→51 min(max(νA(x),μB(x)),max(1-νA(x),νA(x)),max(μB(x),1-μB(x))) max(1-max(νA(x),μB(x)),min(1-νA(x),νA(x)),min(μB(x),1-μB(x)))
→52 1-(1-min(νA(x),μB(x))).sg(1-νA(x)-μB(x)) 1-min(νA(x),μB(x)).sg(1-νA(x)-μB(x))
→53 νA(x)+(1-νA(x))2.μB(x) (1-νA(x)).νA(x)+(1-νA(x))2.(1-μB(x))
→54 μB(x)sg(νA(x))+sg(νA(x)).(sg(1-μB(x))+νA(x).sg(1-μB(x))) (1-μB(x)).sg(νA(x))+(1-νA(x)).sg(νA(x)).sg(1-μB(x))
→55 1-sg(1-νA(x)-μB(x)) 1-sg(1-νA(x)-μB(x))
→56 max(sg(1-νA(x)),μB(x)) min(sg(1-νA(x)),1-μB(x))
→57 max(sg(1-νA(x)),sg(μB(x))) min(sg(1-νA(x)),sg(μB(x)))
→58 max(sg(1-νA(x)),sg(1-μB(x))) 1-max(νA(x),μB(x))
→59 max(sg(1-νA(x)),μB(x)) 1-max(νA(x),μB(x))
→60 max(sg(1-νA(x)),sg(1-μB(x))) min(1-νA(x),sg(μB(x)))
→61 max(μB(x),min(νB(x),νA(x))) min(νB(x),μA(x))
→62 sg(νB(x)-νA(x)) μA(x).sg(νB(x)-νA(x))
→63 1-(1-νA(x)).sg(νB(x)-νA(x)) μA(x).sg(νB(x)-νA(x))
→64 μB(x)+νB(x).νA(x) νB(x).μA(x)
→65 1-(1-min(μB(x),νA(x))).sg(νB(x)-νA(x)) max(νB(x),μA(x)).sg(νB(x)-νA(x)).sg(μA(x)-μB(x))
→66 μB(x)+νB(x)2νA(x) νB(x).μB(x)+νB(x)2μA(x)
→67 νA(x).sg(1-νB(x))+sg(1-νB(x)).(sg(1-νA(x))+μB(x).sg(1-νA(x))) μA(x).sg(1-νB(x))+νB(x).sg(1-νB(x)).sg(1-νA(x))
→68 1-(1-νA(x)).sg(νB(x)-νA(x)) μA(x).sg(νB(x)-νA(x)).sg(μA(x)-μB(x))
→69 1-(1-νA(x)).sg(νB(x)-νA(x))-μA(x).sg(νB(x)-νA(x)).sg(μA(x)-μB(x)) μA(x).sg(μA(x)-μB(x))
→70 max(sg((νB(x)),νA(x)) min(sg(νB(x)),μA(x))
→71 max(μB(x),νA(x)) min(νB(x).μB(x)+νB(x)2,μA(x))
→72 max(μB(x),νA(x)) min(1-μB(x),μA(x))
→73 max(1-max(sg(νB(x)),sg(1-μB(x))),νA(x)) min(sg(1-μB(x)),μA(x))
→74 max(sg(νB(x)),sg(νA(x))) min(sg(νB(x)),sg(νA(x)))
→75 max(μB(x),νA(x).(νA(x)+μA(x))) min(νB(x).(νB(x)+μB(x)),μA(x).(νA(x)2+μA(x))+νA(x).μA(x))
→76 max(μB(x),1-μA(x)) min(1-μB(x),μA(x))
→77 1-min(sg(1-μB(x)),sg(1-μA(x))) min(sg(1-μB(x)),sg(1-μA(x)))
→78 max(sg(1-μB(x)),νA(x)) min(sg(νB(x)),μA(x))
→80 max(sg(1-μB(x)),νA(x)) min(νB(x),μA(x))
→81 max(sg(1-μB(x)),sg(1-νA(x))) min(νB(x),sg(1-μA(x)))
→82 max(1-νB(x),min(νB(x),1-μA(x))) min(νB(x),μA(x))
→83 sg(νB(x)+μA(x)-1) μA(x).sg(νB(x)+μA(x)-1)
→84 1-μA(x).sg(νB(x)+μA(x)+1) μA(x).sg(νB(x)+μA(x)+1)
→85 1-νB(x)+νB(x)2.(1-μA(x)) νB(x).(1-νB(x))+νB(x)2
→86 (1-μA(x)).sg(1-νB(x))+sg(1-νB(x))sg(μA(x)+min(1-νB(x),sg(μA(x)))) μA(x).sg(1-νB(x))+νB(x).sg(1-νB(x)).sg(μA(x))
→87 max(sg(νB(x)),1-μA(x)) min(sg(νB(x)),μA(x))
→88 max(sg(νB(x)),sg(1-μA(x))) min(sg(νB(x)),sg(1-μA(x)))
→89 max(sg(νB(x)),1-μA(x)) min(νB(x),μA(x))
→90 max(sg(νB(x)),sg(μA(x))) min(νB(x),sg(1-μA(x)))
