jeiks.netjeiks.net/wp-content/uploads/2013/10/apostila_logica_fuzzy.pdf · p ! []$9t5x $98...

75

Upload: lytuong

Post on 28-Dec-2018

220 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

������������ �����������

�������! #"�$&%#$' #(*)��,+-��.0/21!�435.6�!�7 �8��9)�:; "��435.<�9=��,�> #��.@?>.<��.6�7 #1A��$CBD�!$E�F.G%5 H"�.<�,��$

IKJ �F.ML9)F"�N�$O��.0P#Q5Q#R

Page 2: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

���

Page 3: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

����� �����

�� ������������������ � �!" �#%$�&('*)�+-,�.�/ �103254 +�6�)�7"$98 2;:=<?> )�@�@3AB,�$�/ �10DCFEFG !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �!LKNM � 8FO?.�& �12 !H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! K �! P Q � 8ROS+3T 2 8 4 +36U)�7=$*8F$*8V+3TW#X&?$ZYZ)ZOS$98\[]$9T5+�& 0 �"2 � 8^!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! P

_a`cb9d%��ef��g�b*h5i5j�h�elk�m�h nKZ!" �oqp*O?&S$ZYZ)Ur3s�$t!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! uKZ!LKNvw+�x�&?+y8F+�p*O 2 r�sz${YZ+JoqpZ|}$�&ST 2 r�sz$~!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! uKZ! P []$�pz�R)�p*O?$98\[]71��8S8 �"0 $98�!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �

KZ! P�!" ��x�+3& 2 r3�*+y8��V�98 �1032 8V+3T�[]$9p��R)�p*O?$*8�[]7"�98?8 �10 $98�!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �KZ! P�!LK #�&S$�xU& � +�Y 2 YZ+�8�YZ$*8\[]$�p��R)Up9OS$98\[]71��8S8 �10 $*8�!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �

KZ! � []$�pz�R)�p*O?$98 4 +36�)U7=$*8F$*8�!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �KZ! �U!" > )�pUr���+y8wYZ+Hoqp 0 7")U8Ssz$�!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �KKZ! �U!LK ��+3&ST � p�$97=$9/ �12 Y�+-[]$9p��R)�p*O?$*8 4 +�6�)�7"$98?$98~!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! y�

KZ!Lu ��x�+3& 2 r���+y8 0 $�T�[]$9p��R)�p*O?$*8 4 +�6�)�7"$98?$98�!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! y�KZ!LuZ!" #�&S$�xU& � +�Y 2 YZ+�8�YZ+�[]$9p��R)�p*O?$*8 4 +�6�)�7"$98?$98~!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK��KZ!LuZ!LK []$9T5x 2 & 2 pUYZ$���x�+3& 2 r���+y8w[]71��8S8 �"0�2 8 0 $�T 4 +�6�)�7"$98 2 8�!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK9K

KZ! ���V��p�$�&ST�8�+HQ���p�$9&?T�8 !H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK9uKZ!�����x�+3& 2 r���+y8w[]$9T{x�+�pU8 2 O?.9& �12 8V+ 4 s�$�  2 YZ+�¡¢!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK��

KZ!���!" ��x�+3& 2 r3�*+y8�£(7"/�¤36U& �1032 8¥!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK*�K�!L�Z!= 9!= ��x�+3& 2 YZ$9&�#�&?$ZYZ)�O?$¦!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK��K�!L�Z!= 9! K ��x�+3& 2 YZ$9&?+y8�§¨¤�Y �12 !J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK��K�!L�Z!= 9! P Q�$9T 2 , � T � O 2 Y 2 !J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP9�

KZ!���!LK ��x�+3& 2 YZ$9&?+y8]� 2 8F+ 2 Y 2 8]+�T > )UpUr���+�8\!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aPU K�!L�Z! K�!= ��x�+3& 2 YZ$9&?+y8([]$�T{x�+�pU8 2 O?.9& � $98VYZ+J© 2 /�+3&ª!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP*K

KZ! ��«(T 2 oqp*O?+�&?xU&?+3O 2 r�sz${¬I+3$�T{¤3O?& �"0�2 Y�+-[]$9p��R)�p*O?$*8 4 +�6�)�7"$98?$98­!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP*u®N¯Bb�°lm���g±b�h�²�°fj�hyh�elk�m�h{b¨¯³b*°lm���g�b*h{´¨b*µ%��°l��hym�h ¶��P�!" �oqp*O?&S$ZYZ)Ur3s�$t!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�� P�!LKa#�&S$ZYZ)ZO?$³[ 2 &FOS+�8 �"2 p�${!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�� P�! P�vw+�7 2 r���+�8\[]71��8S8 �1032 8¸!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·��K

P�! P�!" [ 2 &SY � p 2 7 � Y 2 Y�+IYZ+-v(+37 2 r���+y8w[]71��8S8 �"0�2 8¹!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�*PP�! P�!LK #�&S$�xU& � +�Y 2 YZ+�8�YZ+-v(+37 2 r���+y8V� � pU�z& �12 8([]7"�98?8 �1032 85!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�9�

PU! PU! K�!= vw+�7 2 r���+�8wY�+HºX'*) �=»�2 7=¼�p 0 �12 +J��$�7"+3&D½zp 0 �12 !H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·��uP�! P�! P ��x�+3& 2 r3�*+y8 0 $�T¾vw+37 2 r3�*+y8\[]7"�98?8 �1032 8~!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�*�P�! P�! � []$9T5x�$98 � r3sz${YZ+-vw+�7 2 r3s�$�[]71��8S8 �1032 8;!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�*�

PU! PU! ��!= []$9T5x�$98 � r3sz${§ 2z¿ �q§ � pÀ!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�*�PU! PU! ��! K []$9T5x�$98 � r3sz${§Á� ¿ � T{$z��#�&S$�Y�)ZO?$ !J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�*�PU! PU! ��! P []$9T5x�$98 � r3sz${§Á� ¿ � T{$z�q§¨¤�Y �"2 !I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu��

P�! P�!Lu vw+�7 2 r���+�8 4 +36�)U7=$*8 2 8�!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu��P�! P�! � #�&S$�xU& � +�Y 2 YZ+�8�YZ+-v(+37 2 r���+y8 4 +36�)U7=$*8 2 8Â!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu� P�! P�!�� ��x�+3& 2 r3�*+y8 0 $�T¾vw+37 2 r3�*+y8 4 +�6�)�7"$98 2 8�!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu9K

� ���

Page 4: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�"» �����������

P�! P�! � []$9T5x�$98 � r3sz${YZ+-vw+�7 2 r���+�8 4 +36�)�7"$98 2 8�!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu�PPU! PU! �U!= []$9T5x�$98 � r3sz${§ 2z¿ �q§ � pÀ!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu�PPU! PU! �U! K []$9T5x�$98 � r3sz${§Á� ¿ � T{$z��#�&S$�Y�)ZO?$ !J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu9uPU! PU! �U! P []$9T5x�$98 � r3sz${§Á� ¿ � T{$z�q§¨¤�Y �"2 !I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu9u

¶� HmZ��efj���b�efh�´¨b*µ%��°l��hym�h n���U!" �� 2 & � � » + � 8 4 +�6�)�7"$98 2 8�, � p�/�)��"8FO �1032 8�!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu*�

�U!" �!" ��+3&ST � p�$97=$9/ �12 Y�+�� 2 & � � » + � 8!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu��n���efh���b�� m�h���bÁ 3����b*���*��k�elm ´¨b*µ%��°l��hy��h ���uZ!" �oqp*O?&S$ZYZ)Ur3s�$ !J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�U uZ!LK []$�p 0 + � O?$*8w����8 �"0 $98�YZ+tQ � 8ROS+3T 2 8 4 +36�)U7=$*8F$*8w!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�U

� ��efh���b�� m�h! #"}µ���el����h5 �����b�°fe%$�b����yb*h ��n��!" ·[ 2 & 2�0 O?+�&&�18RO �"0�2 8VYZ+tQ � 8ROS+3T 2 8�oqp*OS+37 � /9+3p*O?+y8�!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�*u��!" �!" £w'*) � 8 � r�sz$�YZ+�[]$9p�¡�+ 0 � T{+�p9OS$98 !H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�*u

��!" �!LK > & 2 / � 7 � Y 2 Y�+ !J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�9���!" �! P � 2 �=¿ $�+-£\7 OS$ 4 � » +37±YZ+Hv 290 � $ 0 �"p � $ !J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�9���!" �! � [ 2 x 2�0 � Y 2 YZ+JYZ+Hº ¿ x�7 �"0�2�0 s�$ !H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�9�

Page 5: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� ������� �� ���¸�^��� �

KZ!" �«(T 2 x�$98S8 � » +�7±YZ+���p � r�sz${YZ$ 0 $9p��R)�p*O?$ 0 7"�98?8 �"0 $5Y 2 8�x�+�8S8F$ 2 8 2 7=O 2 83!�!H!H!I!H!J!H!J!H!J!H!H! �KZ!LK���+��Up � r3s�$�p�+36U)�7=$*8 2 YZ$ 0 $�p��R)�p*OS$5pU+36�)�7"$98?${Y 2 8�x�+�8S8F$ 2 8 2 7 O 2 8�!~!J!H!H!I!H!J!H!J!H!J!H!H! �KZ! P���+3T{x�+�& 2 O?)�& 2 � p 0 7") �"Y 2 +�T¾YZ$ � 8 0 $�p��R)Up9OS$98wY � |})U8?$98 2 $5T{+y8FT{$5O?+�T5x�$U! !H!J!H!J!H!J!H!H! y�KZ! � []$�pz�R)�p*O?$ 4 +�6�)�7"$98?${x 2 & 2 º]8RO 2 O?)U& 2 §¨¤yY �"2 !Â!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �KKZ!Lu :f2*E�> )�p�r3sz${)�p � T{$ZY 2 7�+ : 6 Ew> )�pUr3s�${6 � ��T{$ZY 2 7³!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �KKZ! � > )UpUr3s�$�Q � p�/9)�7 2 &y! !H!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! yPKZ!�� > )UpUr3s�$ 0 7"�98?8 �1032 !�!J!H!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! yPKZ! � > )UpUr3s�$57 � p�+ 2 & 0 &S+�8 0 +3p*OS+�! !I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! ��KZ! � > )UpUr3s�$�OS& �12 pU/�)�7 2 &y!¦!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! ��KZ!" �� > )UpUr3s�$�OS& 2 x�+�@3$ � Y 2 7l!�!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �uKZ!" � > )UpUr3s�${8 � /�T{. � YZ+ 0 &S+�8 0 +�p9OS+�!~!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! �uKZ!" yK []$�pz�R)�p*O?$�p�+�6�)�7"$98?$�)U8?) 2 7=T{+3p*OS+�!Á!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! y�KZ!" �P > )UpUr3s�${�V+3O 2 ! !H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! y�KZ!" 3� > )UpUr3s�$5x�+3&?O?$�Y�+Hu�! !J!H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! ��KZ!" yu Q�)�x�$�&?O?+ 0 $�T{x 290 O?$5+HpUs�$ 0 $�T{x 290 O?$�!-!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! ��KZ!" �� > )UpUr3s�$ 0 $�T�8F)�x�$�&?O?+ 0 $�T{x 2�0 O?$5+ 0 $9&FOS+ 2 7 | 2 !­!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! y�KZ!" �� > )UpUr3s�$ 0 $�T�8F)�x�$�&?O?+JpUs�$ 0 $�T{x 290 OS$�+ 0 $�&?O?+ 2 7=| 2 ! !J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H! y�KZ!" ���� � 8FOS½�p 0 �12 +3p*O?&S+J)�TWx�$�p*OS$5+ 2 YZ+��Up � r3sz$ � YZ+ 2 7�YZ$ 0 $9p��R)�p*O?$�!À!H!J!H!H!I!H!J!H!J!H!J!H!H! y�KZ!" ���º ¿ +3T{x�7"$98\YZ+H$�x�+3& 2 r3�*+y8X+�p*O?&S+ 0 $�p��R)Up9OS$98�!¦!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK� KZ!LKz� []$�pz�R)�p*O?$98wY�+ 2 YZ)�7=O?$*8�+HpUsz$ 2 Y�)�7 OS$98�! !H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK9KKZ!LKZ []$�pz�R)�p*O?$98 2 7 OS${+HT5+ �"2 � Y 2 Y�+Hp � O � Y�$98�! !H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK9KKZ!LK�K���+��Up � r3s�$�YZ$ 0 $9p��R)�p*O?$ : pU+36�)�7"$98?$ E Y 2 8�x�+y8?8?$ 2 8 2 7=O 2 83!w!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NKz�KZ!LKzP []$�pz�R)�p*O?$�YZ+Jx�+�8S8F$ 2 8wYZ+JT{+ �12 � Y 2 YZ+9!U!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NKz�KZ!LK�� ��x�+3& 2 YZ$9&�£ 4 � YZ+t  2 YZ+�¡±! !H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK��KZ!LK�u ��x�+3& 2 YZ$9&(�Iv YZ+�  2 YZ+3¡�!¨!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!NK��KZ!LKz� ��x�+3& 2 YZ$9&�£ 4 � 0 $�TWx�&?$ZYZ)�O?$U!;!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP9�KZ!LK9� ��x�+3& 2 YZ$9&(�Ivx�&S$ZYZ)ZO?$�!%!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aPU KZ!LKz� ��x�+3& 2 YZ$9&�£ 4 � 0 $�TWT5¤yY �12 ! !J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP*KKZ!LKz� ��x�+3& 2 YZ$9&(�Iv 0 $9TWT{¤�Y �"2 ! !H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP*KKZ! P�� ��x�+3& 2 YZ$9&�£ 4 � 0 $�T¾8F$9T 2 7 � T � O 2 Y 2 !Z!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP9PKZ! P� ��x�+3& 2 YZ$9&(�Iv 0 $9T¾8?$�T 2 7 � T � O 2 Y 2 !�!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP��KZ! P9K ¬I&D��� 0 $98VT{$98FO?& 2 pUYZ$ 2 � p��U)�¼�p 0 �12 YZ$�x 2 &D½zT{+3O?&S$

kp 2 |})�pUr�sz$�YZ+Hp�+�/ 2 r�sz$�Y�+J© 2 /�+�&�!�!aP��

KZ! P�P []$�pz�R)�p*O?$98wv(+3x�&S+�8?+3p*O 2 YZ$98 0 $�T{${x�$9p9OS$98�!Z!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP9�KZ! Pz� #�& � p 0 �"x � $*8(YZ+H£(& � 8ROS.zO?+�7=+y8�p�${ºX8?x 2 r3$U!Â!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP9�KZ! P9u [ 2 &DY � p 2 7 � Y 2 YZ+ 0 $�T{$!� +3O?$9&�!�!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!aP9�P�!" ¬I& 2 |}$�Y 2 &?+�7 2 r3s�$5PU! P !H!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�*PP�!LK�¬I&D��� 0 ${Y 2 + ¿ x�&?+y8?8Ssz$

y = x/2!H!J!H!I!H!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!I!H!H!a���

P�! P ¬I& 2 |}$98 � 7")U8ROS& 2 pUYZ$ 2 8�x�&S$�xU& � +�Y 2 YZ+�8�YZ+H&S+���+ ¿ �=» � Y 2 Y�+��Z8 � T{+�OS& �12 +JO?& 2 pU8 � O �=» � Y 2 Y�+�! !J!H!H!·��uP�! � ¬I& 2 |}$�YZ+H)�T 2 &?+�7 2 r3s�$5YZ+J+y'9) �=»�2 7"¼3p 0 �12 ! !H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�*�P�!Lu�¬I& 2 |}$�YZ+H)�T 2 &?+�7 2 r3s�$5YZ+IOS$�7"+3&D½zp 0 �"2 !J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�Z�P�! � []$�T{x�$98 � r�sz${YZ+-vw+�7 2 r���+�8 !I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!·�*�

»

Page 6: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

» � � ��������� �� ��� ���

P�!��avw+�7 2 r���+�8R+

S!]!J!H!I!H!H!J!H!J!H!J!H!J!H!H!I!H!J!H!J!H!H!J!H!J!H!I!H!H!J!H!J!H!J!H!J!H!H!·�*�P�! � � �"2 /�& 2 T 2 Y 2 &S+37 2 r�sz$

R ◦ S!J!H!J!H!J!H!H!I!H!J!H!J!H!H!J!H!J!H!I!H!H!J!H!J!H!J!H!J!H!H!Nu��

P�! ��vw+�7 2 r���+�8R+

S!]!J!H!I!H!H!J!H!J!H!J!H!J!H!H!I!H!J!H!J!H!H!J!H!J!H!I!H!H!J!H!J!H!J!H!J!H!H!Nuz�

P�!" ���� �"2 /�& 2 T 2 Y 2 &S+37 2 r�sz$T!H!I!H!J!H!J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu9u

�U!" �� 2 & � � » +37�p�+�6�)�7"$98 2 O?+3T{x�+3& 2 OS)�& 2 ! !J!H!J!H!H!J!H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu*��U!LK � 2 & � � » +37�p�+�6�)�7"$98 2�0 $�&S&?+y8Fx�$�p�YZ+3p*O?+ 2�0 7 2 8?8 � � 032 r3sz$5YZ+J)�T 2 7=)�pU$5+�TÀ|})�pUr�sz${YZ+H8F) 2 pU$zO 2 !Ju���U! P�� 2 & � � » +37�p�+�6�)�7"$98 2{0 $�T{x�7"+�O 2 +Ip�sz$ 0 $9T5xU7=+3O 2 ! !I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu���U! � Q�$96�&?+�x�$*8 � r3s�$5Y�+Hu����ª+�T 0 $�p��R)�p*OS$98�p�+�6�)�7"$98?$98�!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu��uZ!" ¬I&D��� 0 $98VT{$98FO?& 2 pUYZ$ 0 $�T{$ 2 )�T{+3p*O 2 & 2 x�&?+ 0 � 8?s�$5Y 252 x�&S$ ¿ � T 2 r�sz$-p�+�6�)�7"$98 2 ! !J!H!J!H!H!a�*KuZ!LK�� �"2 /�& 2 T 2 +�TW6�7"$ 0 $*8wYZ+J)�T¾8 � 8FO?+�T 2 pU+36�)�7"$98?$U!¨!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�*KuZ! P []$�pz�R)�p*O?$98�pU+36�)�7"$98?$98w'*)�+HYZ+��Up�+�T 2 O?+�T5x�+3& 2 OS)�& 2 ! !H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�9PuZ! � []$�pz�R)�p*O?$98�pU+36�)�7"$98?$98w'*)�+HYZ+��Up�+�T 25» +�7=$ 0 � Y 2 YZ+�! !J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!a�9P

Page 7: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� ��������� � �

��� � � ��������

����� ����� �"!$#&%('�)�*,+.-0/1#324!45,��67-98�:<;=!4>?>?@A%B�C)�*,+EDGF<H4 $\Tt)�pUY�$\&S+ 2 7�)U8 2 T{$98 x 2 7 2�» & 2 8 0 $�T{$JI T�) � OS$ ��x�$�) 0 $ �y/�& 2 pUY�+��yx�+y'9)U+3p�$ ��|}&S+�'LK�+�p*O?+3T{+�p9OS+���& 2 & 2 T{+3p*O?+ �+�O 0 ! x 2 & 2 YZ+�8 0 &S+ » +3&{8 � O?) 2 r���+y83!�ºX8FO 2 858 � O?) 2 r���+�8�pUsz$�8Ssz$ p � O � Y 2 T{+3p*O?+;YZ+���p � Y 2 85+³p�sz$ x�$ZYZ+3T�8F+�&x�&S+ 0 � 8 2 T{+3p*OS+JY�+�8 0 & � O 2 8�!X#%$�&(+ ¿ +3T{x�7"$ � 2 8�|}& 2 8?+�8 2 6 2 �=¿ $�8Ssz$�YZ+t)U8F$ 0 $�&S&?+�p*O?+J+3T�p�$98S8 2 7 � p�/�) 2 /�+�T �U+2 x�+�8 2 &wYZ+JOS& 2 pU8?T � O � &S+3T � pZ|}$�&ST 2 r3�*+y8 � T{x�&S+ 0 � 8 2 8V+37 2 8w8Ssz${)U8 2 Y 2 8V+3TWO?$�T 2 Y 2 8�YZ+-YZ+ 0 � 8F��+y8MI

•� 032 &?&S$t+y8ROD� 2 pUY 2 pUYZ${Tt) � OS$�&S��x � Y�$ ��x � 8F+Hp�$5|}&S+ � $U!

•º�7=+H¤J)�T 2 x�+�8S8F$ 2 T�) � O?${|}+37 � @�!

•ºX8RO 2 8 2 7 2 ¤Jx�+y'*)�+3p 2 x 2 & 2 OS$�Y�$98�$98 2 7=)Up�$98 �Z&?+y8F+�& » +\$9)ZO?& 2 T 2 � $9&�!

•4 +�8FO 250 � Y 2 YZ+ 2 O?+�T{x�+�& 2 O?)�& 2 |}&S+�'LK�+�p*O?+3T{+�p9OS+�+y8ROD�tT�) � O?$ 2 7=O 2 ��x�$�&�+�8FO 2 & 2 @�s�$t/ 2 8FO 2 T{$*8�Tt) � O 2+3p�+�&?/ �"2 p�$98 0 $�pUY �"0 � $9p 2 YZ$9&?+y8]Y�+ 2 &y!

£ YZ+�8 0 & � r�sz$ 0 $�T{x�7"+�O 2 YZ+I)�T¢8 � 8FO?+�T 2 &S+ 2 7�+3TÀT�) � OS$98 032 8F$*8�&S+�'*)�+�&]Y 2 YZ$98]+ ¿ O?&S+3T 2 T5+�p*O?+�Y�+�O 2 7=¡ 2 YZ$*8+ITt) � O?$ 2 7"¤3T¢YZ${'*)�+I)�T¢8F+�&]¡�)�T 2 p�$�x�$ZYZ+3& �12 8 � T�)�7 O 2 p�+ 2 T{+3p*OS+���x�&S$ 0 +�8S8 2 &X+I+3p*O?+�pUYZ+�&�! 4 $5+3p*O 2 p*O?$ �|}&S+�'LK�+3p*OS+3T{+3p*O?+B+ ¿ + 0 )ZO 2 T5$*8HO 2 &S+�| 2 8 0 $�T{x�7"+ ¿�2 8-'*)�+Bx 2 & 2 8?+3&S+3T­YZ+y8 0 & � O 2 8Hx�&S+ 0 � 8 2 T­YZ+�8FO?+y8HO?+�&?T{$*8� T5xU&?+ 0 � 8F$*8]'*)�+JpUs�$�x�$ZYZ+H8F+�&VT{$ZYZ+�7 2 YZ$98Vx�+37 2 T 2 OS+3T���O �"0�2 OS& 2 Y �10 � $�p 2 7�Y�+ 0 $�p��R)Up9OS$98�!%#�$ZYZ+�T{$98 0 � O 2 &+ ¿ +�T5xU7=$*8w8 � T{x�7=+y8���'*)�+H+�p 0 $9p*O?& 2 T{$98]p�$�p�$*8?8?$5Y �"2{2 Y �12 I

•#�&?+�x 2 & 2 r�sz$5YZ+Jx�& 2 O?$*8 2 x 2 &?O � &�Y�+H&?+ 0 + � O 2 83!

•ºX8RO 2�0 � $9p 2 &�)�T 032 &S&S$U!

