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Subtractions over intuitionistic fuzzy sets: Difference between revisions

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For the various definitions of subtraction of over [[intuitionistic fuzzy sets]], the functions sg and {{overline|sg}} have been used:
<math>
\text{sg}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x > 0 \\
0 & \text{if } x \leq 0
\end{array},</math>&nbsp;&nbsp;&nbsp;<math>
\overline{\text{sg}}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x < 0 \\
0 & \text{if } x \geq 0
\end{array}.</math>
== List of intuitionistic fuzzy subtractions of &#8212;<sub><i>i</i></sub> type ==
=== Alternative separated view ===
== List of intuitionistic fuzzy subtractions of &#8212;<sub><i>i</i></sub>&#8242; type ==
{| width="100%" class="wikitable sortable" style="font-family:Courier; font-size:120%;"
|- valign="top"
! width="5%" | No.
! width="5%" | Ref.
! width="5%" | Year
! width="85%" | Subtraction<br/>
|- valign="top"
| &#8212;<sub>01</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>02</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(μ<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), sg(μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>03</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).ν<sub>B</sub>(x) +  μ<sub>B</sub>(x)<sup>2</sup>)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>04</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>, <font color=red>max(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>05</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), sg(1 - ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>06</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), sg(μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>07</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>08</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x))</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>09</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(μ<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>10</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>11</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), sg(ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>12</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x).(μ<sub>B</sub>(x) + ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).(ν<sub>B</sub>(x)<sup>2</sup> + μ<sub>B</sub>(x) + μ<sub>B</sub>(x).ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>13</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), sg(1 - μ<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>14</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), sg(ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>15</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>16</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(μ<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>17</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>18</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x), sg(μ<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), min(μ<sub>B</sub>(x), sg(ν<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>19</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x), sg(μ<sub>B</sub>(x)))</font>, <font color=red>ν<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>20</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>, <font color=red>ν<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>21</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x), sg(μ<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), min(μ<sub>B</sub>(x), sg(1 - μ<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>22</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x), sg(μ<sub>B</sub>(x)))</font>, <font color=red>ν<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>23</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x))</font>, <font color=red>ν<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>24</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x), sg(1 - ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), min(1 - ν<sub>B</sub>(x), sg(ν<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>25</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x), sg(1 - ν<sub>B</sub>(x)))</font>, <font color=red>ν<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>26</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>, <font color=red>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>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>27</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x))</font>, <font color=red>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>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>28</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>, <font color=red>max(ν<sub>A</sub>(x), (1 - ν<sub>B</sub>(x)).ν<sub>B</sub>(x) + {{overline|sg}}(ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>29</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), max(0, μ<sub>B</sub>(x).ν<sub>B</sub>(x) + {{overline|sg}}(1 - ν<sub>B</sub>(x))))</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x) + {{overline|sg}}(1 - ν<sub>B</sub>(x))) + {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>30</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), μ<sub>B</sub>(x).