→91 max(μB(x),min(1-μB(x),νA(x))) 1-max(μB(x),νA(x))
→92 sg(1-μB(x)-νA(x)) min(1-νA(x),sg(1-μB(x)-νA(x)))
→93 1-min(1-νA(x),sg(1-μB(x)-νA(x))) min(1-νA(x),sg(1-μB(x)-νA(x)))
→94 μB(x)+(1-μB(x))2.νA(x)) (1-μB(x)).μB(x)+(1-μB(x))2.(1-νA(x))
→95 min(νA(x),sg(μB(x)))+sg(μB(x)).(sg(1-νA(x))+min(μB(x),sg(1-νA(x)))) min(1-νA(x),sg(μB(x)))+min(min(1-μB(x),sg(μB(x))),sg(1-νA(x)))
→96 max(sg(1-μB(x)),νA(x)) min(sg(1-μB(x)),1-νA(x)
→97 max(sg(1-μB(x)),sg(νA(x))) min(sg(1-μB(x)),sg(νA(x)))
→98 max(sg(1-μB(x)),νA(x)) 1-max(μB(x),νA(x))
→99 max(sg(1-μB(x)),sg(1-νA(x))) min(1-μB(x),sg(νA(x)))
→100 max(min(νA(x),sg(μA(x))),μB(x)) min(min(μA(x),sg(νA(x))),νB(x))
→101 max(min(νA(x),sg(μA(x))),min(μB(x),sg(νB(x)))) min(min(μA(x),sg(νA(x))),min(νB(x),sg(μB(x))))
→102 max(νA(x),min(μB(x),sg(νB(x)))) min(μA(x),min(νB(x),sg(μB(x))))
→103 max(min(1-μA(x),sg(μA(x))),1-νB(x)) min(μA(x),sg(1-μA(x)),νB(x))
→104 max(min(1-μA(x),sg(μA(x))),min(1-νB(x),sg(νB(x)))) min(min(μA(x),sg(1-μA(x))),min(νB(x),sg(1-νB(x))))
→105 max(1-μA(x),min(1-νB(x),sg(νB(x)))) min(μA(x),min(νB(x),sg(1-νB(x))))
→106 max(min(νA(x),sg(1-νA(x))),μB(x)) min(min(1-νA(x),sg(νA(x))),1-μB(x))
→107 max(min(νA(x),sg(1-νA(x))),min(μB(x),sg(1-μB(x)))) min(min(1-νA(x),sg(νA(x))),min(1-μB(x),sg(μB(x))))
→108 max(νA(x),min(μB(x),sg(1-μB(x)))) min(1-νA(x),min(1-μB(x),sg(μB(x))))
→109 νA(x)+min(sg(1-μA(x)),μB(x)) μA(x).νA(x)+min(sg(1-μA(x)),νB(x))
→110 max(νA(x),μB(x)) min(μA(x).νA(x)+sg(1-μA(x)),νB(x))
→111 max(νA(x),μB(x).νB(x)+sg(1-μB(x))) min(μA(x).νA(x)+sg(1-μA(x)),νB(x).(μB(x).νB(x)+sg(1-μB(x)))+sg(1-νB(x)))
→112 νA(x)+μB(x)-νA(x).μB(x) μA(x).νA(x)+sg(1-μA(x)).νB(x)
→113 νA(x)+(μB(x).νB(x)-νA(x).(μB(x).νB(x)+sg(1-μB(x))) (μA(x).νA(x)+sg(1-μA(x))).(νB(x).(μB(x).νB(x)+sg(1-μB(x)))+sg(1-νB(x)))
→114 1-μA(x)+min(sg(1-μA(x)),1-νB(x)) μA(x).(1-μA(x))+min(sg(1-μA(x)),νB(x))
→115 1-min(μA(x),νB(x)) min(μA(x)(1-μA(x))+sg(1-μA(x)),νB(x))
→116 max(1-μA(x),(1-νB(x)).νB(x)+sg(νB(x))) min(μA(x).(1-μA(x))+sg(1-μA(x)),νB(x).((1-νB(x)).νB(x)+sg(νB(x)))+sg(1-νB(x)))
→117 1-μA(x)-νB(x)+μA(x).νB(x) (μA(x).(1-μA(x))+sg(1-μA(x))).νB(x)
→118 (1-μA(x)).sg(νB(x))+μA(x).νB(x).(1-νB(x)) (μA(x)-μA(x)2+sg(1-μA(x))).((1-νB(x)).νB(x)2+sg(1-νB(x)))+sg(1-νB(x))(x))
→119 νA(x)+min(sg(νA(x)),μB(x)) (1-νA(x)).νA(x)+min(sg(νA(x)),1-μB(x))
→120 max(νA(x),μB(x)) min((1-νA(x)).νA(x)+sg(νA(x)),1-μB(x))
→121 max(νA(x),μB(x).(1-μB(x))+sg(1-μB(x))) min((1-νA(x)).νA(x)+sg(νA(x)),(1-μB(x)).(μB(x).(1-μB(x))+sg(1-μB(x)))+sg(μB(x)))