•[]$�p*O?&S$�7 2 &V)�T 2{032 &?O?+ � & 2 YZ+ � p » +�8FO � T5+�p*O?$98�!

4 +y8RO 2 8 2 O �"» � Y 2 YZ+�8V+�8?x�+ 0 �12 7 � 8RO 2 8�)U8 2 & �"2 T |}& 2 8F+y8 0 $�T{$NI•[]$�7"$�'*)�+Ip�${|}$9&?p�$ 2 OS¤ � 032 &�p�$�x�$9p*O?$U!

•� � &?+J$ » $97 2 p9OS+�)UTWx�$�) 0 $5x 2 & 2PO Y � &S+ � O 2 !

•Q�+ 2 O 2�¿�2 YZ+��R)�&S$98w8?)�6 � &wTt) � O?${+H$�YZ¤�� 0 � Ow|}$�& 2 7 OS$5x�$ZYZ+3&S+3T{$*8VO?+3&w)UT 2 &S+ 0 +�8S8?s�$t6�& 2 p�Y 2 !

£ ,±.�/ �"0�2(4 +�6�)�7"$98 2 x�&S$ » $9)-8F+�& 032 x 2 @]Y�+ 2 x�&?$ ¿ � T 2 & 0 $�T{x�7"+ ¿ $98�8 � 8FO?+3T 2 8�pUs�$\7 � pU+ 2 &S+�8 � 0 $9T¦x�$9) 032 8&S+3/�& 2 8VYZ+y8ROS+JO � x�$�!�§¨$ZYZ+37"$98 4 +36�)�7"$98?$98VOS+3T 2 8�8F+�/�) � p*O?+y8 032 & 2�0 O?+�&&�18RO �"0�2 8�I

•«wO � 7 � @ 2 T�&S+3/9& 2 8{'9)U+ 0 $�p�8F+�/�)�+�T~+ ¿ x�&S+�8S8 2 & 2 8 � T5xU&?+ 0 � 8F��+�8{+ 2 x�&S$ ¿ � T 2 r3�*+y8{YZ$98{T{¤�OS$ZYZ$98�Y�+YZ+ 0 � 8?��+�8�YZ$98�+y8Fx�+ 0 �"2 7 � 8FO 2 8�!

•QZsz$5T 2 � 8V|f� 0 + � 8\YZ+ 0 $�p�8ROS&?) � & �Z+3p*O?+�pUYZ+3& �ZT 2 p*O?+�& ��OS+�8FO 2 &�!

Page 8: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

K � ����� ��� � ����� ��� � � ���� �

•#�$ZYZ+�TW8?+3&wx�&S$zOS$zO � x 2 YZ$98V+3T¾T{+3p�$*8VO?+3T{x�$U!

•QZsz$5T 2 � 8�&?$96�)U8FO?$98�+ 0 $�pU8?+3/9)�+3TÀOS& 2 6 2 7"¡ 2 & 0 $�T¢| 2 7 O 2 YZ+H&?+�/�& 2 8�!

•#�$ZYZ+�TÀO?& 2 6 2 7"¡ 2 & 0 $9T � pZ|}$�&ST 2 r���+y8 � T{x�&S+ 0 � 8 2 8�!

•#�$ZYZ+�T 0 ¡�+�/ 2 & 2{0 $9p 0 7")U8?��+�8wYZ+JT 2 p�+ � & 2 x 2 & 2 7"+37 2 !

•£ 0 )�T�)�7 2 T¢+ » � YZ¼3p 0 �12 8 0 $9p*O?& 2 + 2 | 2�» $9&�Y�+Hx�&?$9x�$*8 � r���+y83!

£(7"/�)UT 2 8wYZ+�8 »�2 p*O 2 /�+�pU8]Y�+tQ � 8FO?+�T 2 8w'*)�+H)ZO � 7 � @ 2 T¾7=.9/ �1032 p�+36�)U7=$*8 2 8?s�$JI• � T 2 � 8wY � | � 0 � 7�YZ+�8?+3p » $�7 » +3&V)�T¾T5$ZYZ+�7=$ 2 x 2 &FO � &(YZ+H)UT¾8 � 8FO?+�T 2 p�+�6�)�7"$98?$��•º�Tt6�$�& 2 8?+R� 2 T�T 2 � 8(|f� 0 + � 8�YZ+ 0 $�pU8FO?&S) � &\+-x�&S$zOS$zO � x 2 &�'*)�+t8 � 8ROS+3T 2 8 0 $�p » +3p 0 � $9p 2 � 8��U+37"+�8�p�+ 0 +�8R�8 � O 2 TÂ'*)�+{8?+R� 2 T + ¿ + 0 )�O 2 Y 2 8�T 2 � 8I8 � T�)�7 2 r���+�8�+5p�+ 0 +�8S8 � O 2 T YZ+�T 2 � 8I8 � p*O?$9p �"2³2 p*O?+y8JYZ+58?+3&S+3TYZ+��Up � O �=»�2 T{+�p9OS+ 2 x�&S$ »�2 YZ$*8��

•4 sz$�O?+�TW)�T 2 YZ+��Up � r3s�$5T 2 O?+3T��zO �1032 x�&S+ 0 � 8 2 +Hp �=O � Y 2{0 $9T{$5$*8w8 � 8FO?+3T 2 8VO?& 2 Y �10 � $�p 2 � 8�!

����� � *M6��7'�� *,-� +�8FO?)UYZ$�Y 2 p�+36U)�7=$*8 � Y 2 YZ+ :������������! #"�"RE±0 $�T{+yr�$�)Jp�$ �Up 2 79YZ$\8?¤ 0 )U7=$%$�o&$ :('*)%+�'*) �� ��9�*� E !�«(T 0 $9p 0 + � OS$¤\pU+36�)�7"$98?$ �*$�) »�2 /�$ �*'*) 2 pUYZ$-+�7=+I¤ � pUYZ+3O?+3&ST � p 2 YZ$ � � pUYZ+��Up � YZ$ ��$�)³8F+F� 2 $98V8?+3)U8X7 � T � OS+�8�8?s�$-p�+�6�)�7"$98?$98 �x�$�) 0 ${Y�+��Up � YZ$*83!º�T�6�$9& 2-0 $�p��R)Up9OS$98 0 71��8S8 �10 $*8�8F+F� 2 T 2 6 2 8F+IYZ+(OS$ZY 2 O?+�$�& �"2 T 2 O?+�T{�zO �1032 T{$ZYZ+�&?p 2 �9+�7=+y8 2 x�&S+�8?+3p*O 2 T

Y � � 0 )U7"Y 2 YZ+�8V'*) 2 p�YZ$ 2 xU7 �1032 YZ$98 2 )�T 2 +�p�$�&ST{+ 0 7 2 8?8?+\YZ+�x�&?$96�7"+3T 2 8]YZ$tT�)�pUYZ$�&?+ 2 7 �9+�p9OS&?+\+�8FO?+�8]+�8FOSs�$$98tx�&S$�6�7"+3T 2 8-p�+36�)U7=$*8F$*83!Á£W)ZO � 7 � Y 2 YZ+BY 2 7=.9/ �1032 +BY 2 O?+�$�& �"2 YZ$*8 0 $�p��R)�p*OS$98-x 2 & 2¨2 T{$�Y�+37 2 /�+�T Y�+8 � 8FO?+�T 2 8�¤ � p�Y � 8?x�)ZO 2 Y 2 ! 4 $�+�p9O 2 p*O?$ ��¡U�\8 � 8FO?+�T 2 8 0 $�T{x�7"+ ¿ $*8���$�p�YZ+XpUs�$(¤Vx�$*8?8&� » +37 0 $�p�8F+�/�) � &�+y'9) 2 r���+y8'*)�+w$98XYZ+�8 0 &S+ »�2 T 0 $�T{x�7"+�O 2 T{+3p*OS+V$9) 032 8?$98%$9pUYZ+ 2 Y �10 $�O?$�T �12 � p*OS&&�"pU8?+ 032 Y 2 OS+3$9& �12 YZ$98 0 $�p��R)�p*OS$98 pUs�$0 $9pU8?+3/�)U+IYZ+y8 0 &?+ » +�& 2 8 � OS) 2 r�sz$�![]$�p�8 � YZ+3&S+¨$ x�&S$�6U7=+�T 2 YZ+ YZ+��Up � &³$ 7 � T � OS+ YZ+ )�T YZ+�OS+3&ST � p 2 YZ$ 0 $9p��R)�p*O?$�! £ Y�+��Up � r3s�$ YZ+y8ROS+

7 � T �12 &-x�$ZYZ+ 0 $9pUYZ)�@ � & 2 x 2 & 2 YZ$ ¿ $*8\+ 2 '*)�+�6�& 2 Y 2 7=+ � YZ+B£\& � 8FO?.zOS+37"+�8 0 $�p�¡U+ 0 � Y 2;0 $9T{$ 2 7=+ � Y 2 pUs�$0 $9p*O?& 2 Y � r3s�$U!�º]8RO 2 7"+ �±2 �U&?T 2 '9)U+ 2 � p*O?+�& 0 +�8S8?s�$�YZ+I)�T 0 $�p��R)�p*OS$ 0 $�T¢8?+3) 0 $�T{x�7"+3T{+3p*OS$�¤I$ 0 $�p��R)�p*OS$»�2 @ � $ :

A ∩A ≡ ∅E �Z$9)c8?+R� 2 )�T¢+�7=+�T{+3p*O?$5pUsz$5x�$�Y�+Hx�+�&FOS+3p 0 +3& 2 )UT 0 $9p��R)�p*O?${+ 2 ${8F+�) 0 $�T{x�7"+3T{+3p*O?$2 ${T{+�8?T{$5O?+3T{x�$U!

� x�&?$96�7"+3T 2 Y 2 YZ+��Up � r3sz$�Y�$�7 � T �"2 &(7"+ »�2�2 $-,�.�/0.�1�2435276�2�/98;: 9" � 2 OS& � 6�) �1YZ$ 2 $�Y �"2 7=¤3O �10 $ �Uº�)�6�)�7 � YZ+�8YZ+H§ � 7=+3O?$ ��'9)U+I+�& 2 )�T 2 Y » +�&S8S�z& � $tYZ+H£(& � 8FO?.�O?+�7=+y83!X� x 2 & 2 YZ$ ¿ $t8?+I+�p�)�p 0 �1250 $�TW$98�8F+�/�) � p*O?+y8XOS+3&ST{$98�I

<= � . � 1�2 �?>@> 2 � : 1 .A/ 8B.C1 8�35.D1 E"� / �?>@> 2 � : 1 .�/ 8B.GF%H0. " 27/ :I8;/ #> 2 "J�?>LK /0M�21 .�/ 8B.N1 H�.�1�.PO ��RQ4S

� +�8FO?)�YZ$cY 2 x�&S$�6U7=+�T 2 Y 2 � pUYZ+��Up � r3s�$³p 2 7=.9/ �1032 � p*O?+�&?+y8?8?$�)¨x�+3p�8 2 YZ$�&S+�8 � T{x�$�&?O 2 p9OS+�8 0 $9T5$c�V+3&?�O?& 2 p�Y³vw)�8?8?+37±'*)�+J|}+�@ 2 8w8?+3/�) � p*O?+y8 2 �U&ST 2 r���+y8MI

<0T 2U1�.PVW8 �XK�� . KX �>ZY O�. K .�[ S <]\ � 1 _^`� 8(aUM�2GF�H#Vb.�/c. >d �� : F YE�?> .De �� #" :fM�2N1 gK /c. � [ S <0T 241�.V]h K 8BH�.i:_/c.�1�8BH�8B2 � .�V�jR.�k#8;: � .�V >l �� : . "�"��?>l e �� *"9m;> k02AV]2 " ,!/ H�8 " 2 "g #" :&M�2 "� #� 1�2 #> ,!/ IK .�1�2 " [n 2�/9:&. � :f2�F � M�2 Y .0,oVW8BH�p�O VrqsO48B1�.E: /9/ 9" :I/ %> . "%" 2 >d �� : ,�.�/0. �?> .J8 > . K 8 � pA/98B. 3�8 " :ft � H�8B.H V #" :_8B.AV][ S

vw)U8S8?+37X+�p 0 $9p*O?&S$�)�)�T­$�)ZOS&?$ x 2 & 2 YZ$ ¿ $;p 2 OS+3$9& �12 YZ$*8 0 $9p��R)�p*O?$*8��%$Áx 2 & 2 YZ$ ¿ $¨YZ$ÁT{+3p*O � &S$98?$98-Y�+[]&S+�O 2 !BºX8FO?+Bx 2 & 2 YZ$ ¿ $ � O 2 Tt6�¤3T 0 $�pU¡�+ 0 � YZ$ÁY�+�8SYZ+ 2¨2 p*O � / K � Y 2 YZ+���x�$�Y�+B8F+�&-+3p�)�p 0 �12 YZ$¨Y 2 8?+3/9) � p9OS+T 2 p�+ � & 2 IV$�x�$�+�O 2B0 &S+�O?+�pU8?+tY � @�'*)�+HOS$�Y�$98($*8 0 &?+3O?+3p�8F+y8wT{+3p*O?+�Tc!\ºX8RO 2 &D��+37"+tY � @3+�pUYZ$ 2�» +�&SY 2 YZ+ G QZ++37"+BT{+�p9OS+�� 0 $�T{$¨O?$ZYZ$*8�� +�p*OSsz$ +37"+³Y � @ 2 » +3&DY 2 YZ+�! QZ+B+37"+³Y � @ 2Á» +�&SY 2 YZ+�+�p9ODsz$Á+�7=+³+�8FOS� T5+�p*O � pUYZ$U!��& 2 pU8R|}+�& � pUYZ$ +�8FO?+ x 2 & 2 YZ$ ¿ $ x 2 & 2 $98cY �"2 8 2 O?) 2 � 8³+ 2 O?+�$�& �"2 YZ$98 0 $9p��R)�p*O?$*8Bx�$ZYZ+3T{$98 � pUY 2 / 2 & I�$0 $9p��R)�p*O?$5YZ+�O?$ZYZ$98]$98 0 $9p��R)�p*O?$*8]'*)�+�pUsz$58?s�$-T{+3Tt6U&?$*8]YZ+I8 � T{+�8?T{$t¤�)�TÀT{+3Tt6U&?$tY�+I8 � T{+y8FT{$ G QZ++37"+-pUs�$�¤-+�p9ODsz$�+�7=+t¤�!�[ 2 8?$�+37"+t8?+R� 2 +�p*OSsz$B+37"+-pUsz$�¤9!\£w'*) � OS+3T{$98�'*)�+

A e nao A8Ssz$�x�$98S8&� » + � 8w+ 2

Page 9: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�����?� �� ��� � � ���D ��� � � �� �� � � � � � � � ��� � � � � � � � � P7"+ � Y 2 pUsz$ 0 $9p9OS& 2 Y � r3sz$J| 2 7=¡ 2 !%º�T y�9KzP�vw)U8S8F+�7�xU&?$9x��*8]'*)�+J+�8FO 2 7=+ � YZ+ » +3& �"2 8F+�&V&S+37 2�¿�2 Y 2 x 2 & 2 '9)U+�$*8x 2 & 2 YZ$ ¿ $98]+ 2 � pUYZ+��Up � r3sz$�Y 2 8 2 �U&ST 2 r3�*+y8Vx�)UYZ+y8?8?+3T�8F+�&]OS& 2 O 2 YZ$98 :��� +X+� � �� y�9KzP E !� T 2 O?+�T���O �10 $³x�$97=$9p�¼�8�� 2 p ,±)�� 2 8 � +�� �10 @t+3T­ ��*Kz��YZ+y8F+�p » $�7 » +�) 2 x�& � T{+ � & 2 7=.9/ �1032 T�)�7=O �"»�2 7=$9& 2 Y 2

$�) p�+36�)U7=$*8 2 !�� 2 x�7 2 p¨+�Q 0 ¡�$zO?OI+�$�)ZOS&?$*8\T 2 O?+3T��zO �10 $98\x�&S$�x�)U8?+3& 2 T ��p 2 YZ¤ 0�2 Y 2 Y�+{u�� � 2 8�$�x�+3& 2 r3�*+y8T 2�¿ +HT � p;x 2 & 2 Y�+��Up � & 2 �z7"/�+�6�& 2 YZ$*8 0 $�pz�R)�p*O?$98�pU+36�)�7"$98?$98�!��x�+�8S'*) � 8 2 Y�$�&{'9)U+ 0 )�p�¡�$9)�$ O?+�&?T{$�� > )�@3@�A ,±$9/ �10 ����'*)�+cOS& 2 YZ)�@ � &S+3T{$98�x�$9&�,±.9/ �1032 4 +36�)U7=$*8 2 �

|}$ � $�x�&S$z|}+y8?8?$�&�,±$zO?|}A¨  2 YZ+�¡ÁY 2 «\p �=» +3&D8 � Y 2 YZ+-Y 2 [ 2 7 � |}.�&Sp �12 '*)�+t+�T y���*utxU)�6�7 �"0 $�)Á$�x 2 x�+3&�� > )U@3@3AQ�+3OS8�� :������� � �� ��9�9u E !�ºX8ROS+ 2 &?O � /�$5)U8?$�) �����0��� x 2 & 2 8 � /�p � � 032 & OG. KA�� �ZO?+�&?T{$�'*)�+J+3& 2 +3T{x�&S+3/ 2 YZ$t)�8F)Z�2 7"T{+3p*O?+5p 2 7 � O?+�& 2 O?)�& 2 OS¤ 0 p �"0�2 ! �\+y8?Y�+�+�p9ODsz$ �����0��� OS+3T 8 � Y�$B$³O?+3&ST{$BT 2 � 8 0 $�T�)�T{+3p*O?+5+3T{x�&S+3/ 2 YZ$�!�¢+�8FO?)UYZ$ÁY 2 7=.9/ �1032 Tt)�7=O �"»�2 7"$�& 2 Y 2 ¤ 2 p9OS+3& � $9& 2 +�8FO 2 ¤3x�$ 032 ��T 2 8-$¨x�&?$�|}+�8S8F$9&t  2 YZ+�¡ ��xU& 2 O �1032 T{+�p9OS+8?$�@ � p�¡�$ �zOS&?$9) ¿ +(p�$ »�2 T{+3p*OS+(x 2 & 2 $-xU& � T{+ � &?$-xU7 2 p�$-$t+y8ROS)UYZ$tY�+�8FO 2 OS+3$9& �12 !%£ 7".�/ �"0�2 p�+36�)U7=$*8 2 +y8ROS+3pUY�+2 7".�/ �"0�250 $�p » +�p 0 � $�p 2 7 � p 0 7") � pUYZ$�$ 0 $�p 0 + � O?$�YZ+ » +3&DY 2 YZ+Hx 2 & 0 �12 7 � »�2 7"$�&S+�8�+�p*O?&S+HO?$�O 2 7"T{+3p*O?+ » +3&DY 2 YZ+J+O?$�O 2 7=T{+3p*OS+�| 2 7"8?$U!%º�T¹ �����P-+�7=+I|}$9&?T�)�7=$9)B$�'*)�+Jx 2 8S8?$�) 2 8?+3& 0 ¡ 2 T 2 YZ${YZ+Jx�& � p 0 �=x � $�Y 2 � p 0 $�T{x 2 O � 6 � �7 � Y 2 YZ+�!! ��ef��k "�"�el���$#f�&% >d 1�8B1�.de �� .-H02 > ,oV 3�8B1�.�1 1 �?>�" 8 " : #> .-. �?>l #� :f.�F � 2 "0" . jR.�k#8;VW8B1�.�1 ,�.�/c. � . �U /. ^ / > .�a(' #" ,!/ H�8 " . "N e �� d"� �) . > " 8 KA� 8 ^ H�.�:I8;O�. " .�H /cH�. 1 9" : l" 8 " : #> . 1A8 > 8 � � 8 .�: Y e �� l�?> VW8 > 8B.�/ Y.�:I8 �XK 8B1�2D.�V Y#> 1�2Ne � .AVX,!/ H�8 " M�2 s" 8 KA� 8 ^ H+* � H#8B.-,2 � / V O.* � H#8B.0/7:f2A/ � . >21f"� e � . "� e �� H�.�/c.�H#: / m " :I8BH�. ">-� : � . >d �� : s 3XH�V �5" 8;O�. " [� x�+y8?'*) � 8 2 YZ$9&�º�6�& 2 ¡ � T¦M-!�§ 2 T�Y 2 T � YZ$�3�)�+�+3pt§ 2 &SAH[]$97=7"+3/9+�+3T ,�$�pUYZ&S+�8 �yp�$ 2 p�$t y�*����� 2 x�7 �"0 $�)

2 7".�/ �1032 p�+�6�)�7"$98 2 x 2 & 2I0 & �12 &%)�T 8 � 8FO?+�T 2 6 2 8F+ 2 YZ$I+3T &?+�/�& 2 8�p�+�6�)�7"$98 2 8 0 $9T 2 �Up 2 7 � Y 2 YZ+wYZ+ 0 $9p9OS&?$97 2 &)�T 2 T��9'9) � p 2 Ow»�2 x�$�& :�45�647�8�898: �� y�*��� E ! ºX8FO?+�8 � 8FO?+�T 2 x�$ZYZ+V8F+�& 0 $9pU8 � YZ+�& 2 YZ$w$ � p � 0 � $\Y 2 +3p�/9+3p�¡ 2 & �12p�+�6�)�7"$98 2c:B'*)%+A'*) �� y���*� E !