ν<sub>B</sub>(x)</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x) + {{overline|sg}}(1 - ν<sub>B</sub>(x))) + {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>31</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), (1 - μ<sub>B</sub>(x)).μ<sub>B</sub>(x) + {{overline|sg}}(μ<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).((1 - μ<sub>B</sub>(x)).μ<sub>B</sub>(x) + {{overline|sg}}(μ<sub>B</sub>(x))) + {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>32</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), (1 - μ<sub>B</sub>(x)).μ<sub>B</sub>(x)</font>, <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).((1 - μ<sub>B</sub>(x)).μ<sub>B</sub>(x) + {{overline|sg}}(μ<sub>B</sub>(x))) + {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>33</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x).(1 - ν<sub>B</sub>(x)) + {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), (1 - ν<sub>B</sub>(x)).(ν<sub>B</sub>(x).(1 - ν<sub>B</sub>(x)) + {{overline|sg}}(1 - ν<sub>B</sub>(x))) + {{overline|sg}}(ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|- valign="top"
| &#8212;<sub>34</sub>&#8242;
|
|
| &#123;&#60;x, <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x).(1 - ν<sub>B</sub>(x)))</font>, <font color=red>max(ν<sub>A</sub>(x), (1 - ν<sub>B</sub>(x)).(ν<sub>B</sub>(x).(1 - ν<sub>B</sub>(x)) + {{overline|sg}}(1 - ν<sub>B</sub>(x))) + {{overline|sg}}(ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
|}
=== Alternative separated view ===
{| width="100%" class="wikitable" style="font-family:Courier; font-size:120%;"
|- valign="top"
! width="5%" | No.
! width="5%" | Ref.
! width="5%" | Year
! width="85%" | Subtraction:
&#123;&#60;x, <font color=green>Subtraction MEMBERSHIP expression</font>, <font color=red>Subtraction 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%" | Subtraction MEMBERSHIP expression<br/>
! width="45%" | Subtraction NON-MEMBERSHIP expression<br/>
|- valign="top"
| &#8212;<sub>01</sub>&#8242;
|
|
| style= "color: green;" | min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))
| style= "color: red;"  | max(ν<sub>A</sub>(x), μ<sub>B</sub>(x))
|- valign="top"
| &#8212;<sub>02</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(μ<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), sg(μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>03</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>
| <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).ν<sub>B</sub>(x) +  μ<sub>B</sub>(x)<sup>2</sup>)</font>
|- valign="top"
| &#8212;<sub>04</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>
| <font color=red>max(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x))</font>
|- valign="top"
| &#8212;<sub>05</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), sg(1 - ν<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>06</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), sg(μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>07</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x))</font>
|- valign="top"
| &#8212;<sub>08</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x))</font>
| <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x))</font>
|- valign="top"
| &#8212;<sub>09</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(μ<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x))</font>
|- valign="top"
| &#8212;<sub>10</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x))</font>
|- valign="top"
| &#8212;<sub>11</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), sg(ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(ν<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>12</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x).(μ<sub>B</sub>(x) + ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).(ν<sub>B</sub>(x)<sup>2</sup> + μ<sub>B</sub>(x) + μ<sub>B</sub>(x).ν<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>13</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), sg(1 - μ<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>14</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), sg(ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>15</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>16</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(μ<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>17</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), {{overline|sg}}(ν<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>18</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x), sg(μ<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), min(μ<sub>B</sub>(x), sg(ν<sub>B</sub>(x))))</font>
|- valign="top"
| &#8212;<sub>19</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x), sg(μ<sub>B</sub>(x)))</font>
| <font color=red>ν<sub>A</sub>(x)</font>
|- valign="top"
| &#8212;<sub>20</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>
| <font color=red>ν<sub>A</sub>(x)</font>
|- valign="top"
| &#8212;<sub>21</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x), sg(μ<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), min(μ<sub>B</sub>(x), sg(1 - μ<sub>B</sub>(x))))</font>
|- valign="top"
| &#8212;<sub>22</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x), sg(μ<sub>B</sub>(x)))</font>
| <font color=red>ν<sub>A</sub>(x)</font>
|- valign="top"