→122 νA(x)+μB(x)-νA(x).μB(x) ((1-νA(x)).νA(x)+sg(νA(x))).(1-μB(x))
→123 νA(x)+μB(x).(1-μB(x)-νA(x).(μB(x).(1-μB(x))+sg(1-μB(x))) ((1-νA(x)).νA(x)+sg(νA(x))).(((1-μB(x)).(μB(x).(1-μB(x))+sg(1-μB(x))))+sg(μB(x)))
→124 μB(x)+min(sg(1-νB(x)),νA(x)) νB(x).μB(x)+min(sg(1-νB(x)),μA(x))
→125 max(μB(x),νA(x)) min(νB(x).μB(x)+sg(1-νB(x)),μA(x))
→126 max(μB(x),νA(x).μA(x)+sg(1-νA(x))) min(νB(x).μB(x)+sg(1-νB(x)),μA(x).(νA(x).μA(x)+sg(1-νA(x)))+sg(1-μA(x)))
→127 μB(x)+νA(x)-μB(x).νA(x) (νB(x).μB(x)+sg(1-νB(x))).μA(x)
→128 μB(x)+νA(x).μA(x)-μB(x).(νA(x).μA(x)+sg(1-νA(x))) (νB(x).μB(x)+sg(1-νB(x))).(μA(x).(νA(x).μA(x)+sg(1-νA(x)))+sg(1-μA(x)))
→129 1-νB(x)+min(sg(1-νB(x)),1-μA(x)) νB(x).(1-νB(x))+min(sg(1-νB(x)),μA(x))
→130 1-min(νB(x),μA(x)) min(νB(x).(1-νB(x))+sg(1-νB(x)),μA(x))
→131 max(1-νB(x),(1-μA(x)).μA(x)+sg(μA(x))) min(νB(x).(1-νB(x))+sg(1-νB(x)),μA(x).((1-μA(x)).μA(x)+sg(μA(x)))+sg(1-μA(x)))
→132 1-μA(x).νB(x) (νB(x).(1-νB(x))+sg(1-νB(x))).μA(x)
→133 1-νB(x)+(1-μA(x)).μA(x)-(1-νB(x)).((1-μA(x)).μA(x)+sg(μA(x))) (νB(x).(1-νB(x))+sg(1-νB(x))).(μA(x).((1-μA(x)).μA(x)+sg(μA(x)))+sg(1-μA(x)))
→134 μB(x)+min(sg(μB(x)),νA(x)) (1-μB(x)).μB(x)+min(sg(μB(x)),1-νA(x))
→135 max(μB(x),νA(x)) min((1-μB(x)).μB(x)+sg(μB(x)),1-νA(x))
→136 max(μB(x),νA(x).(1-νA(x))+sg(1-νA(x))) min((1-μB(x)).μB(x)+sg(μB(x)),(1-νA(x)).(νA(x).(1-νA(x))+sg(1-νA(x)))+sg(νA(x)))
→137 μB(x)+νA(x)-μB(x).νA(x) ((1-μB(x)).μB(x)+sg(μB(x))).(1-νA(x))
→138 μB(x)+νA(x).(1-νA(x))-μB(x).

((1-μB(x)).μB(x)+sg(μB(x))).(1-νA(x).(νA(x).(1-νA(x)) + sg(1-νA(x)) + sg(νA(x))

→139 (νA(x) + μB(x))/2 (μA(x) + νB(x))/2
→140 (νA(x) + μB(x) + min(νA(x), μB(x)))/3 (μA(x) + νB(x) + max(μA(x), νB(x)))/3
→141 (νA(x) + μB(x) + max(νA(x), μB(x)))/3 (μA(x) + νB(x) + min(μA(x), νB(x)))/3
→142 (3 - μA(x) - νB(x) - max(μA(x), νB(x)))/3 (μA(x) + νB(x) + max(μA(x), νB(x)))/3
→143 (1 - μA(x) + μb(x) + min(1 - μA(x), μB(x)) )/3 (2 + μA(x) - μB(x) - min(1 - μA(x), μB(x)))/3
→144 (1 + νA(x) - νb(x) + min(νA(x), 1 - νB(x)) )/3 (2 - νA(x) + νB(x) + min(νA(x), 1 - νB(x)))/3
→145 (νA(x) + μB(x) + min(νA(x), μB(x)))/3 (3 - νA(x) - μB(x) - min(νA(x), μB(x)))/3
→146 (3 - μA(x) - νB(x) - min(μA(x), νB(x)))/3 (μA(x) + νB(x) + min(μA(x), νB(x)))/3
→147 (1 - μA(x) + μb(x) + max(1 - μA(x), μB(x)) )/3 (2 + μA(x) - μb(x) - max(1 - μA(x), μB(x)) )/3


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