���<; ="*M6��7#?> -�6 / #32!$5M��6G��6 #?> � � ��@$!P�7��6BA �C> #3� +.*,-�*,6£ 7".�/ �1032 p�+�6�)�7"$98 2 ¤�+�T5xU&?+�/ 2 Y 2 +3T Y �"» +3&D8?$98�x�&?$ZYZ)�O?$98 0 $�T{+3& 0 �12 � 8 �±YZ+ 2 8?x � & 2 Y�$�&S+�8IYZ+{x�. 2 O?¤ 2 7=O?$*8|}$�&Sp�$*8�+�T¹8 � Y�+3&ED�&?/ �"0�2 8 �±YZ+�8 � 8FO?+3T 2 8Jx 2 & 2³2 ) ¿ � 7 �12 &Hp 2 +�8 0 $97=¡ 2 YZ+{O 2�0 $98HYZ+{/�$97 |}+ 2 O?¤�8 � 8FO?+�T 2 8Ix 2 & 2p�+�/�$ 0 �12 & 2 r3��+�8V+3T¾6�$97"8 2 !º�T ������ � 2 x�& � T{+ � & 2;2 x�7 �1032 r3s�$ 0 $�T{+3& 0 �12 7�YZ+B7=.9/ �1032 p�+36U)�7=$*8 2 |}$ � YZ+y8F+�p » $�7 » � Y 2 x�+37 2 8H+�T5xU&?+y8 2 8

�V7=)U+¨[ � & 0 7=+Á[]+3T{+�p9O�+¨Q�oRv(£�p 2 � � p 2 T 2 & 0�2 :GF6 9IH �$��9�J6�8��: �] y����� E !��Â8 � 8ROS+3T 2 0 $�p*O?&S$�7 2�»�2 )�T|}$�&Sp�$ YZ+ 0 � T{+�p9OS$ � p 0 $9&?x�$�& 2 p�YZ$Á$ 0 $�pU¡�+ 0 � T{+3p*OS$ YZ+c$�x�+3& 2 YZ$9&?+y8H+ ¿ x�+3& � T{+�p9O 2 YZ$*8tx 2 & 2 T{+37"¡�$�& 2 &�$YZ+y8F+�T5x�+3pU¡�$�YZ$ 032 & » sz$�8 � YZ+3&ED�&S/ �10 $ x�$�&cT{+ � $�YZ+ )�T 2 T{$ 2 /�+�T�T 2 � 8³¡�$�T{$�/9¼3p�+ 2 ! ºX8ROS+ 8 � 8ROS+3T 2$�x�+3&S$�) 2 OS¤{ ����*KZ!«(T 8 � 8ROS+3T 2 +�Tt6�7"+3T��zO �10 $B¤5$;'9)U+ 0 $�p*O?&S$�7 2 $BT{+3O?&S$³p 2c0 � Y 2 YZ+(� 2 x�$9p�+�8 2 Y�+{Q�+�pUY 2 � !�� 8 � 8ROS+3T 2

$�x�+3& 2 )�T 2 7 � pU¡ 2 YZ+B �P � �5��T 0 $�T ��{+y8RO 2 r3�*+y8w+-|}$ � YZ+y8F+�p » $�7 » � Y�$5x�+37 2 M � O 2�0 ¡ � !�� /�$ » +3&Sp�$\� 2 x�$9p�¼�8+ ¿ � / � ){O?+y8ROS+�8]+ ¿�2 )U8FO �"» $*8�'*)�+ � p 0 7") �"& 2 TÀP����U! �9���J8 � T�)�7 2 r���+�8�+�P��9���J&S$ZY 2 Y 2 8 p�$-T{+3O?&S$ »�2 @ � $ � 2 p*OS+�8XY�+2 )�O?$�& � @ 2 & $(|})�p 0 � $�p 2 T{+3p*O?$\+3T ��9�*��!�[]$9pUYZ)ZOS$�&S+�8 ¡�)�T 2 p�$98 2 � p�Y 2(0 $9p*O?&S$�7 2 T $�8 � 8ROS+3T 2 |}$�& 2 Y 2 8±¡�$9& 2 8YZ+wx �10 $Jx 2 & 2 T 2 p*O?+�&?+�T 8?) 2 8%¡ 2 6 � 7 � Y 2 Y�+�8�! ��)ZOS& 2 ¡ � 8ROS.�& �"2 YZ+w8?) 0 +y8?8?$Jp�$K� 2 xUsz$�|}$ � $JY�+�8?+3p » $97 » � T5+�p*O?$x�+37 2³> )�� � º�7"+ 0 OS& �10 YZ+�)UT 8 � 8ROS+3T 2 YZ+tOS& 2 O 2 T5+�p*O?$³Y�+5��/�) 2 !Iº]8ROS+�8JYZ$ � 8I8 � 8ROS+3T 2 8I7"+ »�2 & 2 T 2 pU$98�x 2 & 28?+3&S+3T 0 $97=$ 032 YZ$98(+3TÂ|})�p 0 � $9p 2 T{+�p9OS$�&S+ 2 7%YZ+ » � YZ$ 2�» �z& �"2 8\YLD » � Y 2 8�8?$�6�&S+ 2 +�� 0 � 0 �12 +58F+�/�)�& 2 pUr 2 YZ$*88 � 8FO?+�T 2 8�p�+36U)�7=$*8F$*83!º�T y���9�t$5/�$ » +3&Sp�$(� 2 x�$9p�¼�8V+ ¿ + 0 )ZOS$�)c)�T 0 ) � Y 2 YZ$*8F$�+�8FO?)UYZ$�Y 25» �12 6 � 7 � Y 2 YZ+IY�+ 0 & �"2 &]x�&S$z�R+3O?$*8VY�+

x�+�8S'9) � 8 2 p 2�0 � $9p 2 � 8%+3T 7=.9/ �1032 p�+�6�)�7"$98 2 +3p » $97 » +3pUYZ$J)�p �=» +3&D8 � Y 2 Y�+�8�+ � pUYLDU8FO?& �"2 83!�[]$�T{$J&S+�8?)�7 O 2 YZ$-Y�$ � 8/�& 2 p�YZ+�8�x�&S$z�R+�OS$98�|}$�& 2 T 7 2 pUr 2 YZ$*8�x�+37"$H§ � p � 8FO?¤�& � $�Y 2 oqpUYMDU8ROS& �12 +\[]$9T5¤�& 0 � $�oqp*O?+3&Sp 290 � $�p 2 7�+ 2 £\/�¼�p 0 �12YZ+5��+ 0 p�$97=$9/ �12 +{[ � ¼3p 0 �"2 !JQZ$�T{+3p*O?+�$Bx�&S$z�R+3O?$�7 2 p�r 2 Y�$�x�+�7=$c§ � p � 8FO?¤�& � $ � 0 ¡ 2 T 2 YZ$�YZ+ON .�k�2�/c.A:f2A/ �g� 2A/\ � : / � .A:_8B2 � .AVQP ���0���IR��5K 8 �o 0 /98 �XK�S 9"� .�/cH�j-,GN \ P R /V+3p�/97=$96 2�»�2 u�� 0 $�T{x 2 p�¡ �12 8�+�O � p�¡ 2 )�T $�&Dr 2 T5+�p*O?$YZ+-«IQ�T5uZ! �����U! �9���U! �9��� � ���Ix 2 & 2 8?+3&S+3T¢/ 2 8FO?$98V+�T¾8?+ � 8 2 p�$*83!§ 2 OS8?)U8F¡ � O 2 º�7"+ 0 O?& �"0 oqpUYZ)U8FO?& �12 7t[]$U! : # 2 p 2 8?$�p �"0yE |}$ �J2 x�& � T{+ � & 2 0 $�T{x 2 pU¡ �"2�2�2 x�7 �"0�2 &B7".�/ �"0�2

p�+�6�)�7"$98 2 +3T )UT x�&S$ZYZ)ZOS$Hx 2 & 2 $ 0 $�p�8F)�T � YZ$�& �z)�T 0 ¡�) » + � &S$I+37"¤�O?& �10 $ �9+3T¾ �������!�º�T¾ y���9� ��§ 2 OS8?)U8?¡ � O 27 2 pUr3$�)t)UT 2 T���'*) � p 2 YZ+]7 2�»�2 & 0 $9p9OS&?$97 2 Y 2 x�$�&%)�T 8 � 8ROS+3T 2 pU+36�)�7"$98?$I+]7 2 pUr3$�)�)�T 2 +�p�$�&ST{+ 0�2 T{x 2 pU¡ 2x�)�6U7 �10 � OS��& �12 x 2 & 2 x�$�x�)U7 2 & � @ 2 &]$5x�&S$�Y�)ZO?${'*)�+IO � p�¡ 2 )�T 2 x 2 7 2�» & 2 +y8ROS& 2 p�/�+ � & 2���������� +�T¢8F+�)³pU$�T{+�!X�8?) 0 +y8?8?$5Y 250�2 T{x 2 p�¡ 2 x�$ZYZ+-8?+3&�T{+�Y � YZ${x�+37"$�| 2 O?$�YZ+-'*)�+J+�8FO 2 x 2 7 2�» & 2 x 2 8?8?$�) 2 8?+3&w)�8 2 Y 2 pU$5� 2 x�sz$

Page 10: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� � ����� ��� � ����� ��� � � ���� �0 $9T $Á8 � /9p � � 032 Y�$cYZ+ � p*O?+�7 � /�¼�p 0 �12 ! P ���0��� / 2 p�¡�$�) 2 T{+�Y 2 7=¡ 2 YZ+�$9)�&S$³x 2 & 2 p�$ »�2 8�x 2 7 2�» & 2 8�p�$ 2 p�$YZ+¨ y�����U!{ºX8ROS+B8?) 0 +y8?8?$;7=+ » $�) $�)�O?& 2 8J+3T{x�&S+�8 2 8J'*)�+�x�&S$ZYZ)�@�+3T 6�+3p�8-YZ+ 0 $9pU8?)�T{$ÁYZ)�&D� » + � 8 2 7 2 pUr 2 &)�T 2 7=+ »�2 YZ+;x�&?$ZYZ)�O?$98 0 $�T~7".�/ �"0�2 pU+36�)�7"$98 2 I 2 8Fx � & 2 YZ$9&?+y8�YZ+;x�. �VT���'*) � p 2 8{x 2 & 2 0 $9@ � p�¡ 2 & 2 &?&S$�@ �&S+�|}& � /9+3& 2 YZ$9&?+y8��9T{�9'*) � p 2 8V|}$zOS$�/9&S� � 032 8��*+�O 0 !ºX8FO?+-8?) 0 +y8?8?$�8F+t+�8?x 2 7"¡�$�)cx�+37"$�Tt)�pUY�$�+ 2 º�)�&S$�x 2 +H+�T¾T{+�p�$�&w/9& 2 )³$*8wºX8FO 2 Y�$98(«\p � Y�$98 ��$�pUY�+ 2

7".�/ �1032 p�+�6�)�7"$98 2 O � pU¡ 2 8 � YZ$-� � p » +�p9O 2 Y 2 ��� � p �"0 �"2 & 2 TÀx�&?$��R+�OS$98�p 2 �z&S+ 2 !£ Y � | � 0 � 7 290 + � O 2 r3s�$�Y 2 7".�/ �1032 pU+36�)�7"$98 2 p 2 º�)�&S$�x 2 +�p�$98\ºX8RO 2 YZ$*8\«\p � Y�$98w|}$ ��2 O?& � 6�)��"Y 2 x�$9&�� 2 &?O

��$98<��$ 2 )�T 2 &?+ 2 r�sz$�Y 2 T{+3p*O?+t$ 0 � Y�+3p*O 2 7 2 & � 8ROS$zO?¤�7 �1032 � 2�0 $98FO?)�T 2 Y 252 O?)UYZ$ 0 7 2 8?8 � � 032 &wx�&?+ 0 � 8 2 T{+3p*O?+ �O � pUY�+��Up � r3s�$ � p�+�&?+�p9OS+ 2 $98J8 � 8FO?+�T 2 8Ip�+36�)U7=$*8F$*8 :B'*)%+A'*) �% ��9�z� E !-£ª)�T 2 |}& 2 8?+5p�$ ��p 2 7�YZ$cx�& � T{+ � &S$032 x �=O?)�7"$tY 2 x�& � T5+ � & 2 x 2 &FOS+\YZ+�8FO?+\7 �"» &?$-YZ+��\$*8 �9$I'*)�+ ��OS& 2 YZ)�@ � Y 2 7 �=» &S+3T{+3p*OS+��9Y � @ I��3£ 7=.9/ �1032 p�+�6�)�7"$98 20 $9T{+�r 2 $9pUYZ+ 2 7=.9/ �1032 $ 0 � Y�+3p*O 2 7�O?+3&ST � p 2 ��!

Page 11: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� ��������� � �

� ��� � � ��� � � � � � ��� � �

�"��� �i�G� ��@4! �����4\2 7=.9/ �1032 O?& 2 Y �"0 � $9p 2 7±£\& � 8ROS$zOS¤37 �"0�2 $98($96Z�R+�OS$98(8Ssz$ 0 7 2 8?8 � � 032 YZ$98�+�T 0�2 O?+�/�$9& �12 8�T�) � OS$�6�+3TÂYZ+��Up � Y 2 8��)�T¹$�6Z�R+3O?$¨x�+3&?O?+3p 0 + 2 )�T 2¨032 O?+�/�$�& �12 $�)�pUs�$���)�T 2 �U/�)U& 2 /�+�$�T{¤�OS& �1032 $�)�¤�)�T­'*) 2 Y�& 2 YZ$³$9)�pUsz$ �)�T 2 p � T 2 7%¤{8?+37 »�2 /9+3TÂ$�) pUsz$�! 4 $cTt)UpUYZ$ � Y�+ 2 7 � @ 2 Y�$ ��+�8FO 2 8I8?+3x 2 & 2 r3s�$�+3T 032 OS+3/9$�& �"2 8 � YZ+ 2 � 8Ix�$�Y�+8?+3& » � & x 2 & 2J0 7 2 8S8 � � 0�2 &%$�6��R+�O?$*8%+(8F+�&?+y8���p�$H+3p*O 2 p*O?$Jp�$HTt)UpUYZ$H&S+ 2 7U+37 2 | 2 7=¡ 2 +3T )UT /9& 2 pUYZ+�pQD�T{+3&S$JY�+8 � OS) 2 r3�*+y83!�[]$�T{$tY�+��Up � &X+ ¿�2 O 2 T{+3p*O?+\'*)�+�T ¤ 2 7=O?$ ��$�)³'9) 2 pUYZ$H)UT 2 O?+�T5x�+3& 2 OS)�& 2 +�8FOS�HT�) � O?$t'*)�+�p9OS+ G«\8 2 &S+3T{$*8J$98JOS+3&ST5$*8 0 $9p��R)�p*O?$*8 0 71��8S8 �"0 $98I$�) p �=O � YZ$*8 : H�/98 " , E x 2 & 2 p�$98H&S+�|}+�& � &ST5$*8 2 $*8 0 $9p��R)�p*O?$*8

YZ+��Up � YZ$98 2 x 2 &FO � &XY 2 8�$9x�+�& 2 r���+�8XY 2 7".�/ �"0�2 O?& 2 Y �10 � $�p 2 7l! 4 +y8ROS+ 032 x �=O?)�7"$ 2 x�&?+y8F+�p*O 2 &S+3T{$*8 2 8XYZ+��Up � r���+�8+ 2 8HY � |}+�&?+�pUr 2 8H6U��8 �1032 8I+�p9OS&?+ 0 $�p��R)Up9OS$98 0 7"�98?8 �"0 $98J+�p�+�6�)�7"$98?$98�!�§¨$98FO?& 2 &S+3T{$*8 2 8-YZ+���p � r3��+�8J6U�98 �1032 8YZ+¨7".�/ �"0�2 p�+�6�)�7"$98 2 + 0 $�T{$�&S+3x�&S+�8?+3p*O 2 &{+c$9x�+�& 2 & 0 $9T 0 $9p��R)�p*O?$*8{p�+36�)U7=$*8F$*83! Q�+�&Ss�$ Y � 8 0 )ZO � Y 2 8 2 8» �z& �"2 8�|}$�&ST 2 8 2 7=O?+3&Sp 2 O �"»�2 8%YZ+wYZ+��Up � & 2 8%$�x�+3& 2 r���+y8�6U�98 �1032 8%'9)U+Vx�$ZYZ+3Tª8F+�& &?+ 2 7 � @ 2 Y 2 8�8?$�6U&?+ 0 $9p��R)�p*O?$*8p�+�6�)�7"$98?$98�I�)Up � s�$ � � p*O?+3&D8?+ 0 r�sz$�+Hp�+3/ 2 r3s�$U!

�"��� � #��4� # 6G#�i�7-������ @4#�� � ��� > -�������I8JY 2 Y�$98J'*)�+�&S+3xU&?+y8F+�p9O 2 T � pZ|}$�&ST 2 r3s�$³8Ssz$cYZ+3& �=»�2 YZ$98IYZ+{+3p*O � Y 2 YZ+�8I+ ¿ � 8ROS+3p*O?+y8Ip�$³T�)�pUYZ$c&S+ 2 7�! 4 $+3p*O 2 p*O?$ �y+y8ROS+�8�Y 2 YZ$*8 OS+3T¦x�&S$�x�& � +�Y 2 YZ+�8 0�2 & 2�0 O?+3& �18RO �"0�2 8 �3+X¤]p�+ 0 +�8S8?��& � $ 0 $9p�¡�+ 0 ¼���7 2 8 x 2 & 2 '9)U+Xx�$98S8 2 T5$*8O?& 2 O 2 & 0 $�&S&S+�O 2 T{+3p*O?+ 2 8 � pZ|}$�&ST 2 r3�*+y8BYZ$�Tt)�p�YZ$ &S+ 2 7l! ��+�8FO?+ T{$ZYZ$ x�$ZYZ+3&S+3T{$98�| 2 @�+3& 2 8?8?$ 0 �12 r3�*+y80 $�+�&?+�p9OS+�8V+3p*OS&?+J$98 » ��& � $98]O � x�$98(YZ+-Y 2 YZ$*8]+ 2 8�+3p*O � Y 2 YZ+�8w'*)�+H'*)�+3&S+3T{$*8�&?+�x�&?+y8F+�p*O 2 &y!º ¿ � 8ROS+3T¾'*) 2 O?&S$tO � x�$98(6U��8 �"0 $98�YZ+-Y 2 YZ$98�YZ+ 290 $9&SYZ$ 0 $9T 2 � p�|}$�&ST 2 r3s�$5'*)�+J+37"+�8 0 $�p*O?¤�T;!� e�"X� ´¨��� ef��m�°�� &S+�|}+�&?+3��8?+ 2 )�T 2 +�p9O � Y 2 YZ+�$9) + » +�p*O?$¨p�$ '*) 2 7XpUs�$¨¡U�¨&S+37 2 r3�*+y8-YZ+�$9&SY�+3T +B$

D�p �10 $BO?+y8ROS+�x�$98S8 � » +�7�¤5YZ+ � /�) 2 71Y 2 YZ+9!\#%$�&�+ ¿ +3T{x�7"$ ��$³p�$�T{+�Y�+t)�T 2 x�+�8S8?$ 2 !I«(T x�$98S8 � » +�7 0 $�p��R)�p*OS$YZ+y8ROS+JO � x�${¤H$�YZ$98 2 7=)Up�$98wYZ+H)UT 2 OS)�&?T 2 I

alunos = {maria, ana, pedro}

4 $�O 2 &J'*)�+�pU+�8FO?+ 032 8F$c$¨+37"+3T{+�p9OS$;x�+�&FOS+3p 0 +�$�) pUsz$ 2 $ 0 $�p��R)�p*OS$ ��+�pU+3p�¡�)�T 2 $�)ZOS& 2 � pZ|}$�&ST 2 r�sz$+�8FOS� 0 $�p*O � Y 2 p�$�Y 2 Y�$U!� e�"X��������el��m�°�� )�T~Y 2 YZ$���&DY � p 2 7 � Y�+3p*O � � 032 2 x�$*8 � r3s�$ YZ+c)�T 2 +�p9O � Y 2 YZ+c$�) + » +�p9OS$ +�T )�T 2

¡ � +3& 2 &D'*) �"2I0 )z�R$98 � p9OS+3& »�2 7"$98�pUsz$J8Ssz$HYZ+���p � Y�$98%+ ¿�2 O 2 T{+�p9OS+�!�#�$9& + ¿ +3T{x�7"$ � 2I0 7 2 8?8 � � 032 r3sz$�YZ+w)�T 2 7=)Up�$+3TW&?) � T �ZT{¤�Y � $ ��6�$�T¢+J+ ¿�0 +37"+3p*O?+I¤HYZ+IO � x�$B��&DY � p 2 7�! 4 $zO 2 &w'9)U+I$ � p9OS+3& »�2 7"$�+�p9OS&?+I$98V+�7=+�T5+�p*O?$98VpUs�$x�$�Y�+-8F+�&(YZ+��Up � YZ$ 0 $�T¢x�&S+ 0 � 8Ssz$�!

aluno = {ruim, medio, bom, excelente}: KZ!" E

�I8VY 2 YZ$*8Vx�$ZYZ+�TW8F+�& 0 $�T{x 2 & 2 YZ$98]'9) 2 p*O?$ 2 � /�) 2 71Y 2 YZ+ �*8?)�x�+3& � $9& � Y 2 YZ+\+ � pZ|}+3& � $9& � Y 2 YZ+�!%«(T 2 7=)Up�$+ ¿�0 +37"+3p*O?+t8?+3& �12 $B'*)�+-T 2 � 8\8?+ 2 xU&?$ ¿ � T 2 Y 2 YZ+���p � r�sz$�YZ+ 2 7")�p�$ � YZ+ 2 7�x 2 & 2 $�x�&S$z|}+�8S8?$�&y!]oR8FO?$ � T{x�7 �"0�2

u

Page 12: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� � ���`� ��� � ���R� � � � ���� � � � ���� � ���'*)�+�&S+3x�&S+�8?+3p*O 2 & 0 $ � 8 2 8 � pUYZ+��Up � Y 2 8t+�8FOS� � p9O � T 2 T5+�p*O?+�7 � / 2 YZ$ 2 + ¿ xU&?+y8?8 2 &HY � 8ROD½zp 0 �"2 8-YZ+�)UT¹x 2 YZ&Dsz$� YZ+ 2 7l![]$�pz�R)�p*O?$98tp�+36U)�7=$*8F$*8Hx�$�Y�+3T­&?+�x�&S+�8?+3p*O 2 &tY 2 Y�$98 0 )�� 2 YZ+��Up � r3s�$ YZ+ � p 0 7=)U8Ssz$ 8F+ 0 $�T{x�$�&?O 2Á0 $�T{$

)�T 25»�2 & � � » +37 YZ+�8FO?+JO � x�$U!� e�"X� �����b*� ��m�°l� � &S+3xU&?+y8F+�p9O 2 Y 2 YZ$*8 0 )z�R$98 � p*O?+3& »�2 7"$98�+�p9OS&?+H$*8 »�2 7"$�&S+�8�p 2 +�8 032 7 2 x�$ZYZ+�T�8?+3&�YZ+����

p � YZ$98 0 $�T¾+ ¿�2 O � YUsz$U! 4 $�+3p*O 2 p*O?$ �UpUsz${¡U�{)�T�x�$�p*O?$�@�+3&S$ 0 7 2 & 2 T{+3p*O?+HYZ+��Up � YZ$U!V«(TW+ ¿ +3T{x�7"$�6U��8 �"0 $¤-$ 0�2 7"+3pUY���& � $�$�p�YZ+H$ 2 pU$BKz���9�5+�8FOS�; ��9��� 2 pU$98(YZ+�x�$ � 8�YZ$ 2 p�$¨ ��9����! 4 $�+�p*O 2 p*OS$ � 0 $�T{$�pUsz${OS+3T{$982 YZ+��Up � r3sz$B+ ¿�2 O 2 YZ$ � p�� 0 � $BY�$98�OS+3T{x�$*8���pUsz$³8F+Hx�$ZYZ+ 2 ��&?T 2 &�'9)U+H$ 2 p�$BK����9��+y8ROD��YZ) 2 8 » +�@3+y8wT 2 � 8Y � 8RO 2 p*O?+(Y�+�8FO?+ � p � 0 � $U! � 2 YZ$98�Y�+�8FO?+�O � x�$-x�$ZYZ+�Tª8?+3& 0 $�T{x 2 & 2 YZ$98%'*) 2 p9OS$ 2 � /9) 2 71Y 2 YZ+���8?)�x�+3& � $9& � Y 2 YZ+�+x�$�Y�+3T¾8?+3&w8?)�6ZOS& 2 �"YZ$*8V)�T¾YZ$*8V$�)ZOS&?$*83!X[]$�pz�R)�p*O?$98Vp�+�6�)�7"$98?$98Vx�$�Y�+3T¾8?+3&�x�&S$z�R+3O 2 YZ$98Vx 2 & 2 � p 0 $�&Sx�$9& 2 &+�8FO?+IO � x�$�YZ+ � pZ|}$�&ST 2 r3s�$5Y�+�8SYZ+H'9)U+H$5xU&?$��R+�O � 8FO 252 8S8 � T¢$�YZ+�8?+R�R+9!� e�"X� ! ����"X�U��k9ef����m�°�� � 2 YZ$98 Y�+�8FO?+VO � x�$H8Ssz$Ix 2 &S+ 0 � Y�$98 0 $�T $98%YZ$IO � x�$ � p*O?+3& »�2 7"$ � 0 $�T 2 Y � |}+3&S+3pUr 2

8?+3pUYZ$ 2 +y8 0�2 7 2 �9'*)�+�p�+�8FO?+ 032 8F$-O?+3TÀ)UTª@�+3&S$ 2 6�8F$97=)ZOS$U!�#%$�&V+ ¿ +3T{x�7"$ �*¤ 0 $�&S&?+3O?$ 2 �U&?T 2 &]'*)�+IK��-/9& 2 )U8�J+37 » � p�8Ssz$JYZ) 2 8 » +�@3+�8%T 2 � 8�'*)�+�p*O?+�8 '*)�+H y�I/�& 2 )U8�J+�7 » � p±! � T{+�8?T{$Ip�sz$H8F+wx�$ZYZ+ 2 �U&?T 2 &�+�T &S+37 2 r�sz$2 OS+3T{x�+3& 2 O?)U& 2 8tT{+�Y � Y 2 85+3T /�& 2 )�85[]¤3718 � )U8�! 4 $ »�2 T{+�p9OS+�� 0 $�p��R)�p*OS$98�p�+36�)U7=$*8F$*8tx�$ZYZ+3T~8?+3&5)U8 2 YZ$*8x 2 & 2 &?+�x�&S+�8?+3p*O 2 &V+y8ROS+IO � x�$�YZ+-Y 2 YZ$*83!#%$ZYZ+3T{$98 » +3& � � 032 &I+3p*ODsz$¨'*)�+ 0 $�pz�R)�p*O?$98JpU+36�)�7"$98?$98Jx�$ZYZ+3T¹&?+�x�&S+�8?+3p*O 2 &�+y8ROS+�8HYZ$ � 8�D�7=O � T{$98JO � x�$98

YZ+-Y 2 YZ$98V+�pU'*) 2 p*O?$�'*)�+ 0 $9p��R)�p*O?$*8 0 71��8S8 �10 $*8VpUsz${$98�x�$ZYZ+3T¾&?+�x�&S+�8?+3p*O 2 &�!