| &#8212;<sub>23</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x))</font>
| <font color=red>ν<sub>A</sub>(x)</font>
|- valign="top"
| &#8212;<sub>24</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x), sg(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), min(1 - ν<sub>B</sub>(x), sg(ν<sub>B</sub>(x))))</font>
|- valign="top"
| &#8212;<sub>25</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x), sg(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>ν<sub>A</sub>(x)</font>
|- valign="top"
| &#8212;<sub>26</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>
| <font color=red>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>
|- valign="top"
| &#8212;<sub>27</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), 1 - μ<sub>B</sub>(x))</font>
| <font color=red>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>
|- valign="top"
| &#8212;<sub>28</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x))</font>
| <font color=red>max(ν<sub>A</sub>(x), (1 - ν<sub>B</sub>(x)).ν<sub>B</sub>(x) + {{overline|sg}}(ν<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>29</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), max(0, μ<sub>B</sub>(x).ν<sub>B</sub>(x) + {{overline|sg}}(1 - ν<sub>B</sub>(x))))</font>
| <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x) + {{overline|sg}}(1 - ν<sub>B</sub>(x))) + {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>30</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), μ<sub>B</sub>(x).ν<sub>B</sub>(x)</font>
| <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x) + {{overline|sg}}(1 - ν<sub>B</sub>(x))) + {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>31</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), (1 - μ<sub>B</sub>(x)).μ<sub>B</sub>(x) + {{overline|sg}}(μ<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).((1 - μ<sub>B</sub>(x)).μ<sub>B</sub>(x) + {{overline|sg}}(μ<sub>B</sub>(x))) + {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>32</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), (1 - μ<sub>B</sub>(x)).μ<sub>B</sub>(x)</font>
| <font color=red>max(ν<sub>A</sub>(x), μ<sub>B</sub>(x).((1 - μ<sub>B</sub>(x)).μ<sub>B</sub>(x) + {{overline|sg}}(μ<sub>B</sub>(x))) + {{overline|sg}}(1 - μ<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>33</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x).(1 - ν<sub>B</sub>(x)) + {{overline|sg}}(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), (1 - ν<sub>B</sub>(x)).(ν<sub>B</sub>(x).(1 - ν<sub>B</sub>(x)) + {{overline|sg}}(1 - ν<sub>B</sub>(x))) + {{overline|sg}}(ν<sub>B</sub>(x)))</font>
|- valign="top"
| &#8212;<sub>34</sub>&#8242;
|
|
| <font color=green>min(μ<sub>A</sub>(x), ν<sub>B</sub>(x).(1 - ν<sub>B</sub>(x)))</font>
| <font color=red>max(ν<sub>A</sub>(x), (1 - ν<sub>B</sub>(x)).(ν<sub>B</sub>(x).(1 - ν<sub>B</sub>(x)) + {{overline|sg}}(1 - ν<sub>B</sub>(x))) + {{overline|sg}}(ν<sub>B</sub>(x)))</font>
|}
== List of intuitionistic fuzzy subtractions of &#8212;<sub><i>i</i></sub>&#8242;&#8242; type ==
=== Alternative separated view ===
== Approaches to defining intuitionistic fuzzy subtractions ==
== References ==
== References ==
* [[Issue:Remark on operation "subtraction" over intuitionistic fuzzy sets|Remark on operation "subtraction" over intuitionistic fuzzy sets]], [[Krassimir Atanassov]], 2009
* [[Issue:Remark on operation "subtraction" over intuitionistic fuzzy sets|Remark on operation "subtraction" over intuitionistic fuzzy sets]], Krassimir Atanassov, 2009
* [[Issue:On intuitionistic fuzzy subtraction, generated by an implication from Kleene-Dienes type|On intuitionistic fuzzy subtraction, generated by an implication from Kleene-Dienes type]], Lilija Atanassova, 2009
* [[Issue:On intuitionistic fuzzy subtraction, generated by an implication from Kleene-Dienes type|On intuitionistic fuzzy subtraction, generated by an implication from Kleene-Dienes type]], Lilija Atanassova, 2009
* [[Issue:On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬11|On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬<sub>11</sub>]],  Beloslav Riečan, Diana Boyadzhieva, Krassimir Atanassov, 2009
* [[Issue:On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬11|On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬<sub>11</sub>]],  Beloslav Riečan, Diana Boyadzhieva, Krassimir Atanassov, 2009
* [[Issue:On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬4|On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬<sub>4</sub>]], Beloslav Riečan, Magdaléna Renčová, Krassimir Atanassov, 2009
* [[Issue:On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬4|On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬<sub>4</sub>]], Beloslav Riečan, Magdaléna Renčová, Krassimir Atanassov, 2009
* [[Issue:Equalities with intuitionistic fuzzy subtractions and negations|Equalities with intuitionistic fuzzy subtractions and negations]], Krassimir Atanassov, Magdaléna Renčová, Dimitar Dimitrov, 2010
* [[Issue:On Łukasiewicz's intuitionistic fuzzy subtraction|On Łukasiewicz's intuitionistic fuzzy subtraction]], Beloslav Riečan, Krassimir Atanassov, 2011
* [[Issue:On Zadeh's intuitionistic fuzzy subtraction|On Zadeh's intuitionistic fuzzy subtraction]], Beloslav Riečan, 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 subtraction of over intuitionistic fuzzy sets, the functions sg and sg have been used:

[math]\displaystyle{ \text{sg}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x \gt 0 \\ 0 & \text{if } x \leq 0 \end{array}, }[/math]   [math]\displaystyle{ \overline{\text{sg}}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x \lt 0 \\ 0 & \text{if } x \geq 0 \end{array}. }[/math]

List of intuitionistic fuzzy subtractions of —i type

Alternative separated view

List of intuitionistic fuzzy subtractions of —i′ type

No. Ref. Year Subtraction
—01′ {<x, min(μA(x), νB(x)), max(νA(x), μB(x))>|x ∈ E}
—02′ {<x, min(μA(x), sg(μB(x))), max(νA(x), sg(μB(x)))>|x ∈ E}
—03′ {<x, min(μA(x), νB(x)), max(νA(x), μB(x).νB(x) + μB(x)2)>|x ∈ E}
—04′ {<x, min(μA(x), νB(x)), max(νA(x), 1 - νB(x))>|x ∈ E}
—05′ {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(1 - νB(x)))>|x ∈ E}
—06′ {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(μB(x)))>|x ∈ E}
—07′ {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), μB(x))>|x ∈ E}
—08′ {<x, min(μA(x), 1 - μB(x)), max(νA(x), μB(x))>|x ∈ E}
—09′ {<x, min(μA(x), sg(μB(x))), max(νA(x), μB(x))>|x ∈ E}
—10′ {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), 1 - νB(x))>|x ∈ E}
—11′ {<x, min(μA(x), sg(νB(x))), max(νA(x), sg(νB(x)))>|x ∈ E}
—12′ {<x, min(μA(x), νB(x).(μB(x) + νB(x))), max(νA(x), μB(x).(νB(x)2 + μB(x) + μB(x).νB(x)))>|x ∈ E}
—13′ {<x, min(μA(x), sg(1 - μB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E}
—14′ {<x, min(μA(x), sg(νB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E}
—15′ {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E}
—16′ {<x, min(μA(x), sg(μB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E}
—17′ {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(νB(x)))>|x ∈ E}
—18′ {<x, min(μA(x), νB(x), sg(μB(x))), max(νA(x), min(μB(x), sg(νB(x))))>|x ∈ E}
—19′ {<x, min(μA(x), νB(x), sg(μB(x))), νA(x)>|x ∈ E}
—20′ {<x, min(μA(x), νB(x)), νA(x)>|x ∈ E}
—21′ {<x, min(μA(x), 1 - μB(x), sg(μB(x))), max(νA(x), min(μB(x), sg(1 - μB(x))))>|x ∈ E}
—22′ {<x, min(μA(x), 1 - μB(x), sg(μB(x))), νA(x)>|x ∈ E}
—23′ {<x, min(μA(x), 1 - μB(x)), νA(x)>|x ∈ E}
—24′ {<x, min(μA(x), νB(x), sg(1 - νB(x))), max(νA(x), min(1 - νB(x), sg(νB(x))))>|x ∈ E}
—25′ {<x, min(μA(x), νB(x), sg(1 - νB(x))), νA(x)>|x ∈ E}
—26′ {<x, min(μA(x), νB(x)), max(νA(x), μB(x).νB(x) + sg(1 - μB(x)))>|x ∈ E}
—27′ {<x, min(μA(x), 1 - μB(x)), max(νA(x), μB(x).(1 - μB(x)) + sg(1 - μB(x)))>|x ∈ E}
—28′ {<x, min(μA(x), νB(x)), max(νA(x), (1 - νB(x)).νB(x) + sg(νB(x)))>|x ∈ E}
—29′ {<x, min(μA(x), max(0, μB(x).νB(x) + sg(1 - νB(x)))), max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))>|x ∈ E}
—30′ {<x, min(μA(x), μB(x).νB(x), max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))>|x ∈ E}
—31′ {<x, min(μA(x), (1 - μB(x)).μB(x) + sg(μB(x))), max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(x))) + sg(1 - μB(x)))>|x ∈ E}
—32′ {<x, min(μA(x), (1 - μB(x)).μB(x), max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(x))) + sg(1 - μB(x)))>|x ∈ E}
—33′ {<x, min(μA(x), νB(x).(1 - νB(x)) + sg(1 - νB(x))), max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sg(νB(x)))>|x ∈ E}
—34′ {<x, min(μA(x), νB(x).(1 - νB(x))), max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sg(νB(x)))>|x ∈ E}