�"�<; A ����<! i�7��6 A 5���6G67*,+.��6«(T 0 $�p��R)�p*OS$ 0 71��8S8 �"0 $ �U$9)¨p � O � Y 2 T{+3p*OS+�YZ+��Up � YZ$ �

A¤�)�T 2B0 $�7"+�r3s�$�YZ+t+�7=+�T{+3p*O?$*8($�) $�6Z�R+3O?$98

x+ ¿ � 8R�

O?+�p*O?+�8\+3TÂ)�T�)�p �"» +�&S8?$�YZ+�Y � 8 0 )�&D8?$X:A ⊂ X

E !\[ 2 Y 2 +37"+3T{+3p*OS$xYZ$B)�p �"» +3&D8?$�x�$ZYZ+-x�+3&?O?+�p 0 +�& 2 $

0 $9p��R)�p*O?$A:x ∈ A

E $�)cpUsz$ :x /∈ A

E !«(T 0 $�p��R)�p*OS$ 0 71��8S8 �10 $Hx�$ZYZ+\8?+3&VYZ+�8 0 & � OS$tYZ+\Y �=» +3&D8 2 8 T 2 p�+ � & 2 8�!�#%$ZYZ+3T{$*8�+3T{x�&S+3/ 2 &�$*8�Y �12 /9& 2 T 2 8

YZ+#� +�p�p ��'9)U+¨T5$*8ROS& 2 T /�& 2 � 0�2 T{+3p*OS+³$*8{+37"+3T{+3p*O?$*8�'*)�+;x�+�&FOS+3p 0 +3T 2 )�T YZ+�OS+3&ST � p 2 YZ$ 0 $�pz�R)�p*O?$U!��)ZOS&?$*8±T{$ZYZ$98�8?s�$ ��x�$�&%+ ¿ +3T{x�7"$ ��x�$ZYZ+3T{$98�7 � 8FO 2 &�$*8�+�7=+�T5+�p*O?$98�'*)�+Vx�+�&FOS+3p 0 +3T 2 $ 0 $�pz�R)�p*O?$ �zY�+�8 0 &S+ » +�&2 p 2 7 � O �"0�2 T{+3p*O?+w+y8ROS+�8 +�7=+�T{+3p*O?$*8�$�){)U8 2 &X)�T 2 |})�pUr�sz$-'*)�+ 2 $ 2 8S8F)�T � &�$ »�2 7"$�&( � pUY �"0�2 '9)U+($-+37"+3T{+�p9OS$x�+3&?O?+3p 0 + 2 $ 0 $9p��R)�p*O?$�! # 2 & 2 � pUY �1032 &('9)U+I$�+�7=+�T{+3p*O?${pUs�$5x�+3&?O?+�p 0 +Jx�$ZYZ+��q8F+J)U8 2 &w$ »�2 7"$�&w�U!#%$�&�+ ¿ +�T5xU7=$NI

frutas iniciando com letra l = {laranja, limao, . . .}

numeros menores que 20 = {x ∈ < | x < 20}

alto =

{

1 se altura ≥ 1, 800 se altura < 1, 80

4 $;x�& � T5+ � &S$ 0 $9p��R)�p*O?$¨$;)�p �"» +�&S8?$¨YZ+�Y � 8 0 )�&D8F$;¤ 0 $9T{x�$*8ROS$cx�$�&JOS$ZY 2 8 2 8I|}&?)�O 2 8J+ 2 8J|}&S)ZO 2 8H'9)U+x�+3&?O?+3p 0 +3T 2 $ 0 $�p��R)Up9OS$ 0 $�T{+�r 2 T 0 $�T 2 7=+3O?& 2 7�! �Â8?+3/9)�pUYZ$ 0 $9p��R)�p*O?$�¤ 0 $�T{x�$98FO?$ x�$9&5O?$ZYZ$985$*8pQD�T{+3&S$98V&S+ 2 � 8]T5+�p�$�&S+�8�'*)�+-Kz�U! > � p 2 7"T{+3p*O?+Jp�$5D�7=O � T{${+ ¿ +3T{x�7"$tOS+3T{$98V$ 0 $9p��R)�p*O?${YZ+IO?$ZY 2 8Vx�+y8?8?$ 2 80 )z� 2 8 2 7=O?)U& 2 8w8?s�$�T 2 � $�&S+�8V$9) � /�) 2 � 8 2 � �9�tT;![ 2 8?$�+ ¿ � 8RO 2 )�T 2 |})UpUr3s�$B'*)�+tYZ+��Up 2 8?+-)�TÂ+�7=+�T5+�p*O?$Bx�+3&?O?+�p 0 +t$�)ÁpUs�$ 2 $ 0 $�p��R)Up9OS$ ��+�8FO 2 |})�pUr3s�$0 $*8ROS)�T 2 8F+ 0 ¡ 2 T 2 Y 2 YZ+J|})�pUr�sz$ 032 & 290 O?+�&&�18FO �1032 $9)¨YZ+ � p 0 7=)�8?s�$U!

`cb9d%��ef���U� _ #l������������ ��b� ���k�°f��hy��� b �¥²�����y��������h;²�°lj�h�hyefk9��h � � > H�2 ��)9�?� :&2CH#Vbp "�" 8BH02AF 1 �1

^`� 8B1�2 �> �?> �?� 8;O / " 2D1 1A8 " H � / " 2XF�,�241 i"# /i1 B^�� 8B1�2s, Vb. "9� . ���?� aGM�2C1 8 � H#V �5" M�2

χ → {0, 1}F

e �� l> .0, 8B. H0.�1�. V �>l #� :f2 1 X

�>�� 2 ��� 1 , #� 1 �� 1�2 "� 2 V �>d �� :&2 Y 2 � � M�2 �?> >l #> k�/c2 1�2H�2 � )9�?� :&2�[ \ " :f2 ,R2U1 E"� /g/ ,o/ 9"� �� :&.�1�2 ,�2�/��

χA(x)

{

1 se x ∈ A0 se x /∈ A

: K�! K E«(T 0 $�p��R)�p*OS$ 0 71��8S8 �10 $ � p*O?&S$ZYZ)�@X)UT 7 � T �12 &�'*)�+X+y8Fx�+ 0 � � 0�2 '*) 2 p�YZ$()�T +37"+3T{+�p9OS$\YZ+ � ¿�2 Y�+�x 2 &?O �10 � x 2 &

YZ+�)�T 0 $9p��R)�p*O?$�!B#%$�&H+ ¿ +3T{x�7"$ ��x�$ZYZ+�T5$*8 0 $9pU8 � YZ+�& 2 &H'9)U+ 2 7=O?$98t8?s�$³OS$ZYZ$98H$*8H8F+�&?+�T ¡�)�T 2 pU$98 0 )z� 2

Page 13: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R���?� � � �� � ��� � ����� �� � �� �

µ

Altura

1,0

1,80

> � /9)�& 2 K�!= I%«\T 2 x�$*8?8&� » +37 YZ+��Up � r3sz$�YZ$ 0 $�p��R)Up9OS$ 0 71��8S8 �10 ${Y 2 8�x�+�8S8?$ 2 8 2 7 O 2 8�!

2 7=O?)U& 2 8F+F� 2 T 2 � $�&�$�) � /9) 2 7 2 � �9�tT :f> � /9)�& 2 K�!= E ! «(T 2 Y � � 0 )U7"Y 2 YZ+H'*)�+-8F)U&?/9+�¤ 0 $9T{$5+y8 0 $�7"¡�+3&�+y8ROS+7 � T �12 & ��'9)U+-YZ+ » +J8?+3&�+ ¿Z2 O?$ G ºX8RO 2 Y � � 0 )�71Y 2 YZ+H7=+ »�252 $ ���F�5T{+3p 0 � $9p 2 Y�$ �Zx 2 & 2 YZ$ ¿ $�YZ+-Q�$9& � O?+y83!3�) 2 7X$¨O 2 T 2 p�¡U$ YZ+B)UT­Y�+�O?+�&?T � p 2 Y�$ 0 $9p��R)�p*O?$

AG º]8ROS+BO 2 T 2 p�¡U$Á$�) 2 032 &DY � p 2 7 � Y 2 YZ+�Y�+B)�T0 $9p��R)�p*O?$H¤(Y�+��Up � YZ$Hx�+37 2H0 $9p9O 2 /9+3T YZ+�OS$ZYZ$98 $*8%+37"+3T{+3p*OS$98�'*)�+wx�+3&?O?+�p 0 +�T 2 $ 0 $�p��R)�p*OS$U!�ºX8ROS+(pQD�T{+3&S$

O 2 Tt6�¤3T¾x�$ZYZ+-8?+3& 0 ¡ 2 T 2 YZ$�YZ+JpQD�T{+3&S$ 0�2 &DY � p 2 7 YZ+A!

`cb9d%��ef���U� _ # _²�mZ����ef��m�°fel��m���b���b�²�����y��������hJ²�°lj�h�hyefk9��h � % H�.A/c1�8 � .�VW8B1�.�1 1 *�?> H�2 � )9�?� :&2EH�V]p "01" 8BH�2

AYg>l 1�8B1�. , V].lH�2 � :f. K� �> 1 :f2U1�2 " 2 "J V �>l #� :f2 " e �� , /9: �� H �> . #" : H�2 ��)9�?� :&2GF JY 1 �� 2A:f.�1�.

,�2�/|A|

[£ x 2 &FO � &VYZ+�8FO 2 YZ+��Up � r�sz$HOS+3T{$98]'9)U+ 0 $9p��R)�p*O?$*8XY � 8 0 &S+�O?$*8�'*)�+�8Ssz$ 0 $�T{x�$98FO?$98]YZ+()�T pQD�T{+3&S$ �Up � OS$+ 0 $9p*OS� » +�7*Y�+w+37"+3T{+3p*O?$*8�O?¼�T )�T 2H032 &SY � p 2 7 � Y 2 YZ+ ��p � O 2 � 2 $Jx 2 8S8F$J'*)�+ 0 $9p��R)�p*O?$*8 0 $9p9O �=p�)�$*8 0 $�T{x�$98FO?$*8

YZ+H)�T¾pQD�T{+3&S$ � p��Up � O?$�Y�+H+37"+3T{+3p*O?$*8VO?+�T 032 &SY � p 2 7 � Y 2 YZ+ � p��Up � O 2 !`cb9d%��ef���U� _ #1® ²�����y������� ! ��� ����k9efmÁ��b;��� ²�����y������� ²�°lj�h�hyefk9� ��� .�1�2 �?> H02 ��)9�?� :f2

AF`2lH�2 ��1

)9�?� :&2i,�2�:&t � H#8B. 1 A

Y 2 H�2 ��)9�?� :f2 1 :&2U1�2 " 2 "i"9� k�H�2 ��)9�?� :f2 " 1 A[ R " : H�2 ��)9�?� :f2 Y 1 �� 2A:f.�1�2d,R2A/

2A [

��� b � "�°l� _ #f��� Q�+F� 2 )UT 0 $9p��R)�p*O?$A,0 $�T{x�$98FO?${YZ+IO?&S¼�8V+37"+3T{+3p*OS$98 �

A = {1, 2, 3},YZ+JT 2 p�+ � & 2 '*)�+ 2

032 &SY � p 2 7 � Y 2 YZ+IY�+�8FO?+ 0 $9p��R)�p*O?${¤|A| = 3

!X� 0 $�p��R)Up9OS${x�$�O?¼3p 0 �12 YZ+A¤-Y 2 YZ${x�$�& I

2A = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

£ 0�2 &DY � p 2 7 � Y 2 YZ+ÁYZ$ 0 $�pz�R)�p*O?$ x�$zOS¼3p 0 �"2 YZ+ )�T 0 $�p��R)Up9OS$A¤ YZ+3pU$zO 2 Y 2 x�$�&

|2A|�(+ x�$ZYZ+ 8F+�&

032 7 0 )�7 2 Y 250 $9T5$|2A| = 2|A| = 23 = 8[ 2 8?$ 2I032 &SY � p 2 7 � Y 2 YZ+]YZ$ 0 $9p��R)�p*O?$J8?+R� 2 � p���p � O 2 �z+�p*OSsz$ 2�0�2 &DY � p 2 7 � Y 2 YZ+VYZ$ 0 $�p��R)Up9OS$�x�$zOS¼3p 0 �"2 O 2 Tt6�¤3T

¤ � p��Up � O 2 ![]$�pz�R)�p*O?$98Xx�$ZYZ+�T 0 $9p9OS+3&V8F)�6 0 $9p��R)�p*O?$*83!%«(T 0 $�p��R)Up9OS$

A¤\)�TÀ8?)�6 0 $�p��R)Up9OS$�YZ+

B��YZ+3pU$zO 2 YZ$tx�$�&

A ⊂ B��8F+H+�8F$9T5+�p*O?+-8?+ 032 Y 2 +37"+3T{+3p*O?$�Y�+

A¤-)�T�+37"+3T{+3p*O?$�Y�+

B!(� 0 $�p��R)�p*OS$�x�$�O?¼3p 0 �12 YZ+

B�2B0 $9p*O?¤3T OS$ZYZ$98V$98V8?)�6 0 $�p��R)Up9OS$98�YZ+

B! #%$�&?O 2 p*O?$ �*x�$�Y�+3T{$98�Y � @3+�&�'*)�+I$ 0 $�pz�R)�p*O?$

A¤J8F)U6 0 $�p��R)�p*OS$5YZ+

B8?+

Ax�+3&?O?+�p 0 + 2 $ 0 $�p��R)�p*OS$5x�$zOS¼3p 0 �"2 Y�+

B!

������� �������������������� �"!�#�� �$��% &('*)�+",-)/.0'*�1&(23� �"�"!�#�'*�Q�+F� 2 T

A+

BYZ$ � 8 0 $9p��R)�p*O?$*8tYZ+³)�T «(p �"» +�&S8?$

X!Á£W)�p � sz$¨+3p*OS&?+�+y8ROS+�8�YZ$ � 8 0 $�p��R)�p*OS$98 �%Y�+3p�$�O 2 Y 2

x�$�&A ∪ B

��&S+3xU&?+y8F+�p9O 2 O?$ZYZ$98-$98H+�7=+�T5+�p*O?$98t'9)U+�x�+�&FOS+3p 0 +3T $�) 2 $ 0 $9p��R)�p*O?$A� 2 $ 0 $9p��R)�p*O?$

B��$�)

2c2 Tt6�$98�$98 0 $9p��R)�p*O?$*8 2 $BT{+y8FT{$³O?+3T{x�$U!t£ � p9OS+3&D8F+ 0 r�sz$BYZ$*8JYZ$ � 8 0 $�p��R)Up9OS$98 �±YZ+3pU$zO 2 Y 2 x�$�&A ∩ B

Page 14: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� � ���`� ��� � ���R� � � � ���� � � � ���� � ���&S+3x�&S+�8?+3p*O 2 OS$ZYZ$98 2 '*)�+�7=+y8-+37"+3T{+�p9OS$98�YZ$Á)�p �"» +�&S8?$

X'*)�+c8 � Tt)�7=O 2 p�+ 2 T{+3p*O?+�x�+3&?O?+�p 0 +�T 2 2 T�6�$*8-$*80 $9p��R)�p*O?$*8

A+

B!�� 0 $�T{x�7"+3T{+�p9OS$BYZ$ 0 $�p��R)Up9OS$

A��&?+�x�&S+�8?+3p*O 2 YZ$�x�$9&

A��¤tY�+��Up � YZ$ 0 $�T{$ 2�0 $�7"+�r�sz$

YZ+5+37"+3T{+3p*OS$98IYZ$³)Up �=» +3&D8F$X'*)�+5pUsz$³x�+�&FOS+3p 0 +3T 2 $ 0 $9p��R)�p*O?$

A!-£ªY � |}+3&S+3pUr 2 YZ$ 0 $9p��R)�p*O?$

A0 $�T

&S+�8?x�+ � OS$ 2 $ 0 $9p��R)�p*O?$B��&S+3x�&S+�8?+3p*O 2 Y 2 x�$9&

A | B�U¤�YZ+��Up � Y 2�0 $�T{$ 2�0 $97=+yr3s�$5YZ+-O?$ZYZ$98($98w+�7=+�T5+�p*O?$98

YZ$c)�p �"» +�&S8?$B'*)�+5x�+�&FOS+3p 0 +3T OA+{'*)�+tpUs�$³x�+3&?O?+�p 0 +�T O

B!-£ª)�p � s�$B+ ¿�0 7=)U8 �"»�2 +3p*O?&S+t$*8 0 $�pz�R)�p*O?$98

A+

B��&S+3x�&S+�8?+3p*O 2 Y 2 x�$9&

A ⊕B�U¤JY�+��Up � Y 2 x�+37"$ 0 $�pz�R)�p*O?$�YZ$98�+�7=+�T{+3p*O?$*8w'9)U+Ix�+3&?O?+�p 0 +�T O

A+HpUsz$

x�+3&?O?+3p 0 +3T OB$�)5x�+�&FOS+3p 0 +3T O

B+�pUs�$Jx�+�&FOS+3p 0 +3T O

A!%º�T OS+3&ST5$*8 YZ+(8&�=T�6�$97=$*8%T 2 O?+�T{�zO �10 $*8%+�8FO 2 8

YZ+��Up � r���+�8VOS¼3T 2 8?+3/9) � p*OS+�|}$9&?T 2 I

A ∪ B = {x | x ∈ A ou x ∈ B}: K�! P E

A ∩ B = {x | x ∈ A e x ∈ B}: K�! � E

A = {x | x /∈ A e x ∈ X}: K�! u E

A | B = {x | x ∈ A e x /∈ B}: K�! � E

A ⊕ B = {x | x ∈ A e x /∈ B ou x /∈ A e x ∈ B}: KZ!�� E

��� b � "�°l� _ #1_ � # 2 & 2 T5$*8ROS& 2 &B$ )U8?$�YZ+y8ROS+�8c$�x�+3& 2 YZ$9&?+y8 »�2 T{$98 2 xU7 �10 �z��7"$98 2 $98 0 $9p��R)�p*O?$*8A+

BYZ+��Up � YZ$98 2 6 2 �=¿ $U!�� 2 T{$*8 2 8S8F)�T � &('*)�+J$98w+�7=+�T5+�p*O?$98wY�+A+

Bx�+3&?O?+�p 0 +�T 2 $ 0 $�p��R)Up9OS$�YZ$98�pQD�T{+3&S$98

� p9OS+ � &?$*83!A = {1, 3, 5, 7, 11, 13, 17}B = {1, 2, 3, 5, 8, 13, 21}A ∪ B = {1, 2, 3, 5, 7, 8, 11, 13, 17, 21}A ∩ B = {1, 3, 5, 13}A ⊕ B = {2, 7, 11, 17, 21}A | B = {7, 11, 17}A = {2, 4, 6, 8, 10, 12, 13, 14, 15, 16, 18, . . .}

������ � � ' ����!3���-��� ����� '*��&('*)�+",-)/.0'*�1&(23� �"�"!�#�'*�[]$�p�8 � YZ+3&S+w$98 0 $9p��R)�p*O?$*8

A�B+

CY�+��Up � YZ$*8�p�$ 0 $�p��R)Up9OS$t)�p �=» +3&D8F$

X+\$ 0 $�pz�R)�p*O?$ »�2 @ � $

∅!%£� 2 6�+�7 2

KZ!" JT{$98FO?& 2�2 8�x�& � p 0 � x 2 � 8�xU&?$9x�& � +�Y 2 YZ+y8VY�$98 0 $9p��R)�p*O?$*8�p � O � Y 2 T{+3p*OS+JY�+��Up � YZ$*83!

� 2 6�+37 2 KZ!" I%#�& � p 0 � x 2 � 8�x�&?$9x�& � +yY 2 YZ+y8�YZ+ 0 $9p��R)�p*O?$*8 0 71��8S8 �10 $*8[]$�T�)ZO 2 O �"» � Y 2 YZ+

A ∪ B = B ∪ A A ∩ B = B ∩ A£�8?8?$ 0 �12 O �=» � Y 2 YZ+A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = (A ∩ B) ∩ CoRYZ+�T{x�$�O?¼3p 0 �12

A ∪ A = A A ∩ A = Aº ¿�0 7=)U8Ssz$�� 4 sz$�[]$�p*O?& 2 Y � r3s�$A ∪ A = X A ∩ A = ∅� � 8FO?& � 6�)�O �"» � Y 2 YZ+

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)£\6U8?$�&Dr3sz$A ∪ (A ∩ B) = A A ∩ (A ∪ B) = Aoqp » $�7")Ur3s�$

A = Aº�7=+�T{+3p*O?$*8 4 +3)�O?&S$98A ∪ ∅ = A A ∩ X = AA ∩ ∅ = ∅ A ∪ X = X,±+ � 8wYZ+ ��+�§Á$�&S/ 2 p

A ∩ B = A ∪ B A ∪ B = A ∩ B

ºX8FO 2 8\x�&S$�xU& � +�Y 2 YZ+�8\x�$ZYZ+�T 8?+3&Ix�&?$ »�2 Y 2 8\)U8 2 pUYZ$���8?+-$98JY �12 /9& 2 T 2 8(YZ+ � +3pUp±!t£\8I7"+ � 8JYZ+ ��+�§Á$�&?�/ 2 p 8Ssz$ � T{x�$�&?O 2 p9OS+�8Jp 2 T 2 p � x�)U7 2 r3sz$cYZ+�8FO 2 8J$�x�+3& 2 r3�*+y8\x�$�&D'*)�+{x�+�&?T � O?+�T '*)�+�8F+�O?& 2 p�8R|}$9&?T{+5)�T 2$�x�+3& 2 r�sz$5'9)U+H)U8 2 )�p � s�$ : � p*OS+3&D8F+ 0 r3s�$ E +3T � p*O?+�&S8?+ 0 r3s�$ : )Up � s�$ E !��)ZOS& 2 8 � T{x�$9&FO 2 p*O?+y8�xU&?$9x�& � +�Y 2 YZ+y8I8Ssz$ 2�� b�e���m ´¨��� ²���������mU��el����� :

A ∩ A = ∅E + 2�� b�eJ��m

��� k�°f��hy������� b�ef� :A ∪ A = X

E ! £¦,±+ � YZ+ 4 sz$�[]$�p*OS& 2 Y � r�sz$�+�8FO 2 6�+37"+ 0 +H'9)U+Ip�sz$5¤Ix�$98S8&� » +37 2 )�T

Page 15: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R� �R� � � �� � ���P ��� � � �� �� �+37"+3T{+�p9OS${x�+�&FOS+3p 0 +3& 2 $5T{+�8?T5$5OS+3T{x�$ 2 )�T 0 $�pz�R)�p*O?$5+ 2 $�8F+�) 0 $9T{x�7=+�T{+3p*O?$ � � 8FO?${¤J$98V7 � T � O?+�8(+3p*O?&S+$98 0 $9p��R)�p*O?$*8�8Ssz$cYZ+���p � Y�$98�Tt) � O?$cp � O � Y 2 T{+3p*OS+�!JºX8FO 2 7=+ � +y8RO 2 6�+37"+ 0 +���x�$9&�+ ¿ +3T{x�7"$ � '*)�+ 2 7=/9)�¤3TÂpUs�$x�$�Y�+-8F+�& 2 7=O?${+JpUsz$ 2 7=O?$ 2 $5T{+y8FT{$5O?+�T5x�$U! 4�2 8�x 2 7 2�» & 2 8VYZ+-£(& � 8ROS$zO?+�7=+y8 :��8��:&+������ �$ + �± y����� E

� � � T5x�$98S8&� » +37�'9)U+($tT{+�8?T{$ 2 OS& � 6�)ZOS$-x�+3&?O?+3p�r 2 +�pUsz$tx�+�&FOS+3pUr 2-2 $-T{+y8FT{$t8?)��R+ � O?$ �Z8 � Tt)�7=�O 2 pU+ 2 T{+�p9OS+\+I8F$96 2 T{+�8?T 2 &S+37 2 r�sz$U!"!"! 4 sz$t¤�x�$*8?8&� » +37 � 0 $�T +3|}+ � O?$ � 0 $�p 0 +36�+3&]p*)Up 032 '9)U+ 2T5+y8FT 2{0 $ � 8 2 8F+F� 2 +Hp�sz$�8F+F� 2 ! ��!

£ ,±+ � Y 2 º ¿�0 7")U8?s�$tYZ$5§¨+ � $�YZ+��Up�+J'*)�+\)�TÀ+�7=+�T{+3p*O?$tY�+�)�TÀ)Up �=» +3&D8F$-Y�+ » +�x�+�&FOS+3p 0 +3& 2 )�T 0 $�p��R)�p*OS$$�) 2 $ 8?+3) 0 $9T{x�7=+�T{+3p*O?$ �%�F� '*)�+ 2 )�p � s�$ YZ+ )UT 0 $�p��R)�p*OS$ 0 $9T 8?+3) 0 $�T{x�7"+3T{+3p*OS$ YZ+���p�+¨O?$ZYZ$ $)�p �"» +�&S8?$5+Hx�$9&FO 2 p*O?$5)�T 2�2 �U&ST 2 r�sz$�Y�+ » +H8?+3& » +3&DY 2 YZ+ � & 2 $9)B| 2 7"8 2 �Zp�sz${+ ¿ � 8ROS+IT{+ � ${OS+3&ST5$�!º ¿ � 8ROS+t)�T 2 &S+37 2 r�sz${+3p*OS&?+�$�x�+3& 2 r���+y8(8F$96�&S+ 0 $9p��R)�p*O?$*8(+t$9x�+�& 2 r���+�8(7=.9/ �1032 8\Y 2 ��7=/9+36�& 2 �V$*$97=+ 2 p 2 �

+�8FO 2 &?+�7 2 r3s�$ÁpUsz$ ¤ 290 � YZ+3p*O 2 7l! Q�+A�B+

C&?+�x�&?+y8F+�p*O 2 &S+3T »�2 & � � » + � 8H6�$�$97=+ 2 p 2 8��

∩�∪+

A|}$�&S+3T8?)�6U8FO � O?) �1YZ$98�x�$�&�� ���;+ ����� �U&S+�8?x�+ 0 O �=»�2 T5+�p*O?+��

∅|}$9&�+y'9) �=»�2 7"+3p*OS+ O�������� ��+

X|}$�&�������� � ��� ��O?$ZY 2 8

2 8�xU&?$9x�& � +�Y 2 YZ+y8VY 2 � 2 6�+37 2 KZ!" IO?$�&Sp 2 T��q8?+�x�&S$�xU& � +�Y 2 YZ+�8]6U�98 �1032 8 :f2�¿ � $�T 2 8 E Y 2 �z7"/�+36U& 2 �V$*$97=+ 2 p 2 !«(T 2 x�&S$�x�& � +�Y 2 YZ+ � p*O?+�&?+y8?8 2 p*O?+JY 2 �z7"/�+�6�& 2 �V$�$�7"+ 2 p 2 ¤H${x�& � p 0 �"x � ${Y 2 Y�) 2 7 � Y 2 YZ+H'*)�+H+�8FO 2 6�+�7=+ 0 + I

! ��ef��k "�"�el� _ #f� n .A/c.De � .AV]e �� /sV 8�1�.NpAV KX k�/c.��*2U2�V . � .�F "� 2 "J"9m;> k02AV]2 " 1 �!F���oF ������� � ������� � ���� 2�/ #> 8 � : /cH�. > k�8B.�1�2 " / 9" , H�:I8;OG. >d �� : ,R2A/����F��!F ������� � ��� !������� �RF . e � .�aUM�2d/ #"9� VW:f. � : Yg�?> .iV 8O�p�VW8B1�.D1�.PpAV K� k�/c."� 2U2�V . � .�[£T5+3O 2 YZ+J+�8S'9)U+3&DY 2 Y 2 � 2 6�+37 2 K�!= Ix�$ZYZ+H8?+3&�$�6�O � Y 2�2 x�7 �"0�2 pUYZ$���8?+\+y8ROS+�xU& � p 0 �"x � $ O T{+�O 2 YZ+HY � &?+ � O 2

+ » �"0 +�� » +�&S8 2 !

�"�$# A ����<! i�7��6 / #32!$5M��6G��6º�T 0 $�p��R)�p*OS$98³p � O � Y�$98 2 OS& 2 pU8 � r3s�$ +3p*OS&?+ )�T +37"+3T{+3p*OS$ x�+3&?O?+3p 0 +3& 2 )UT 0 $�pz�R)�p*O?$ $�) pUs�$ $ 0 $9&?&S+2 6U&?)�x�O 2 T{+�p9OS+�� � 8ROS$B¤�� )UT +37"+3T{+3p*O?$cx�+3&?O?+3p 0 +5$�)ÁpUs�$ 2 )�T 0 $9p��R)�p*O?$³+�x�$�p*OS$ �Up 2 7�!IºXT 0 $9p��R)�p*O?$*8p�+�6�)�7"$98?$98�+�8FO 2 O?& 2 pU8 � r�sz$�$ 0 $�&S&?+tY�+�|}$9&?T 2 /�& 2 YZ) 2 7l!Jº�T 0 $�pz�R)�p*O?$98�p�+�6�)�7"$98?$98 2 |})�p�r3sz$;YZ+ � p 0 7")U8?s�$O?+�TW)�T 2 YZ+��Up � r3s�${Y � |}+�&?+�p9OS+-Y 2 '*)�+�7 2 x 2 & 2{0 $�p��R)�p*OS$98�p �=O � YZ$98�!X£�8�|}&?$9p*O?+ � & 2 8V+3p*O?&S+J$98 0 $�p��R)Up9OS$98�pUs�$8Ssz$5p � O � Y 2 T{+3p*O?+HYZ+��Up � Y 2 8�+J)�TW+37"+3T{+3p*O?${x�$ZYZ+Jx�+�&FOS+3p 0 +3& 0 $9T¢)�T 0 +3&?O?${/�& 2 ) 2 )�T 0 $�pz�R)�p*O?$U! º]8ROS+/�& 2 )³x�$ZYZ+ »�2 & �12 &�+3p*OS&?+J@3+�&?${+J)�T � p 0 7=)�8 �"» +9!� 2 T{+�8?T 2 T 2 p�+ � & 2 '*)�+ )UT 0 $�p��R)Up9OS$ 0 7"�98?8 �"0 $ $�) p�� O � YZ$�x�$ZYZ+ 8F+�&³YZ+��Up � YZ$ x�+37 2 8?) 2 |})�pUr3s�$

032 & 290 OS+3& �"8FO �1032 �z)�T 0 $9p��R)�p*O?$�p�+36U)�7=$*8F$ :��L: 4�45 �645��9 9 �� y���9u E x�$ZYZ+J8?+3&]Y�+��Up � YZ$�x�+�7 2 8F) 2 |})�pUr�sz$�Y�+� p 0 7")U8Ssz$

µ(.) ∈ [0, 1]!�oR8ROS$�8 � /9p � � 032 '*)�+�)�T +37"+3T{+3p*O?$tx�$ZYZ+J8F+�&XT{+3T�6�&S$ 2 x�+3p 2 8�x 2 & 0 �"2 7=T{+�p9OS+\YZ+�)�T

0 $9p��R)�p*O?$�! � 2 T{$98 0 $�pU8 � Y�+3& 2 &J$;+ ¿ +3T{x�7"$¨YZ$ 0 $�pz�R)�p*O?$ÁY 2 8Jx�+y8?8?$ 2 8 2 7 O 2 8�!�£ª|})�pUr3s�$¨YZ+ � p 0 7")U8?s�$³pU$0 $9p��R)�p*O?$�YZ+ � pUY �"» �"Y�)�$98 2 7 OS$98�x�$�Y�+���x�$�&w+ ¿ +�T{x�7=$ ��O?+�& 2 |}$�&ST 2 Y 2�> � /�)�& 2 K�! K�!

µ

Altura

1,0

1,70

0,5

1,91,8

> � /�)�& 2 KZ!LK I �\+��Up � r3s�$5pU+36�)�7"$98 2 YZ$ 0 $9p��R)�p*O?${p�+�6�)�7"$98?${Y 2 8�x�+�8S8F$ 2 8 2 7=O 2 83!4 +y8ROS+ 032 8?$J)�T 2 x�+y8?8?$ 2J0 $9T 2 7=O?)U& 2 �� ����T x�+3&?O?+�p 0 + 2 $ 0 $�pz�R)�p*O?$98 Y�$98 2 7=O?$*8 0 $9T /�& 2 )5YZ+ � p 0 7")U8?s�$

YZ+-� � ut+ 2 '9)U+JO?+3T¹ �� ����TWx�+3&?O?+�p 0 + 2 $ 0 $�p��R)�p*OS$ 0 $�T¢/9& 2 ) � �U!

Page 16: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�� � ���`� ��� � ���R� � � � ���� � � � ���� � ���`cb9d%��ef���U� _ #"¶ �����������À��b¦ ���k�°l��hy��� b�� ²t��� ���������h ´¨b*µ���°f��h���h � % ���?� aUM�2 1 8 � H�V �5" M�2 1 �?> H�2 � )9�?� :&2 �! k � Vb2 " 2

AY 1 B^�� 8B1�. � 2 "� �� �?� 8;O / " 2 1 1�8 " H � / " 2 "� �� 1�2 H�.�/c.�H#: /98 � .�1�.-, V]. ���?� aGM�2

µ(.) : X → [0, 1]e �� J> .�, 8B.PH�.�1�. V �>d �� :&2l1

X �> �?>L���?>d /c2-/ .AV � 2-8 � : /9OG.AV]2

[0, 1][ n .�/0. �?>

,�.�/9:I8BH � V].A/ V #>l �� :&2GF . ���?� aUM�27/ ,o/ 9"� �� :&.72 K /c. � 1 8 � H�V �5" M�2D1�2 V �>l #� :f2 � 27H�2 � )9�?� :&2�[> )UpUr���+�8�YZ+ � p 0 7=)U8Ssz$³8?s�$�|}+3&S& 2 T{+3p*O 2 8wT 2 O?+�T���O �1032 8(8 � T{x�7"+�8�x 2 & 2 � pUY �1032 & 2 x 2 &FO �"0 � x 2 r3s�$�+3TÂ)�T

YZ+3O?+3&ST � p 2 YZ$ 0 $�p��R)Up9OS$ +³x 2 & 2 T{$ZYZ+37 2 &�$ 8 � /9p � � 032 Y�$ YZ$98t&?.�O?)�7"$98 2 8S8F$ 0 �"2 YZ$98 2 $98 0 $�pz�R)�p*O?$98�! «\T 2'*)�+�8FOSs�$ � T{x�$9&FO 2 p*O?+ '9)U+ 2 x 2 &S+ 0 + p�+�8FO?+ � p�8RO 2 p*O?+ ¤ I 0 $�T{$�YZ+��Up � & 2 8B|})�pUr���+y8³YZ+ � p 0 7")U8Ssz$ G º�7 2 8x�$�Y�+3T &S+3x�&S+�8?+3p*O 2 & 2 T 2 p�+ � & 2 8F)�6��R+�O �"»�2 x�+37 2 '*) 2 7�)UT � pUY �"» �"Y�)�$³+�p*O?+3p�YZ+t)�T 2 YZ+�OS+3&ST � p 2 Y 2�0 7 2 8?8?+YZ+�$�6��R+�O?$*8�$�)5x�+y8?8?$ 2 83! []$�pU8 � Y�+3&S+���x�$�&%+ ¿ +3T{x�7"$ � 2 YZ+��Up � r3s�$JY�+V)UT 2 x�+�8S8F$ 2�2 7 O 2 T{$*8ROS& 2 Y 2 p 2J> � /�)�& 2KZ!LK���+�8FO 2 YZ+���p � r�sz$;x�$�Y�+{8?+3&J|}+ � O 2 YZ+�Y �"» +3&D8 2 8IT 2 p�+ � & 2 8I+3T |})�pUr�sz$¨YZ+ 032 & 290 O?+�&&�18FO �1032 8 ��/�+�$�/�&D��� 032 8 �+�OD�z& �"2 8���+�O 0 ! > )�p�r���+�8IYZ+ � p 0 7=)U8Ssz$³x�$ZYZ+�T 8F+�&�$96ZO � Y 2 8IYZ+5Y �=» +3&D8 2 8wT 2 p�+ � & 2 8�!\#%$9&�+ ¿ +�T{x�7=$ �U|})UpUr���+�8x�$�Y�+3TÀ8F+�& 0 & �12 Y 2 8 2 x 2 &?O � &]Y 2 8�x�+3& 0 +3x�r���+y8XYZ$*8�+�8?x�+ 0 �12 7 � 8RO 2 8X8?$�6U&?+w)�TÀYZ+3O?+�&?T � p 2 YZ$ 2 8S8F)�p*OS$U!���)ZO?& 2T 2 p�+ � & 2 ¤�$�6ZOS+3&V+�8FO 2 8�|})�p�r���+�8 2 x 2 &?O � &]YZ+IT{+�Y � Y 2 8]+y8RO 2 O �"8FO �1032 83!�ºX8FO 2 8]|})�p�r���+�8 �Z'9)U+(|}$�& 2 TÀYZ+��Up � Y 2 82 x�& � $�& � �*x�$ZYZ+3T�8F+�& 2 7=O?+3& 2 Y 2 8]x�$9&V&S+�8?)�7=O 2 YZ$98V$�6�O � YZ$98�YZ)�& 2 p*OS+�$*8XOS+�8FO?+y8wYZ$�8 � 8ROS+3T 2 p�$�'9) 2 7 2 7".�/ �"0�2p�+�6�)�7"$98 2 |}$ � +3T{x�&S+3/ 2 Y 2 !

� � T{x�$�&?O 2 p9OS+;&?+y8?8 2 7=O 2 &5'*)�+c|})�pUr3��+�8�YZ+ � p 0 7=)U8Ssz$ pUs�$ +y8RODsz$ &?+�7 290 � $�p 2 Y 2 8 0 $9T�x�&S$�6 2 6 � 7 � Y 2 YZ+�8�!��+�8FO?+HT{$ZYZ$58?+J)�T 2 x�+y8?8?$ 2 O?+�T¢)�T 2{2 7=O?)�& 2 � /9) 2 7 2 �� ��+37 2 x�+3&?O?+�p 0 + 2 $ 0 $�p��R)Up9OS$5Y 2 8�x�+�8S8F$ 2 8 2 7=O 2 80 $9T /�& 2 ) � � u � 8F+�/�)�pUY�$ 2 p�$98S8 2 YZ+��Up � r�sz$U!HoR8FO?$cpUs�$³8 � /9p � � 032 '*)�+ 2 x�&?$96 2 6 � 7 � Y 2 YZ+5YZ+�+�p 0 $9p*O?& 2 &\pU$)�p �"» +�&S8?${Y 2 8�x�+�8S8F$ 2 8 2 7"/�)U¤3T 0 $�T 2 7 OS)�& 2 � �{8?+R� 2 � /9) 2 7 2 � � u�!��+ 290 $9&SYZ$ 0 $9T¢+�8FO 2 YZ+��Up � r3s�$ 0 $�p��R)Up9OS$98�p�+�6�)�7"$98?$98�x�+3&ST � OS+3TW$*8w8F+�/�) � p*O?+y8XO � x�$98\YZ+ � p 0 7")U8?s�$JI ���k�°l��hy��� k9� ��$���mU� � )�T +37"+3T{+�p9OS$�x�$ZYZ+Áx�+3&?O?+�p 0 +�& 2 )�T 0 $�pz�R)�p*O?$ 0 $9T~)�T 0 +�&FOS$ /9& 2 ) Y�+

� p 0 7")U8Ssz$U! º]8ROS+³O � x�$�YZ+¨YZ+��Up � r�sz$ x�+3&ST � OS+ � p*O?+�&?xU&?+3O 2 & 2 � p 0 7=)U8Ssz$ +3T )UT 0 $9p��R)�p*O?$ 0 $9T5$ O?+�pUYZ$ $8 � /9p � � 032 Y�${'*)�+ 2 7"/�)�pU8(+37"+3T{+3p*O?$*8(YZ$ 0 $�p��R)�p*OS$�8?s�$�T 2 � 8(&?+�x�&?+y8F+�p*O 2 O �=» $98�Y 2 � YZ¤ �12 /9+3& 2 7±YZ$ 0 $�p��R)�p*OS$YZ$;'*)�+�$9)ZO?&S$98�!J#%$�&I+ ¿ +3T{x�7"$ ��p�$ 032 8F$cYZ$ 0 $�p��R)�p*OS$cY 2 8Ix�+y8?8?$ 2 8 2 7=O 2 8\)UT 2 x�+�8S8F$ 2 YZ+ 2 7=O?)�& 2 T 2 � $9&$�) � /�) 2 7 2 �L�z�5T Ys> .�8 " / ,o/ 9"� �� :&.�:I8;OG.N1�2DH�2 � H 8;:f2N1 , #"�" 24.D.�VW:&.�YZ$�'*)�+J)�T 2 x�+�8S8F$ 2 Y�+� �� �9utTc! ���k�°l��hy���¨b�����e ��b*��hy��hHk�����y��������h � 0 $�Tª+�8FO?+(O � x�$tY�+\YZ+���p � r�sz$-)UT +�7=+�T5+�p*O?$-x�$ZYZ+�8?+3&�T{+�Tt6�&S$x 2 & 0 �"2 7�Y�+wT 2 � 8XYZ+�)�T 0 $�p��R)�p*OS$U! �\+�x�+�pUYZ+3p�YZ$tYZ$J/�& 2 ){YZ+ � p 0 7")U8Ssz$Jp�$98 0 $�p��R)Up9OS$98%¤wx�$98S8 � » +�7 0 $9p 0 7") � &

2 '*) 2 7 0 $�p��R)�p*OS$5${+�7=+�T5+�p*O?${+�8FOS�5T 2 � 8V|}$�&?O?+�T5+�p*O?+J7 � / 2 YZ$U!�#%$�&w+ ¿ +�T{x�7=$ � 0 $9pU8 � YZ+�&?+ 2{> � /�)U& 2 KZ! P5'9)U+T{$98FO?& 2�2 8 YZ+���p � r3��+�8 YZ+(YZ$ � 8 0 $�p��R)Up9OS$98 YZ+VOS+3T{x�+�& 2 O?)�& 2 8�I±T{¤�Y �"2 8 +V|}& �12 8�!�«\T 2 OS+3T{x�+3& 2 O?)U& 2 YZ+

20o [x�+3&?O?+3p 0 + 2 $98 0 $�p��R)�p*OS$98XY 2 8%OS+3T{x�+�& 2 O?)�& 2 8�T5¤yY �12 8�+(|}& �12 8�! 4 $-+3p*O 2 p*O?$ ��+37 2 x�+3&?O?+�p 0 +(T 2 � 8%|}$9&FOS+3T{+3p*OS+2 $ 0 $�pz�R)�p*O?$BY 2 8wOS+3T{x�+�& 2 O?)�& 2 8w|}& �12 8 : /�& 2 )ÁY�+ � p 0 7")U8Ssz$�� � � ��� E YZ$B'*)�+ 2 $�YZ+tO?+3T{x�+3& 2 OS)�& 2 8(T{¤�Y �"2 8: /9& 2 )cY�+ � p 0 7=)U8Ssz$�� � � P�P E � 0 $9T5$5+�8FOS� � pUY �1032 YZ${p 2�> � /9)�& 2 KZ! P�!