Alternative separated view

No. Ref. Year Subtraction:

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

No. Ref. Year Subtraction MEMBERSHIP expression
 Subtraction NON-MEMBERSHIP expression
—01′ min(μA(x), νB(x)) max(νA(x), μB(x))
—02′ min(μA(x), sg(μB(x))) max(νA(x), sg(μB(x)))
—03′ min(μA(x), νB(x)) max(νA(x), μB(x).νB(x) + μB(x)2)
—04′ min(μA(x), νB(x)) max(νA(x), 1 - νB(x))
—05′ min(μA(x), sg(1 - νB(x))) max(νA(x), sg(1 - νB(x)))
—06′ min(μA(x), sg(1 - νB(x))) max(νA(x), sg(μB(x)))
—07′ min(μA(x), sg(1 - νB(x))) max(νA(x), μB(x))
—08′ min(μA(x), 1 - μB(x)) max(νA(x), μB(x))
—09′ min(μA(x), sg(μB(x))) max(νA(x), μB(x))
—10′ min(μA(x), sg(1 - νB(x))) max(νA(x), 1 - νB(x))
—11′ min(μA(x), sg(νB(x))) max(νA(x), sg(νB(x)))
—12′ min(μA(x), νB(x).(μB(x) + νB(x))) max(νA(x), μB(x).(νB(x)2 + μB(x) + μB(x).νB(x)))
—13′ min(μA(x), sg(1 - μB(x))) max(νA(x), sg(1 - μB(x)))
—14′ min(μA(x), sg(νB(x))) max(νA(x), sg(1 - μB(x)))
—15′ min(μA(x), sg(1 - νB(x))) max(νA(x), sg(1 - μB(x)))
—16′ min(μA(x), sg(μB(x))) max(νA(x), sg(1 - μB(x)))
—17′ min(μA(x), sg(1 - νB(x))) max(νA(x), sg(νB(x)))
—18′ min(μA(x), νB(x), sg(μB(x))) max(νA(x), min(μB(x), sg(νB(x))))
—19′ min(μA(x), νB(x), sg(μB(x))) νA(x)
—20′ min(μA(x), νB(x)) νA(x)
—21′ min(μA(x), 1 - μB(x), sg(μB(x))) max(νA(x), min(μB(x), sg(1 - μB(x))))
—22′ min(μA(x), 1 - μB(x), sg(μB(x))) νA(x)
—23′ min(μA(x), 1 - μB(x)) νA(x)
—24′ min(μA(x), νB(x), sg(1 - νB(x))) max(νA(x), min(1 - νB(x), sg(νB(x))))
—25′ min(μA(x), νB(x), sg(1 - νB(x))) νA(x)
—26′ min(μA(x), νB(x)) max(νA(x), μB(x).νB(x) + sg(1 - μB(x)))
—27′ min(μA(x), 1 - μB(x)) max(νA(x), μB(x).(1 - μB(x)) + sg(1 - μB(x)))
—28′ min(μA(x), νB(x)) max(νA(x), (1 - νB(x)).νB(x) + sg(νB(x)))
—29′ min(μA(x), max(0, μB(x).νB(x) + sg(1 - νB(x)))) max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))
—30′ min(μA(x), μB(x).νB(x) max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))
—31′ min(μA(x), (1 - μB(x)).μB(x) + sg(μB(x))) max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(x))) + sg(1 - μB(x)))
—32′ min(μA(x), (1 - μB(x)).μB(x) max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(x))) + sg(1 - μB(x)))
—33′ min(μA(x), νB(x).(1 - νB(x)) + sg(1 - νB(x))) max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sg(νB(x)))
—34′ min(μA(x), νB(x).(1 - νB(x))) max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sg(νB(x)))


List of intuitionistic fuzzy subtractions of —i′′ type

Alternative separated view

Approaches to defining intuitionistic fuzzy subtractions

References

"What Links Here" References

See also

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