5 10 15 20 25 30 35 40 45 50

médiafria

0

1.0

temp

0.66

0.33

> � /9)�& 2 KZ! P I ��+3T{x�+3& 2 O?)U& 2 � p 0 7") �1Y 2 +3T¾Y�$ � 8 0 $�pz�R)�p*O?$98wY � |})U8?$98 2 ${T5+y8FT{$5O?+�T{x�$�!4\2 7 � O?+�& 2 O?)�& 2\2 x 2 &S+ 0 +3T¦Y � |}+3&S+3p*O?+y8�T 2 p�+ � & 2 8�Y�+]&S+3x�&S+�8?+3p*O 2 &±)�T 0 $�pz�R)�p*O?$\pU+36�)�7"$98?$U!�«(T 0 $�p��R)�p*OS$

p�+�6�)�7"$98?$tx�$�Y�+�8?+3&�YZ+3p�$�O 2 YZ$ 0 $�T{$-)�T 0 $�p��R)�p*OS$t$�&DYZ+�p 2 YZ$�YZ+�x 2 &S+�8 �*8?+3pUY�$�'*)�+\$�x�& � T{+ � &S$t+37"+3T{+�p9OS$YZ$(x 2 &�¤X$(+37"+3T{+3p*OS$�YZ$ 0 $�p��R)Up9OS$(x�&S$�x�& �12 T{+3p*O?+�Y � OS$(+X$\8?+3/9)�pUYZ$w$\/�& 2 )-YZ+ � p 0 7=)U8Ssz$�YZ+�8FO?+�+�7=+�T{+3p*O?$(pU$0 $9p��R)�p*O?$�!X[]$�pU8 � Y�+3&S+3T{$98V+3p*ODsz$5)�T 250 $�7"+�r�sz${YZ+J$�6Z�R+3O?$98 � pUY �1032 YZ$98�/9+3p�+�& �1032 T5+�p*O?+Ix�$9&

X!�#%$9&FO 2 p*O?$ �

)�T 0 $�pz�R)�p*O?${p�+36U)�7=$*8F$A+�T

Xx�$ZYZ+-8?+3&(YZ+y8 0 & � O?$ 0 $9T{$5)UT 0 $9p��R)�p*O?$�YZ+Jx 2 &?+y8�$�&DYZ+3p 2 YZ$*8MI

A = {(x, µA(x)) | x ∈ X}

Page 17: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R� �R� � � �� � ���P ��� � � �� �� 9 $�p�YZ+

µA(x)¤]/�& 2 )-YZ+ � p 0 7=)�8?s�$\YZ$(+�7=+�T5+�p*O?$\pU$ 0 $9p��R)�p*O?$ :

µA(x) ∈ [0, 1]E !�# 2 & 2 8 � T{x�7 � � 0�2 & 2 p�$zO 2 r�sz$ �p�$9&?T 2 7"T{+3p*O?+JpUs�$58?+H7 � 8FO 2 $98�+�7=+�T{+3p*O?$*8 0 $�TW/�& 2 )³Y�+ � p 0 7=)�8?s�$ � /�) 2 7 2 @3+�&?$�!

��� b � "�°l�³_ #1® � «\T x�&S$z|}+y8?8?$�&�&?+y8F$97 » +3) 0 7 2 8?8 � � 032 & 0 $�T YZ+��Up � r���+�8�p�+�6�)�7"$98 2 8�$98%8?+3)U8 2 7=)Up�$98%YZ+ 2�0 $�&DYZ$0 $9T 2 8�p�$�O 2 8�'*)�+�+37"+�8�$�6�O �"» +3& 2 T +�T 8F) 2 Y � 8 0 � xU7 � p 2 ! � 2 T{$*8 2 8?8?)�T � & '*)�+V$ 0 $�p��R)Up9OS$IY�+VpU$zO 2 8 0 $�T{+�r 2+3T��5+ »�2 ��2 OS¤5 y�{8F+�T5xU&?+ 2 )UT5+�p*O 2 p�YZ$5Y�+-� �Lut+�T¾� � utx�$9p*O?$98 �Z$�)¨8?+R� 2

X = {0, 0, 5, 1, . . . , 9, 5, 10}

£\8S8 � T ��$ 0 $�pz�R)�p*O?$ p�+36�)U7=$*8F$AYZ+ 2 7")�p�$*8t6�$�p�8�x�$ZYZ+3& �12 8?+3&{YZ+��Up � YZ$ �X8?+3/�)UpUYZ$ 2 x�+�& 0 +3x�r3s�$ YZ+y8ROS+

x�&S$z|}+y8?8?$�& � 0 $�T{$

A = {(6, 5, 0, 25), (7, 0, 5), (7, 5, 0, 75), (8, 1), (8, 5, 0, 75), (9, 0, 5), (9, 5, 0, 25)}

��6U8?+3& »�2 &\'*)�+ ��$cx�&S$z|}+y8?8?$�& �Ux 2 & 2 $BT{+�8?T{$BO � x�$³YZ+{Y 2 YZ$ :aluno

E x�$ZYZ+{YZ+���p � &I$�)�O?&S$98 0 $�pz�R)�p*O?$980 $9T{$�x�$�&�+ ¿ +�T5xU7=$NI�+ ¿�0 +37"+3p*OS+�8 �*&S) � pU8 �*T{¤yY � $*8��Z+3O 0 !�º]8ROS+�8 0 $9p��R)�p*O?$*8Vx�$ZYZ+�TW8F+H8?$�6�&S+3x�$�& �*+�x�$�&?O 2 p9OS$)�T 2 7")�p�${x�$ZYZ+-8?+3& 0 7 2 8?8 � � 032 YZ$t+�T¾YZ$ � 8 0 $�p��R)�p*OS$98 2 ${T{+�8?T{$5O?+3T{x�$U!«(T 2 p�$zO 2 r3s�$t'*)�+I¤\)U8 2 Y 2 T 2 � 8�|}&S+�'LK�+�p9OS+3T{+3p*OS+ � pUY �103252 )�p � sz$5YZ+(OS$ZYZ$98]+37"+3T{+3p*OS$98V'9)U+\|}$9&?T 2 T

$ 0 $9p��R)�p*O?$�YZ+J|}$9&?T 2 + ¿ x�7 � 0 � O 2 �Z¤ 2 8?+3/�) � p*O?+

A =∑

xi∈X

= µA(xi)/xi: K�! � E

� 8 � p 2 7�YZ+ 8?$�T 2 OS.�& � $ p 2 +�'*) 2 r3s�$ K�! ��¤ $ � pUY �"0�2 YZ$9&�YZ+Á)�p � sz$�!º]8RO 2 +�'*) 2 r�sz$ 2 8S8?)�T{+Á'9)U+¨$)�p �"» +�&S8?${YZ+tY � 8 0 )�&D8F$

X¤-Y � 8 0 &S+�OS$U! 4 +�8FO 2 p�$�O 2 r�sz$ �*$ 0 $�p��R)Up9OS$

AYZ+H6�$�pU8 2 7=)�pU$98wx�$ZYZ+-8F+�&(YZ+���p � Y�$0 $9T{$JI

A = 6, 5/0, 25 + 7/0, 5 + 7, 5/0, 75 + 8/1 + 8, 5/0, 75 + 9/0, 5 + 9, 5/0, 25

# 2 & 2 )UTW)�p �"» +�&S8?${YZ+-Y � 8 0 )�&S8?$ 0 $�p*O �=p�)�$ � 2 +y'*) 2 r�sz$�KZ! �t¤H&S+3+y8 0 & � O 250 $9T5$

A =

X

µA(x)/x: KZ! � E

«(T 0 $9p��R)�p*O?$¨pU+36�)�7"$98?$AO 2 Tt6�¤3T x�$�Y�+B8F+�&tYZ+y8 0 & � O?$¨Y � &?+3O 2 T5+�p*O?+�x�$�&t8?) 2 |})�p�r3sz$ÁY�+ � p 0 7")U8?s�$

µA(x)!

��� b � "�°l� _ #=¶ � «(T xU&?$�|}+�8S8F$9&JY�+�+�YZ) 0�2 r�sz$³| �"8 �"0�2 &?+y8F$97 » +3) | 2 @3+�&H)�T 2¨0 7 2 8S8 � � 0�2 r3s�$³YZ+�8?+3)�8 2 7")�p�$*8x�$�& 2 7 OS)�& 2 ��'*)�+{|}$98S8?+�T 2 � 8tYZ+�O 2 7"¡ 2 Y 2 YZ$Á'*)�+ 2 � 7")U8ROS& 2 Y 2 p 2¨> � /9)�& 2 K�! K�!B£À8?) 2¨0 7 2 8S8 � � 0�2 r3s�$³+y8ROD�� 7=)U8FO?& 2 Y 2 p 2 > � /9)�& 2 K�! � + � p 0 7=) �(2 8{8F+�/�) � p*O?+y8 0�2 OS+3/�$9& �12 8tx 2 & 2 +y8RO 2 OS)�& 2 IB6 2 � ¿�2 �XT{¤�Y �12 + 2 7 O 2 ! £+�'*) 2 r3s�$ 0�2 & 2�0 OS+3& �"8FO �1032 YZ$ 0 $9p��R)�p*O?$�p�+�6�)�7"$98 #" :f.A: � /c. >dY 1�8B.H¤

µmedia(altura) =

0 se altura ≤ 1, 505 · altura − 7, 5 se 1, 50 ≤ altura < 1, 70−5 · altura + 9, 5 se 1, 70 ≤ altura < 1, 900 se altura ≥ 1, 90

��6U8?+3& »�2 &('*)�+-)�T � pUY �"» �"YZ)U$ 0 $9T 2 7 OS)�& 2 �� ����+y8ROD� � p 0 7") �"Y�$�+3T YZ$ � 8 0 $�pz�R)�p*O?$98�!wº�7"+tx�+3&?O?+�p 0 + 2 $0 $9p��R)�p*O?$�YZ+H+y8RO 2 OS)�& 2 T{¤�Y �12�0 $9T¢/�& 2 )c��!Lu�+ 2 $ 0 $�pz�R)�p*O?$�YZ+H+�8FO 2 O?)�& 2 6 2 �=¿�2�0 $�T¢/9& 2 )c� � u�!��� b � "�°l� _ #1n � []$9p��R)�p*O?$

AYZ+HpQD�T{+3&S$98V&S+ 2 � 8Vx�+3&?O?$�YZ${pQD�T{+3&S$; ���!

A = {(x, µA(x)) | µA(x) =1

1 + (x − 10)2}

Page 18: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

yK � ���`� ��� � ���R� � � � ���� � � � ���� � ���

µ

Altura

1,0

1,70 1,91,5

altomédiabaixo

> � /�)�& 2 K�! �NIX[]$�p��R)�p*OS$ 4 +36�)U7=$*8F${x 2 & 2 ºX8FO 2 O?)�& 2 §¨¤�Y �12 !�����-��� �-,-)-������� � ��� )�# 2�,�����'£�T 2 � $9&�x 2 &?O?+�Y 2 8�|})�pUr3�*+y8%YZ+ � p 0 7=)�8?s�$\)ZO � 7 � @ 2 Y 2 8�8Ssz$IYZ$�O � x�$I)�p � T{$�Y 2 7�!�ºX8FO?+VO � x�$IY�+X|})�p�r3sz$I/ 2 & 2 p*O?+'*)�+ 2-»�2 & � � » +�7ZO?+�&S�t8F$9T5+�p*O?+ � p��U)�¼�p 0 �12 V]2UH�.�V±pU$-x�&S$�6�7"+3T 2 !�#%$�&]+ ¿ +3T{x�7"$ � 0 $9pU8 � YZ+�&?+w$ 032 8?$-Y 2 |})�pUr3s�$YZ+�YZ+��Up � r3s�$;Y�$ 0 $�pz�R)�p*O?$;+�8FO 2 O?)�& 2 T5¤yY �12 :f> � /9)�& 2 K�! � E !c[ 2 8F$c+�8FO 2 |})UpUr3s�$³|}$*8?8?+�6 � T{$�Y 2 7%O?+�&&� 2 T{$*8YZ) 2 8 2 7=O?)�& 2 8 0 $�T{$ 2 7=O?)�& 2 8�T{¤�Y �"2 x 2 YZ&Ss�$U!`cb9d%��ef���U� _ # n �������������³��e � �±��mU°��?% ���?� aGM�2D1 8 � H#V �5" M�2 Y H�2 �R" 8B1 /c.�1�. �?� 8 > 2U1�.AV�,2 � H�2 � O 35.+/"�

∀x1, x2 ∈ X, ∀λ ∈ [0, 1] : µ(λx1 + (1 − λ)x2) ≥ min(µ(x1), µ(x2)}

£ > � /9)�& 2 KZ!Lu � 7")U8FO?& 2 )�T + ¿ +�T5xU7=$¨YZ+{|})�p�r3sz$;)�p � T{$ZY 2 7 $9) 0 $�p » + ¿�2 +�YZ+{)�T 2 |})�pUr�sz$;6 � T5$ZY 2 7l![]$�p�8 � YZ+3&S+ 2 |})�p�r3sz$ÁY�+ � p 0 7")U8Ssz$¨)�p � T{$ZY 2 7

µA(x)'*)�+ � p�Y �"0�2 '*) 2 p�YZ$¨)�T +37"+3T{+3p*O?$

xx�+3&?O?+3p 0 + 2 $0 $9p��R)�p*O?$

A!\[ 2 8?$

µA(x)8F+F� 2 T 2 � $9&\'*)�+

µA(y)�Ux�$�Y�+3T{$98\+3p*OSs�$ � p*O?+�&?xU&?+3O 2 &('*)�+

x+�8FOS�{T 2 � 8wx�+3&?O?$

Y 2 YZ+��Up � r3sz$ � YZ+ 2 7±YZ$ 0 $9p��R)�p*O?$AYZ$�'*)�+

y!

0

1 1

0

(a) (b)

> � /�)�& 2 KZ!Lu I :f2*E�> )�p�r3sz${)�p � T{$ZY 2 7 + : 6 E�> )UpUr3s�$56 � ��T{$�Y 2 7[]$�pz�R)�p*O?$98 4 +�6�)�7"$98?$98�x�$ZYZ+3T¢8?+3&VYZ+3pU$zO 2 YZ$98Xx�$9&VY � |}+�&?+�p9OS+�8X|})�pUr3�*+y8�� 2 7"/�)UT 2 8XYZ+�7 2 8�O 2 T�6�¤�Tª)�8 2 �

Y 2 8�+�T 0 $9p��R)�p*O?$*8 0 7"�98?8 �"0 $98 0 $�T 2 T{+�8?T 2 8?+3T�½zp*O �1032 !�oR8FO?$I8F+�& » +Xx 2 & 2 � 7")U8FO?& 2 &�$�| 2 OS$IY�+V'*)�+ 0 $9p��R)�p*O?$*8p�+�6�)�7"$98?$98Vx�$ZYZ+�T¾YZ+y8 0 &?+ » +�& 0 $�p��R)Up9OS$98 0 7"�98?8 �"0 $98]O 2 Tt6�¤3T ��+I|}$�&ST 2 T¢)UT¾8 � 8FO?+�T 2 T 2 � 8�/�+�& 2 7l!`cb9d%��ef���U� _ # � ����������� �±el��$���°lmZ� � % ���?� aUM�2 1 8 � H�V �5" M�2 Y H0j�. > .�1�. 1 " 8 �XK�� V].�/ , " 8 �XK V :f2 � /e � . � 1�2

Sing(x) =

{

1 se x = x0 se x 6= x

Page 19: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R� �R� � � �� � ���P ��� � � �� �� yP£ > � /9)�& 2 K�! � � 7=)U8FO?& 2 )�TW+ ¿ +3T{x�7"${Y�+J|})�pUr3s�$�8 � p�/�)U7 2 &�! «(T¾+ ¿ +3T{x�7"$�YZ+J)U8?$5|})�pUr���+y8(8 � p�/9)�7 2 &?+y8V¤&S+3x�&S+�8?+3p*O 2 &]pQD�T{+�&?$*8�p�$�8F+�p9O � Y�$ 0 7"�98?8 �10 $U!

µ

x

1,0

x

> � /�)�& 2 KZ! � I > )�p�r3sz$�Q � p�/9)�7 2 &�!

`cb9d%��ef���U� _ # � ��������g±b�ht²�°fjUh�hyefk9m�h � � > . ���?� aUM�2-H�V]p "0" 8BH�. Y 1 _^`� 8B1�.�,R2A/g1�2A8 " ,�.�/<* >d :_/02 "x1

x2F

O�.�V]2A/s8 � 8BH�8B.�V �^�� .�V 1�2 K /c. � 1 8 � H�V �5" M�2P8 KA� .AV . � [ % "9� .P1 _^`� 8(aUM�2 Y

C(x) =

0 x < x1

1 x1 ≤ x ≤ x2

0 x > x2

QZs�$\|})�p�r���+�8 +�T{x�&?+�/ 2 Y 2 8�x 2 & 2 YZ+���p � r�sz$HYZ+ 0 $�p��R)�p*OS$98 0 71��8S8 �"0 $98�+ 2J> � /�)U& 2 KZ!�� � 7")U8FO?& 2 )�T 2 |})�pUr3s�$0 71��8S8 �"0�2 !�«\T 2 |})�pUr3s�$ 0 71��8S8 �1032 &S+3x�&S+�8?+3p*O 2 )�T 0 $�pz�R)�p*O?$Hp�$H8?+3p*O � YZ$ 0 71��8S8 �"0 $U!���6U8?+3& »�2 & '*)�+(8?+w$Ix�$�p*OS$Y 2 Y � &S+ � O 2 ¤�YZ+y8F7"$ 032 Y�$Hx 2 & 2 $ � p��Up � O?$ 2 |})�p�r3sz$�8F+wOS$�&Sp 2 8?+3T{+37"¡ 2 p9OS+ O � 7")U8ROS& 2 Y 2 p 2t> � /�)U& 2 K�!= 9!�º]8RO 2¤ 2 YZ+���p � r�sz${p�$9&?T 2 7 YZ+ 0 $�p��R)Up9OS$98 0 7"�98?8 �"0 $98�! �¦T{+�8?T5${x�$ZYZ+-8F+�&�|}+ � O?$ 0 $�T¢${x�$�p*O?$�Y 2 +�8S'*)�+3&DY 2 !

x

1,0

x1 x2

C(x)

> � /�)�& 2 KZ!�� I > )UpUr3s�$ 0 7"�98?8 �1032 !

`cb9d%��ef���U� _ #�� ¯³b "���b�h�b�����mU����� � ef��b�mZ� ��% / ,o/ #"# �� :&.AaGM�2NVW8 �o .�/s:&. > k Y#> Y 1 _^`� 8B1�.s,R2A/i1�2A8 " ,�. 1/<* >d :_/02 "E : �> . "� fK�� 8 � : 1 _^`� 8(aUM�2

L(x) =

0 x < x1x−x1

x2−x1

x1 ≤ x ≤ x2

0 x > x2

ºX8FO?+-¤t)�T YZ$98\T 2 � 8\8 � T{x�7=+y8 0 $�p��R)�p*OS$98(pU+36�)�7"$98?$98�!�£ > � /9)�& 2 KZ! � � 7")U8FO?& 2 )�TÂ+ ¿ +3T{x�7"$�YZ+t|})�pUr3s�$7 � p�+ 2 & 0 &?+y8 0 +3p*O?+9! > )UpUr���+�8XYZ+�8FO?+�O � x�$Hp�$�&ST 2 7"T{+3p*O?+w8Ssz$I)U8 2 Y 2 8%x 2 & 2 &S+3x�&S+�8?+3p*O 2 & 0 $9p��R)�p*O?$*8%p�+36�)U7=$*8F$*8p 2 8�+ ¿ O?&S+3T � Y 2 YZ+y8wYZ${)�p �"» +�&S8?$5Y�+-Y � 8 0 )U&S8?$U!

Page 20: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

3� � ���`� ��� � ���R� � � � ���� � � � ���� � ���

x

1,0

x1 x2

L(x)

> � /�)U& 2 KZ! � I > )UpUr3s�$57 � p�+ 2 & 0 &S+�8 0 +3p*OS+�!

`cb9d%��ef���U� _ #�� ��������g±b�h � ��efmU��$���°lmZ��b�h �2% ���?� aGM�2D:I/98B. �5KA� V].A/%,�241 -"� /i/ ,!/ #"� #� :f.�1�.E,�2�/E:I/ct " O�. 1V]2A/ 9" �g2*,R2 � :&2lq 9" e �� /01�.

x1F`2*,R2 � :&2l2 � 1 2 K /c. � 1 8 � H#V �5" M�2dO�.�V �?>

x2 2*,�2 � :f2 > .�8 " ql1�8;/ 8;:f.

x3[ % "9� . e � .�aUM�27: �> . "# fK�� 8 � : `� 2A/ > .

Tri(x) =

0 x < x1x−x1

x2−x1x1 ≤ x ≤ x2

x3−xx3−x2

x2 ≤ x ≤ x3

0 x > x3

£\8-|})�pUr3�*+y8JO?& �12 p�/9)�7 2 &?+y8H8?s�$;+3T{x�&S+3/ 2 Y 2 8I|}&S+�'LK�+�p*O?+3T{+�p9OS+�YZ+ » � YZ$ 2 8 � T{x�7 �"0 � Y 2 YZ+�YZ+B&?+�x�&?+y8F+�p*O 2 r�sz$+�YZ+�)ZO � 7 � @ 2 r�sz$�!B£ > � /9)�& 2 K�! � � 7=)�8ROS& 2;0 $9T{$¨)�T 0 $�p��R)�p*OS$¨p�+36�)U7=$*8F$;¤�YZ+��Up � YZ$¨+3T¹O?+�&?T{$98HY�+�)�T 2|})�pUr�sz$5O?& �12 p�/9)�7 2 &�!

x

1,0

x1 x3

Tri(x)

x2

> � /�)�& 2 KZ! � I > )�p�r3sz$5OS& �12 p�/9)�7 2 &y!

`cb9d%��ef���U� _ #l��� �������9g±b�h � ��mL"Xb�����ef��m�efh �6% ���?� aGM�2J:I/c.�, �� 2�8B1�.AV Y :f. > k Y�> >-� 8;:f. �5" .�1�.�, V]. "%>l 9"01> . " /c. � ' #" 1�. ���?� aGM�2N:I/98B. �XK�� V].A/�[ % ���?� aGM�2P:_/c.�, �� 2�8B1�.�V ,R2U1 i"� /s/ ,!/ #"� #� :f.�1�. �5" . � 1�2 1f"� .0, �� . "��O�.�V]2A/ 9" [ % "9� . e � .AaGM�2P: #> . "� fKA� 8 � : �� 2�/ > .

Tra(x) =

0 x < x1x−x1

x2−x1

x1 ≤ x ≤ x2

1 x2 < x < x3x4−xx4−x3

x2 ≤ x ≤ x3

0 x > x3

£ > � /9)�& 2 K�!= y� � 7=)�8ROS& 2I0 $9T5$J)�T 0 $�pz�R)�p*O?$Jp�+36U)�7=$*8F$J¤wYZ+��Up � YZ$J+3T OS+3&ST{$98%YZ+�)�T 2 |})�pUr3s�$IO?& 2 x�+3@3$ � Y 2 7�!

Page 21: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R� �R� � � �� � ���P ��� � � �� �� �u

x

1,0

x1 x3

Tra(x)

x2 x4

> � /9)�& 2 KZ!" ��JI > )�pUr�sz$5O?& 2 x�+3@�$ � Y 2 7�!

`cb9d%��ef���U� _ #l��� �������9��� �±e%$ � ��el��b�� � > . ���?� aGM�2 " 8 KA> h�8B1 Y 1 _^`� 8B1�. �5" . � 1�2 :_/ct " ,�.�/<* >d :_/02 " �"� #� OG.AV]2�/ � 1 8 � H�V �5" M�2 ,

α/GF "� �� OG.AV]2�/ � 1 8 � H�V �5" M�2 ,

γ/ 2J,R2 � :&2N1 8 � � 35M�2-,

β/GF e �� dY 2J,�2 � :f2

2 � 1 27O�.�Vb2�/E1�. ���?� aGM�2N1 8 � H�V �5" M�27OG.AV � [���[ % "�" 8 > . ���?� aGM�2g,�2U1 E"� /s1 _^`� 8B1�.DH�2 > 2 �

S(x; α, β, γ) =

0 x ≤

2 ·(

x−αγ−α

)2

α ≤ x ≤ β

1 − 2 ·(

x−γγ−α

)2

β ≤ γ

1 x ≥ γ

: KZ!" �� E

£ > � /9)�& 2 K�!= 9 � 7=)�8ROS& 2 )�T¾+ ¿ +3T{x�7"$�YZ+J|})�pUr�sz$�8 � /�T{. � YZ+ 0 &?+y8 0 +3p*O?+9!

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Sig

moi

de

f(x)

> � /�)�& 2 KZ!" � I > )�pUr3s�$�8 � /�T{. � YZ+ 0 &S+�8 0 +�p9OS+�!«(T¸)U8?$ 0 $�Tt)UT YZ+y8RO 2 |})�pUr�sz$ 2 x 2 &S+ 0 +Á'*) 2 pUY�$ &?+�x�&S+�8?+3p*O 2 T5$*85|}&?+y'LK�¼3p 0 �"2 83!¦#%$9&B+ ¿ +�T5xU7=$ pU$

0 $9p��R)�p*O?$ �5"9� .�V >d �� : $�p�YZ+H$�YZ$�T �"p � ${¤J)�T 0 $�pz�R)�p*O?$�YZ+Jx�&S$�x�$�&Dr���+�8 :l> � /�)U& 2 KZ!" yK E !`cb9d%��ef���U� _ #l�Z_ �������9���¦i5b9�ym � � > . ���?� aGM�2 k :f. Y 1 _^`� 8B1�. H�2 > 1�2�8 " ,R.A/<* >l :_/c2 " Fi2 OG.AV]2�/ #>:&2�/ � 2 1�2 e � .AV . H � /9OG. Y H�2 �R" :_/ �?m 1�. ,

γ/ P�?> O�.�V]2A/Pe �� 8 � 1A8BH0. . >l :f.�1 1�. V].�/ K�� /c. 1�. H � /9O�. � 2

,�2 � :&2P1 8 � � 35M�2-,β/�[

% "�" 8 > . ���?� aGM�2J,R2U1 s"� /E1 B^�� 8B1�.NH�2 > 2 �

B(x; γ, β) =1

1 +(

x−γβ

)2

£ > � /9)�& 2 K�!= yP � 7=)�8ROS& 2 )�T¾+ ¿ +3T{x�7"$�YZ+J|})�pUr�sz${6�+3O 2 !ºX8FO 2 |})�pUr3s�$ 2 x 2 &S+ 0 +�'9) 2 pUYZ$ )U8 2 T{$98t$98HOS+3&ST{$98-pU+36�)�7"$98?$98 #> :&2�/ � 2���, /9:f2 1 �� �*.0,!/02�3�8 > .�1�. 1

>l #� : �X+3O 0 ! �Âx 2 &D½zT{+�OS&?$βY�+��Up�+ 2 |}$�&ST 2 Y 2 0 )�& »�2 +��]x�$9&�+ ¿ +3T{x�7"$ �]$ '9) 2 p*O?$ x�+�&FOS$ Y�$ »�2 7=$9&

0 +�p*O?& 2 7γ)�T »�2 7=$9&

x+y8ROD��!

Page 22: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�� � ���`� ��� � ���R� � � � ���� � � � ���� � ���

00

1

10.5

> � /�)�& 2 KZ!" yK IX[]$9p��R)�p*O?${p�+�6�)�7"$98?$5)�8F) 2 7=T{+�p9OS+�!

0

0.5

γγ−β γ+β

> � /�)�& 2 K�!= yP I > )UpUr3s�${�V+3O 2 !

��� b � "�°l� _ # � � � 0 $�pz�R)�p*O?$�YZ$98�pQD�T{+�&?$*8�&?+ 2 � 8]x�+3&?O?$�YZ$�pQD�T{+�&?$�u :f> � /�& 2 KZ!" 3� E !

C(x) =1

1 − (x − 5)2

�����-�� � ��� % ! ) '*2�'��*!�� �-� &('*)�+",-)/.0'*��� ��� ,-2�'*� '*�`cb9d%��ef���U� _ #l��® �±� "X������b ��b ����²���� y�����y��´¨b*µ%��°l��hy� ��� "9� ,R2A/9: 1 l�?> H�2 ��)9�?� :&2 �! k � V]2 " 2 Y�?> H�2 ��)9�?� :&2 � m :_8B1�2

A1 _^`� 8B1�2N1�. "� fK�� 8 � : J> . �! 8;/c. �

SA = {x ∈ X | µA(x) > 0}

«(T 0 $�p 0 + � O?$�&?+�7 290 � $�p 2 YZ$ 2 $ YZ+ 8?)�x�$�&?O?+ ¤ $ YZ+�8F)�x�$�&?O?+ 0 $9T5x 2�0 OS$U!W� 8?)�x�$9&FOS+ ¤ 0 $9T5x 2�0 OS$'*) 2 p�YZ$B8?+3);O 2 T 2 p�¡�$B¤tT{+3p�$9&�'*)�+-$B)�p �"» +3&D8?$BYZ+tY � 8 0 )�&D8F$B$�& � / � p 2 7 � 0 $�T{$�+y8ROD� � 7=)U8FO?& 2 Y�$�p 2�> � /�)�& 2KZ!" yu�!{[ 2 8F$ 2 |})�pUr3s�$¨YZ+ � p 0 7=)�8?s�$cp�sz$;8F+F� 2c0 $�T{x 290 O 2 � » +�&?+�T5$*8IT 2 � 8�O 2 &DYZ+�'9)U+ » ��& �12 8I&S+3/�& 2 8I8?+3&Dsz$2 O �=»�2 Y 2 8]x�$9& 0�2 Y 2 +3p*O?& 2 Y 2�032 )U8 2 pUYZ$t'*)�+�$58 � 8ROS+3T 2 8?+R� 2 8F$96�&?+ 032 &?&S+3/ 2 YZ$I+�$ � T{x�$�&?O 2 p9OS+ 0 $�p 0 + � O?$�Y�+2 &ST 2 @�+3p 2 T5+�p*O?$t+H&S+ 0 )�x�+3& 2 r3s�$�YZ+ 0 $�p�¡U+ 0 � T{+3p*O?${7"$ 032 7 x�$98S8 2 8?+3&wx�+3&DY � YZ$U!

Page 23: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R� �R� � � �� � ���P ��� � � �� �� ��

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

1/(1+(x-5)*(x-5))

> � /9)�& 2 K�!= ��JI > )�p�r3sz${x�+3&?O?$�YZ+tu�!

00

1

0

1

S S

Suporte compactoSuporte nao compacto

> � /�)U& 2 KZ!" yu I QZ)�x�$9&FOS+ 0 $�T{x 290 OS$5+JpUs�$ 0 $�T{x 290 OS$U!

��� b � "�°l��_ # � � []$9pU8 � YZ+�& 2 pUYZ$�$³+ ¿ +3T{x�7"$¨KZ! �Bx�$�Y�+3T{$98 » +�& � � 0�2 &�'*)�+5$³8?)�x�$�&?O?+{YZ$ 0 $�pz�R)�p*O?$A¤�$0 $9p��R)�p*O?$

SA = {6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5}!

`cb9d%��ef���U� _ #l�*¶¦²���� y�����y� k9�U����bα� � H�2 � )9�?� :&2 ��m :I8B1�271 s V #>l #� :f2 " e �� , /9: �� H #> .�27H�2 � )9�?� :&2

�! k � V]2 " 2A.�: Y , V]2 >d �� 2 " 2 K /c. �

αY H0j�. > .�1�2 1 H�2 ��)9�?� :&2PH�2A/9:

α[ � H�2 ��)9�?� :&2PH�2A/9:

αY- �� :&M�2

1 _^`� 8B1�2DH�2 > 2 �Aα = {x ∈ X | µA(x) ≥ α}

: KZ!" � EA,

α = {x ∈ X | µA(x) > α}: KZ!" yK E

� 0 $�p��R)Up9OS$A, ¤ 0 ¡ 2 T 2 YZ${YZ+ 0 $9p��R)�p*O?$ 0 $�&?O?+ α

|}$�&?O?+9!£ > � /�)�& 2 KZ!" ��{T{$*8ROS& 2 )UT 0 $�p��R)Up9OS$�p�+�6�)�7"$98?$ 0 $9TW|})�p�r3sz$³YZ+ � p 0 7")U8Ssz$�OS& �12 pU/�)�7 2 & 0 $�T{x 2�0 O 2 +t$0 $9&FOS+

α!

3�) 2 7"'*)�+�&w|})�pUr�sz$BYZ+ � p 0 7")U8?s�$�)�p � T{$�Y 2 7 0 $�TÂ8?)�x�$9&FOS+-pUs�$ 0 $9T5x 2�0 OS$�x�$�Y�+�8?+3&wOS& 2 pU8R|}$9&?T 2 Y 2 +3T)�T 2 |})�pUr�sz$ 0 $9T{x 2�0 O 2 ��T 2 8wY�+�8 0 $�p*O �=p�) 2 � 0 $�TW$ 0 $�&?O?+

α� 0 $9T5${+y8ROD��T{$*8ROS& 2 YZ$�p 25> � /�)�& 2 KZ!" ��Z!

��� b � "�°l�¨_ # � � []$�p�8 � YZ+3& 2 p�YZ$Hp�$ »�2 T{+3p*OS+w$ 0 $9p��R)�p*O?$-YZ$t+ ¿ +3T{x�7"$�KZ! �JO?+�T{$98�$*8�8?+3/�) � p*O?+y8 0 $�p��R)�p*OS$98�I

A0.5 = {7.0, 7.5, 8.0, 8.5, 9.0}

Page 24: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�� � ���`� ��� � ���R� � � � ���� � � � ���� � ���

x

1

nivel alfa

0

α

> � /9)�& 2 K�!= y� I > )�p�r3sz$ 0 $�TW8?)�x�$�&?O?+ 0 $9T{x 2�0 O?$5+ 0 $�&?O?+ 2 7=| 2 !

0

1

nivel alfa

> � /�)�& 2 KZ!" �� I > )�pUr3s�$ 0 $�T¾8F)�x�$�&?O?+JpUs�$ 0 $�T{x 290 OS$�+ 0 $�&?O?+ 2 7=| 2 !

A,0.5 = {7.5, 8.0, 8.5}

A0.75 = {7.5, 8.0, 8.5}

`cb9d%��ef���U� _ #l�Zn ²�mZ����el��m�°lef��m���b � n .A/c. �?> H02 ��)9�?� :f2 �! k � Vb2 " 2 ^`� 8;:f2A

.DH�.A/c1�8 � .�VW8B1�.�1 |A|

Y 1 �1^`� 8B1�.NH02 > 2

| A |=∑

x∈X

µ(x)

.PH�.�/c1A8 � .AVW8B1�.�1 / V].A:_8;O�.N1 AY 1 B^�� 8B1�.NH�2 > 2

‖A‖ =| A |

| X |

��� b � "�°l� _ # � � £ 0�2 &DY � p 2 7 � Y 2 YZ+JYZ$ 0 $�pz�R)�p*O?$�YZ$�+ ¿ +�T5xU7=$�KZ! �5¤ � /9) 2 7 2| A |= 0.25 + 0.5 + 0.75 + 1.0 + 0.75 + 0.5 + 0.25 = 4.0

+ 2{032 &SY � p 2 7 � Y 2 Y�+�&S+37 2 O �=»�2 ¤‖A‖ =

4.0

21= 0.19

Page 25: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R� � � ��� � � �s��� � � � � � � � �� � ���P ��� � � �� �� y�£ 032 &DY � p 2 7 � Y 2 YZ+�&?+�7 2 O �"»�2 x�$ZYZ+H8F+�& � p*O?+�&?x�&S+�O 2 Y 2�0 $9T5$ 2 |}& 2 r�sz$5YZ+J+37"+3T{+3p*O?$*8�YZ+

X'*)�+�+y8RODsz$5+3T

A�*x�$�pUYZ+�& 2 YZ$98%x�+37"$98�/�& 2 )U8 YZ+ � p 0 7=)U8Ssz$H+�T

A!%# 2 & 2J0 $9p��R)�p*O?$*8

A� p��Up � O?$*8 2H032 &SY � p 2 7 � Y 2 YZ+�¤(YZ+��Up � Y 2

0 $9T{$A =

x

µA(x)dx

[]7 2 &S$5'*)�+ 2{032 &SY � p 2 7 � Y 2 YZ+�p�sz${+ ¿ � 8ROS+J8?+3T{x�&S+�!`cb9d%��ef���U� _ #l�����B°f������m ��% .�V�: � /c.P1 i�?> H�2 � )9�?� :&2 �! k � V]2 " 2

AY 1 B^�� 8B1�.NH�2 > 2

HA = maxx∈X

{µA(x)}: KZ!" �P E

E�?> H�2 � )9�?� :&2 Y 1 B^�� 8B1�2NH�2 > 2 � 2�/ > .�V "# HA = 1

J"9� k 1&� 2A/ > .AV � 2PH�. " 2NH�2 � :I/cpA/98B2�[`cb9d%��ef���U� _ #l� � `³elh3������k�efm ��� 1 . 1�8 " : * � H#8B. e �� d�?> O�.�V]2A/ 9" :&pD1�. 1 _^`� 8(aUM�2C8B1 .AV�1�2 H�2 � )9�?� :&2 -Y 1 B^�� 8B1�.DH�2 > 2

d(A, x) =

{

∞ se µA(x) = 01

µA(x) − 1 caso contrario

£ > � /9)�& 2 K�!= y� � 7=)�8ROS& 2 +�8FO?+ 0 $�p 0 + � O?$�!

ideal

A1

0x

d(A,x)

d(A,x)

x

> � /�)�& 2 KZ!" ��JI � � 8FOS½�p 0 �12 +3p*OS&?+J)�T¾x�$9p9OS${+ 2 Y�+��Up � r3s�$ � YZ+ 2 7�YZ$ 0 $�p��R)�p*OS$U!

�"��� �� #3� -��� # 6 +.�C> A ����<! i�7�C6�/ #32!45,��6G�C6£\8-$�x�+3& 2 r���+y8 0 $�T 0 $�p��R)Up9OS$98Jp�+�6�)�7"$98?$98H8Ssz$;YZ+���p � Y 2 8H+�T O?+�&?T{$*8HY 2 8I|})�p�r���+�8HYZ+ � p 0 7=)U8Ssz$�!³  2 Y�+3¡:������� � � y���9u E +�T� y���9u 2 xU&?+y8F+�p9OS$�) )�T x�&S$�x�$98FO 2 YZ+ 0 $�T{$ 2 x�7 �"0�2 &J+�8FO?+y8 0 $9p 0 + � OS$98 2 $*8 0 $9p��R)�p*O?$*8p�+�6�)�7"$98?$98�!`cb9d%��ef���U� _ #l� � �³��ef�U� � % ���?� aGM�2 1 8 � H�V �5" M�2

µU (x)1�. �?� 8BM�2 1�2 " H�2 � )9�?� :&2 "l�! k � V]2 " 2 "

A

B,U = A ∪ B

/ Y 1 B^�� 8B1�.PH�2 > 2µU (x) = max(µA(x), µB(x)), x ∈ X

: KZ!" 3� E`cb9d%��ef���U� _ #l��� �����b*��hyb*k9��������% ���?� aGM�2 1 8 � H�V �5" M�2

µI (x)1�. 8 � : / "� H0aGM�2 1�2 " H02 ��)9�?� :f2 "

A

B,I = A ∩ B

/ Y 1 _^`� 8B1�.DH02 > 2µI (x) = min(µA(x), µB(x)) x ∈ X

: KZ!" yu E

Page 26: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

Kz� � ���`� ��� � ���R� � � � ���� � � � ���� � ���`cb9d%��ef���U� _ # _ � ²�� � "%°fb�� b����y�Á��� ´¨b�$�m������ ��% ���?� aUM�2P1 8 � H#V �5" M�2

µC(x)1�2NH�2 > ,!V #>l #� :f2N1�2

H�2 � )9�?� :&2A (C = A)

Y 1 _^`� 8B1�.DH02 > 2

µC(x) = 1 − µA(x), x ∈ X: KZ!" �� E

ºX8FO 2 8 |}.�&STt)U7 2 8X8Ssz$H+�'*) �"»�2 7"+3p*O?+y8 2 8�$�x�+3& 2 r3��+�8 0 7"�98?8 �"0�2 8�'*) 2 pUYZ$-$ 0 $�pz�R)�p*O?$[0, 1]

¤\&?+yYZ)�@ � YZ$-x 2 & 2{0, 1}

!Hº ¿ � 8FO?+�T $�)ZOS& 2 8\+ ¿ O?+3p�8F��+�8Ix 2 & 2³2 8I$�x�+3& 2 r3��+�8\Y�+ � p*O?+�&S8?+ 0 r3sz$ :∩E ��)�p � sz$ :

∪E +{p�+3/ 2 r�sz$B'*)�+8?+3&Dsz$ 2 p 2 7 � 8 2 Y 2 8]T 2 � 8 O |}&?+�p*O?+�!

��� b � "�°l� _ #l��� � QZ+R� 2 $ 0 $�p��R)�p*OS$ YZ+¨6�$9pU8 2 7=)�pU$98AYZ$ + ¿ +3T{x�7"$ KZ! � ��'9)U+Á&?+�x�+3O � T{$98 2 '*) � x 2 & 2| 2�0 � 7 � O 2 &y!%£\8S8F)�T 2 '*)�+($JT{+�8?T{$Hx�&?$�|}+�8S8F$9&�O 2 T�6�¤�T 0 7 2 8?8 � � 0 $�){8?+3)U8 2 7=)Up�$98%x�$�& 2 8?8 � YZ) � Y 2 YZ+9!�£�8?8?)�T � &

'*)�+($ 0 $�p��R)�p*OS$-YZ+(T{+�Y � Y 2 8�YZ+ 2 8?8 � Y�) � Y 2 YZ+ 0 $9T{+�r 2 +�Tª�J+ »�2 ��2 OS¤H ��H8?+3T{x�&S+ 2 )�T{+�p9O 2 pUY�$JY�+\��!LuI+3T��!Lu{x�$�p*O?$�!J��pUY�+B �� 0 $�&S&S+�8?x�$9pUYZ+ 2 y��� �¢YZ+�x�&S+�8?+3pUr 2 +�� 2 ���¾YZ+�x�&?+y8F+�pUr 2 !\£\8S8 � T ��$98 0 $9p��R)�p*O?$*8p�+�6�)�7"$98?$98

A+BYZ+ 2 7=)�pU$98±6�$�p�8�+VYZ+ 2 7=)�pU$98 2 8?8&�1YZ)�$98�&?+y8Fx�+ 0 O �"»�2 T{+3p*O?+�|}$�& 2 T YZ+��Up � YZ$*8�x�+37"$\xU&?$�|}+�8S8F$9&

0 $9T{$A(bons alunos) = {(6.5, 0.25), (7, 0.5), (7.5, 0.75), (8, 1), (8.5, 0.75), (9, 0.5), (9.5, 0.25)}

B(alunos assiduos) = {(6, 0.25), (6.5, 0.5), (7, 0.75), (7.5, 1), (8, 1), (8.5, 1), (9, 1), (9.5, 1), (10, 1)}

A ∪ B = {(6, 0.25), (6.5, 0.5), (7, 0.75), (7.5, 1), (8, 1), (8.5, 1), (9, 1), (9.5, 1), (10, 1)}

A ∩ B = {(6.5, 0.25), (7, 0.5), (7.5, 0.75), (8, 1), (8.5, 0.75), (9, 0.5), (9.5, 0.25)}

B = {(0, 1), (0.5, 1), (1, 1)(1.5, 1), (2, 1), (2.5, 1), (3, 1), (3.5, 1), (4, 1),

(4.5, 1), (5, 1), (5.5, 1), (6.0.75), (6.5, 0.5), (7, 0.25)}

£ > � /9)�& 2 K�!= y� � 7=)�8ROS& 2 +�8FO?+y8 0 $9p��R)�p*O?$*8�+ 2 8�$�x�+3& 2 r3��+�8�!��������� � � ' ����!3���-��� ����� � & '*)�+",�)*.0'*��� � ��,-2�'*� '*�� 2 T{$*8(YZ+���p � &\$98 0 $9p��R)�p*O?$*8wp�+�6�)�7"$98?$98

A�B+

CpU$ 0 $�pz�R)�p*O?$�)�p �"» +�&S8?$

X+H$ 0 $�pz�R)�p*O?$ »�2 @ � $

∅!V£\88?+3/�) � p*O?+y8�$�x�+3& 2 r3�*+y8 0 $�p*O � p�) 2 T »�2 7=+�pUYZ${x 2 & 2{0 $9p��R)�p*O?$*8�p�+36U)�7=$*8F$*8MI

²�� � ����mZ�ye �±ef��m���b��A ∪ B = B ∪ A

�A ∩ B = B ∩ A

��Bhyhy�±k�elmZ��e �±ef��m���b��

A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C�

���b�� "X��� ����k9efm �A ∪ B = A, A ∩ A = A

�`cefh��3��e}µ%���ye �±el��m���b��

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∩ C), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C);

��µ%hy�U�����U� �A ∪ (A ∩ B) = A, A ∩ (A ∪ B) = A

� �� ����°l���9��� �

A = A�

� °fb�� b�������ht´¨b����3����h �A ∪ ∅ = A, A ∩ X = A

���µ%hy�U�����U� �

A ∩ ∅ = ∅, A ∪ X = X

`cb ��� $�m�� �(A ∩ B = A ∪ B,

º ¿ � 8ROS+H)�T 2 Y � |}+�&?+�pUr 2 6U�98 �1032 +3p*OS&?+J$98 0 $�p��R)Up9OS$98wpU+36�)�7"$98?$98V+ 0 $�p��R)�p*OS$98�p �=O � YZ$98�!XM��5Y�) 2 8w$9x�+�& 2 �r���+y8]'*)�+IpUsz${8?+\T 2 p*O?¤�T¢'*) 2 pUY�$ 2 x�7 �1032 Y 2-2 $98 0 $�pz�R)�p*O?$98Vp�+�6�)�7"$98 2 !�ºX8FO 2 Y � |}+�&?+�pUr 2 +�8FOS�t&?+�7 290 � $�p 2 Y 20 $9T 2 8�7"+ � 8(Y 2{4 sz$B[]$�p*O?& 2 Y � r�sz$ :

A ∩ A ≡ ∅E +-Y 2 º ¿Z0 7=)�8?s�${Y�${§Á+ � $ :

A ∪ A ≡ XE !

[]$�T{$Hp 2 7".�/ �"0�2 p�+36U)�7=$*8 2 $*8 0 $9p��R)�p*O?$*8 pUs�$-8?s�$Hp � O � Y 2 T{+3p*O?+�YZ+��Up � YZ$*8 +37"+�8XpUsz$H$96�+yYZ+ 0 +3T 2 +�8FO 2 87"+ � 83!�[]$�T{$³+ ¿ +3T{x�7"$ � »�2 T{$98 0 $�pU8 � Y�+3& 2 &I$ 0 $�p��R)�p*OS$cYZ$*8 2 YZ)U7 OS$98I+5Y�$98JpUs�$ 2 Y�)�7 OS$98HYZ+��Up � YZ$98 0 $�T{$T{$98FO?& 2 YZ$³p 2;> � /9)�& 2 KZ!LKz�U! � 2 T5$*8 0 ¡ 2 T 2 &I$ 0 $9p��R)�p*O?$¨YZ$*8 2 YZ)�7=O?$*8HYZ+

A+{$;8F+�) 0 $�T{x�7"+3T{+3p*OS$ �±$*8

YZ$*8�pUsz$ 2 Y�)�7 OS$98wYZ+A!

Page 27: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R� � � ��� � � �s��� � � � � � � � �� � ���P ��� � � �� �� K�

> � /�)�& 2 KZ!" ��JI�º ¿ +�T5xU7=$*8(YZ+J$�x�+3& 2 r3�*+y8V+3p*O?&S+ 0 $�pz�R)�p*O?$98�!

£ ��&?+ 2 8F$9Tt6�&S+ 2 Y 2 T{$98FO?& 2�2�2 � p*O?+�&S8?+ 0 r3sz$�Y�$98 0 $�p��R)Up9OS$98�!I��6U8?+3& » +-'*)�+-$ 0 $9p��R)�p*O?$�p�sz$�¤ »�2 @ � $+ 2 7"/�)U¤3T¢YZ+tK�� 2 p�$98�YZ+ � Y 2 YZ+Jx�$ZYZ+-8?+3& 0 $�pU8 � Y�+3& 2 YZ$ 2 YZ)U7 OS$5+JpUs�$ 2 YZ)�7=O?$ 2 $5T{+y8FT{$5O?+�T5x�$U![]$�T{$ )�T 0 $�p��R)�p*OS$ p�� O � YZ$ 8 2 O � 8F| 2 @ 2 857"+ � 8�Y 2�4 s�$ []$9p9OS& 2 Y � r3sz$ +¨Y 2 º ¿�0 7")U8?s�$ Y�$�§Á+ � $ +;$*8

0 $9p��R)�p*O?$*85p�+36�)U7=$*8F$*8�p�sz$ � �\$*8 �9$ :F:B'*)%+A'*) �w y����� EFE x�&S$�x$�98{'*)�+¨8F+;)U8 2 8S8F+³+�8FO?+y85| 2 O?$*8{x 2 & 2 T{+yY � &'*)Usz${p�+�6�)�7"$98?$�)UT 0 $9p��R)�p*O?${¤9! º�7=+-Y�+3p�$9T � p�$9)c+y8RO 2 T5+yY � Y 2 YZ+ #� :_/020,!8B. �! k � Vb2 " .�!`cb9d%��ef���U� _ # _ � � ��������"%efm ´¨b*µ%��°l��hym ��� 2 ��" 8B1 / �?> H�2 ��)9�?� :f2 �! k � Vb2 " 2

A[ % #� :_/020,!8B. 1 #" :

H�2 � )9�?� :&2 Y 1 B^�� 8B1�. , Vb. � hA/ >i� V]. �E(A) =

c(A ∩ A)

c(A ∪ A)

: KZ!" �� E2 � 1

c/ _�# / 1f"� . �?> .PH�2 � :f. K� �> ,.�1�8(aGM�2C2 � 8 � : fK /0.AaGM�2+/ " 2�k#/ 2 "9� ,�2�/9: 1�2PH�2 � )9�?� :&2�[

��6U8?+3& »�2 &('*)�+tx 2 & 2 )�T 0 $�pz�R)�p*O?$Bp�� O � YZ$B$Bp�)�T{+3& 2 YZ$9&(¤t8?+3T{x�&S+t��+t$BYZ+3pU$�T � p 2 YZ$9&\¤t8?+3T{x�&S+� ��x�$�&?O 2 p*OS$ 2 +�p9OS&?$9x �12 YZ+H)UT 0 $9p��R)�p*O?${p �=O � YZ$ »�2 7"+H8F+�T{x�&?+H�U!£¾+3p*O?&S$�x �12 ¤c)�T 2 T{+�Y � Y 2 '*)�+B&S+�|}+�&?+3��8?+³8?$�T{+�p9OS+ 2 $ 0 $9p��R)�p*O?$ ��+3T�6�$9& 2 � pUY � &S+�O 2 T{+3p*O?+;8F$�|}& 2 +

&S+ 0 +�6 2 � p���)�¼3p 0 �"2 Y�${8 � 8ROS+3T 2{0 $�T{x�7"+�OS$ 2 O?& 2�» ¤y8�Y 2 � p���)�¼3p 0 �"2 YZ$*8 0 $9p��R)�p*O?$*8 » � @ � p�¡�$*83!� 25> � /�)�& 2 KZ!LKz��x�$ZYZ+�T{$98VO � & 2 &w$98w8?+3/9) � p9OS+�8 »�2 7"$�&S+�8�I

c(A ∩ A) = 5

c(A ∪ A) = 20 + 20− 5 = 35

E(A) =5

35= 0.14

Page 28: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

K�K � ���`� ��� � ���R� � � � ���� � � � ���� � ���

0 10 20 30 40

0

1

adultonao adulto

idade

adulto adulto e nao adulto

> � /�)�& 2 KZ!LKz�JI []$9p��R)�p*O?$*8wYZ+ 2 YZ)U7 OS$98�+HpUs�$ 2 YZ)�7=O?$98�!

�������� &('*% �-��� � ) �-' $����� � ��� � � &(2����"�"!�#�� ��#�'/% � ��� ,-2�'*� ���Q � 8ROS+3T 2 8 4 +36�)�7"$98?$98 � 0 $9T5${8?+3&D� » � 8FO?$5T 2 � 8 2 Y �12 p*OS+��Z8?s�$ 0 $�pU8FO?&S) �1YZ$98 0 $9T¢6 2 8?+I+3T¢&S+3/9& 2 8X'*)�+�O?+3T 28?+3/�) � p*O?+I|}$9&?T 2 /�+�& 2 7 I

IF x IS X AND y IS Y THEN r IS R

4 +y8RO 2 &?+�/�& 2 $98 2 p*O?+ 0 +�YZ+�p*O?+�8w8Ssz$x IS X AND y IS Y

oqp*O?+�&S8?+ 0 r3sz$ :AND

E ¤ 2 $9x�+�& 2 r3s�$�6U��8 �"0�2 x 2 & 2�2�0 $�pU8FO?&S)Ur3s�$�YZ$98 2 p*O?+ 0 +yYZ+3p*O?+y8\Y 2 &?+�/�& 2 8�p�+36U)�7=$*8 2 83!#%$�&w+ ¿ +�T{x�7=${)�T 2 &S+3/�& 2 x�$ZYZ+-8?+3&IF water temperature IS low AND air temperature IS low THEN switch on water heater longer

� 2 T{$*8 0 $9pZ|}+3& � &B$98B+�|}+ � OS$98cY 2 $9x�+�& 2 r3sz$ Y�+ � p9OS+3&D8F+ 0 r�sz$ 0 $9T{x 2 & 2 pUY�$ $98�+3|}+ � O?$*8³Y 2 8BYZ+��Up � r���+�8p �=O � Y 2 8�+�p�+�6�)�7"$98 2 8 p 2 8F$97=)Ur�sz$(YZ+X)�TxU&?$96�7=+�T 2 + ¿ +3T{x�7"$U!�# 2 & 2 � 8FO?$ »�2 T{$*8 0 $9pU8 � YZ+�& 2 &±YZ$ � 8 0 $�p��R)�p*OS$98�IT{+ �12 � Y 2 YZ+H+ 2 7=O?$�!%£¦Y�+��Up � r3s�${p � O � Y 2 YZ+�8FO?+y8 0 $�pz�R)�p*O?$98�+y8ROD�5T5$*8ROS& 2 Y 2 p 2�> � /9)�& 2 KZ!LKZ 9!

25 30 35 40 45 50 550

1MeiaIdade

1,5 1,6 1,7 1,80

1Alto

idade altura

> � /�)U& 2 KZ!LKZ I�[]$9p��R)�p*O?$*8 2 7=O?${+HT{+ �12 � Y 2 YZ+Jp �=O � YZ$98�!£|})�pUr�sz$�YZ+ � p 0 7")U8Ssz$�YZ$ 0 $9p��R)�p*O?${T{+ �12 � Y 2 YZ+J¤ 2 8F+�/�) � p*O?+ I

µmeia idade(idade) =

0 se idade < 351 se 35 ≤ idade ≤ 450 se idade > 45

Page 29: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R� � � ��� � � �s��� � � � � � � � �� � ���P ��� � � �� �� K�P£|})�pUr�sz$�YZ+ � p 0 7")U8Ssz$�YZ$ 0 $9p��R)�p*O?$ 2 7=O?${¤ 2 8?+3/�) � p*O?+ I

µalto(altura) =

{

0 se altura < 1.701 se altura ≥ 1.70

[]$�p�8 � YZ+3&S+Á$*8³Y 2 YZ$98cY 2 � 2 6�+�7 2 KZ!LK &S+�O � & 2 YZ$ Y�+ )�T 6 2 p 0 $ YZ+�Y 2 YZ$*8 0 $�T � pZ|}$�&ST 2 r���+y8B8?$�6�&S++3T{x�&S+3/ 2 YZ$*8�YZ+J)�T 2 YZ+�OS+3&ST � p 2 Y 2 +3T{x�&S+�8 2 !

� 2 6�+37 2 K�! K I%º�T{x�&?+�/ 2 YZ$98 �Z8F) 2 8 2 7=O?)�& 2 8V+ � Y 2 Y�+�8�!4 $�T{+ £(7=O?)�& 2 oRY 2 YZ+£(p 2 �!���� P��£(p*O<�9p � $ �! ��P uz���$*sz$ �! ��� �z���$*8F¤ �! �*� P9K,±) � @ �! �z� �9�§ 2 & �"2 �!Luz� K�K# 2 )�7"$ �!���� ���#�+yYZ&S$ �! ��P K�u

Q�)�x�$�p�YZ$H'9)U+(YZ+�8?+R� 2 T5$*8%&?+y8Fx�$�p�YZ+3& �z8F$96�&S+V$*8 Y 2 Y�$98 2 &ST 2 @�+3p 2 YZ$98�p�$J6 2 p 0 $ � 2 8F+�/�) � p*O?+�x�+�&?/9)�p*O 2 I3�) 2 � 8�+3T{x�&S+3/ 2 YZ$*8\8Ssz$BYZ+5T{+ �12 � Y 2 YZ+�+ 2 7=O?$*8 2 $BT{+�8?T{$BOS+3T{x�$ G ��6U8?+3& »�2 pUYZ$B$98IY 2 YZ$*8�Y 2 � 2 6�+37 2KZ!LK�+ 2 8wYZ) 2 8V|})�pUr3�*+y8(YZ+ � p 0 7")U8Ssz$ :f> � /�)�& 2 KZ!LKZ E x�$ZYZ+3T{$98�/9+3& 2 & 2 � 2 6�+�7 2 K�! PU!

� 2 6�+37 2 KZ! P I%º�T{x�&S+3/ 2 YZ$*8w£(7=O?$98�º YZ+-§Á+ �12 oRY 2 YZ+4 $�T{+ £\7 OS$ §¨+ �"2 oRY 2 YZ+ £\7 OS$�º §Á+ �12 oRY 2 YZ+£\p 2

£\p*O<��p � $ � ���$*sz$ � � ��9$98?¤ � �,±) � @ § 2 & �12 � � �# 2 )�7"$ � �#%+yYZ&?$ � �

� &?+y8F)�7=O 2 YZ$JT{$98FO?& 2 YZ$Jp 2 � 2 6�+37 2 KZ! P � pUY �1032 '*)�+\YZ$ � 8XYZ$98%|})Up 0 � $�pU��& � $98 : £(p 2 +(,±) � @ E +�8FOSsz$-Y�+3p*O?&S$YZ$*8 0 & � OS¤3& � $*8]+y8RO 2 6�+37"+ 0 � YZ$*8�x�+�7 2{0 $�pU8?)�7=O 2{2 ${6 2 p 0 $�YZ+HY 2 YZ$98�x�+37"$98 0 & � O?¤�& � $98Vp �=O � YZ$98�!# 2 & 2 &S+�| 2 @3+�&H+�8FO 2 x�+3&S/�)�p*O 2 )ZO � 7 � @ 2 p�YZ$¨$ 0 & � OS¤3& � $*8Jp�+36U)�7=$*8F$*8Jx�&S+ 0 � 8 2 T{$98JYZ+��Up � &-$Á'*)�+�8 � /�p � � 0�2p�+�6�)�7"$98 2 T{+3p*O?+

alto+

meia idade!

� 0 $�pz�R)�p*O?$BY 2 8wx�+�8S8F$ 2 8 2 7 O 2 8w+y8ROD��YZ+��Up � YZ$Bp 2�> � /9)�& 2 K�! K�!]# 2 & 2 | 290 � 7 � O 2 &($�+3p*OS+3pUY � T{+3p*O?$B&?+�x�+3�O � T{$98V$ 0 $�p 0 + � O?$5p 25> � /9)�& 2 KZ!LK�K�! � 0 $�p��R)Up9OS$5Y�+�x�+�8S8F$ 2 8VYZ+IT5+ �"2 � Y 2 YZ+I¤J&?+�x�&S+�8?+3p*O 2 YZ$-x�+37 2�> � /�)�& 2KZ!LKzPU! £\8�&S+�8?x�+ 0 O �"»�2 8V|})�p�r���+�8(Y�+ � p 0 7=)U8Ssz$�8Ssz$ 2 8w8?+3/9) � p*OS+�8�I

µalto(altura) =

0 se altura ≤ 1.705 · altura − 8.5 se 1.70 < altura < 1.901 se altura ≥ 1.90

µmeia idade(idade) =

0 se idade ≤ 30idade

10 − 3 se 30 < idade ≤ 40−idade

10 + 5 se 40 < idade ≤ 500 se idade > 50

Page 30: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

K�� � ���`� ��� � ���R� � � � ���� � � � ���� � ���

µ

Altura

1,0

1,70

0,5

1,91,8

> � /9)�& 2 K�! K9K I �\+��Up � r3sz$�Y�$ 0 $�p��R)Up9OS$ : p�+�6�)�7"$98?$ E Y 2 8�x�+�8S8?$ 2 8 2 7 O 2 8�!

1MeiaIdade

25 30 35 40 45 50 550

idade

> � /�)U& 2 KZ!LKzPJI []$9p��R)�p*O?$�YZ+Hx�+�8S8?$ 2 8�YZ+HT{+ �"2 � Y 2 YZ+9!

��6U8?+3& »�2 &%'9)U+w)ZO � 7 � @ 2 pUYZ$ 0 $9p��R)�p*O?$*8 0 71��8S8 �10 $*8%8F$9T5+�p*O?+wYZ$ � 8%+3T{x�&S+3/ 2 YZ$*8�+y8RODsz$HYZ+�p9OS&?$HYZ$*8 0 & � OS¤3& � $*8+�8FO 2 6�+�7=+ 0 � YZ$98 ��+3pU'*) 2 p*O?$ '*)�+Áx�+37"$98 0 & � O?¤�& � $98{p�+�6�)�7"$98?$98{OS+3T{$98B'9) 2 O?&S$U! £ )ZO � 7 � @ 2 r�sz$ YZ$98 0 & � OS¤3& � $*8p�+�6�)�7"$98?$98Vx�+�&?T � O?+-'*)�+H+�7=+�T5+�p*O?$98wY�$98 0 $9p��R)�p*O?$*8w'*)�+ 2 p*O?+y8�pUsz${8?+3& �"2 TW8F+�7=+ 0 � $�p 2 YZ$98Vx 2 8?8?+3T 2 | 2 @3+�&x 2 &FOS+JY�$ 0 $�p��R)Up9OS$�8F$97=)Ur�sz$c!§ 2 8X'*) 2 7�¤ 2t»�2 p*O 2 /9+3T YZ+�8FO 2 &?+y8Fx�$98FO 2-2 T{x�7 �12 Y 2 !�[]& � O?¤�& � $98 & �"/ � YZ$98]YZ+ �=¿�2 TÀYZ+(|}$9& 2 $98�+�7=+�T{+3p*O?$*8

'*)�+t+y8RODsz$�x�&S. ¿ � T{$*8 2 $*8(7 � T � O?+�8IY 2 8w|})�pUr3��+�8IYZ+ � p 0 7")U8Ssz$�+5'*)�+-x�$�Y�+3T 8?+3& � T{x�$9&FO 2 p*O?+y8(+y8 0 $�7"¡ 2 8(Y�+8?$�7")Ur3s�$¨!w#�$9&\+ ¿ +�T{x�7=$ �U$*8w|})�p 0 � $9pU�z& � $98?�9$98?¤H+-# 2 )�7"$B'*)�+�pUsz${|}$9& 2 T � p 0 7") �1YZ$98\p 2 8?$�7")Ur3s�$ 0 71��8S8 �1032 �x�$�Y�+3TÂ8F+�&(8F$97=)Ur3��+�8�& 2 @3$*� » + � 8Vx 2 & 2{2 '*)�+�8FOSs�$U!V�¦|})�p 0 � $9pU�z& � $5�9$98?¤ 0 +�&FO 2 T{+3p*O?+J¤ 2 7 OS$ : �� �*��T E T 2 8� 0 $�)³Y�+\|}$9& 2 YZ+ » � YZ$ 2 $-| 2 O?${YZ$ 0 & � O?¤�& � $�YZ+IT{+ �12 � Y 2 YZ+�+ ¿ � / � & � Y 2 YZ+IT 2 � $�&�'9)U+IP*u 2 pU$98 �9+I+37"+\OS+3TWP*K2 pU$98�! ��& 2-2 8?) 2 � p 0 7=)�8?s�$-p 2 &S+�8?x�$98FO 2 p�+�6�)�7"$98 2 +Ix�+�&F|}+ � O 2 T{+3p*O?+�& 2 @3$*� » +37Z+JYZ+�p9OS&?$�YZ$�8?+3pU8?$ 0 $�Tt)UTc!�|})�p 0 � $�pU��& � $t# 2 )U7=$tO 2 T�6�¤�T¢¤ 2 7=O?$ : ������tT E �ZT 2 8 � 0 $9)³YZ+I|}$9& 2 x�$9&]OS+3&V��� 2 p�$*8���YZ$ � 8 2 T 2 � 8V'*)�+ 2� Y 2 YZ+JT�� ¿ � T 2 !

� 2 6�+37 2 KZ! �JI%º�T{x�&S+3/ 2 YZ$98 2 7 OS$98�+-YZ+JT{+ �12 � Y 2 YZ+��Up�+36�)U7=$*8 2 T5+�p*O?+�!4 $9T5+ £(7=O?)�& 2µalto(altura)

oRY 2 YZ+µmeia idade(idade)

£\7 OS$�º §¨+ �12 oRY 2 YZ+£(p 2 �!���� �U! K�� P�� ��! � ��!LKz�£(p*O<�9p � $ �! ��P �U! �*u uz� ��! � ��! �����$*sz$ �! ��� �U! �9� �z� ��! � ��! �����$*8F¤ �! �*� �U! �*u P9K ��!LK ��!LKz�,±) � @ �! �z� �U!L�z� �9� �! � ��!����§ 2 & �12 �!Luz� �U! �9� K�K ��! � ��! ���# 2 )�7"$ �!���� �U! ��u ��� ��! P ��! P��#%+�YZ&S$ �! ��P �U! �*u K�u ��! � ��! ���

Page 31: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R���X� ���B � � � � � ���B � � � � K9u# 2 & 2B0 $�&S& � / � & 2 8\| 2 7=¡ 2 8JY 2 8JYZ+��Up � r���+�8HY�$98 0 $9p��R)�p*O?$*8 0 7"�98?8 �10 $98 ��pUsz$c6 2 8RO 2 T{$ » +�&�$*8I7 � T � OS+�8HYZ$*80 $9p��R)�p*O?$*8 YZ+�|}$�&ST 2J2 � p 0 7=) � &�$*8 032 8?$98%'*)�+�� 0�2 & 2 T YZ+�|}$9& 2 !�Q�+�T{x�&?+('*)�+w8?+wT{$ » +V$H7 � T � OS+���x�$ZYZ+3&S+3T{$*8

YZ+ � ¿�2 &�YZ+ |}$9& 2 +�7=+�T{+3p*O?$*8�'*)�+Xx�$ZYZ+3& �12 T 8F+�& � p 0 7") �1YZ$98 ��$�) � p 0 7") � & 0 $�T|}$�&Dr 2 O?$�O 2 79p�$ 0 $�pz�R)�p*O?$(&S+�8?x�$*8RO 2+37"+3T{+�p9OS$98w'*)�+H8?.5T 2 &?/ � p 2 7=T{+�p9OS+ 2 O?+�pUYZ+3T 2 8 0 $�pUY � r���+y8wYZ+�8?+R� 2 Y 2 8�!,±+�Tt6�& 2 &5'*)�+;p 2 8?$�7")Ur�sz$Áp�+�6�)�7"$98 2 $*85+37"+3T{+3p*O?$*8�8?+37"+ 0 � $9p 2 Y�$98�&?+ 0 +36�+3T )UT~/�& 2 ) '*)�+ � pUY �1032 2

|}$�&Dr 2³0 $9TÂ'*)�+ 2 &S+�8?x�$98FO 2 8?+ 2 8?8?+3T{+�7=¡ 2B2 � YZ¤ �12 '*)�+5$B)�8F)U��& � $�| 2 @5Y 2 &?+y8Fx�$98FO 2 � Y�+ 2 7�!J#%$�Y�+3T{$98 » +�&p 2 � 2 6�+�7 2 KZ! �;'9)U+�$c+3T{x�&S+3/ 2 YZ$c,±) � @{¤ 2 '9)U+37"+5'*)�+{T 2 � 8J8F+ 2 x�&S$ ¿ � T 2 Y 2 Y�+��Up � r3s�$ � YZ+ 2 7�YZ+ 2 7=O?$c+T{+ �12 � Y 2 YZ+9! 4 $-+�p9O 2 p*O?$ �9# 2 )�7=$�'9)U+\pUs�$ 0 $�p�8RO 2 Y 2 &S+�8?x�$*8RO 2-0 71��8S8 �"0�2 �zOS+3T )�T /�& 2 )�YZ+ � p 0 7")U8?s�$ : �U! P ET 2 � $�&X'*)�+�$tYZ+\£\p 2B: ��!LK E �*'*)�+ 2 x 2 &S+ 0 +wp 2 &S+�8?x�$*8RO 2H0 71��8S8 �1032 ! ��6�8F+�& »�2 &�'*)�+�$98�Y�$ � 8�+�8FOSs�$-p�$*8�7 � T � O?+�8Y 2 8wYZ+��Up � r3�*+y83!�"��� �� $��� > 6�# =� $���> 6£(7"/�)UT 2 8('*)�+�8FO?��+y8wx�$ZYZ+�T�8?)�&S/ � & 2 x�.98 2�2 x�&S+�8?+3p*O 2 r3sz${Y 2 8(YZ+���p � r3��+�8(Y 2 8($9x�+�& 2 r���+�8V+3p*OS&?+ 0 $�p��R)�p*OS$p�+�6�)�7"$98?$98�I 9!w#�$9&t'*)�+�+y8RO 2 8H$9x�+�& 2 r���+�8I|}$9& 2 T +�8 0 $97=¡ � Y 2 8Hx 2 & 2 � T5xU7=+�T5+�p*O 2 & 2 8H$9x�+�& 2 r���+�8HY�+ � p*OS+3&D8F+ 0 r3s�$;+)�p � sz$ GK�!�3�)�+V�R)U8RO � � 032 O �=»�2 8�T 2 O?+�T{�zO �1032 8�x�$ZYZ+�T¾8?+3&(Y 2 Y 2 8Vx 2 & 2 +�8FO 2 +y8 0 $�7"¡ 29GPU!wº ¿ � 8FO?+�TW$�)�O?& 2 8V$�x�+3& 2 r3�*+y8��Z'*)�+Jx�$�Y�+3& �"2 TÀO 2 Tt6�¤3T¾8?+3&w+�T{x�&?+�/ 2 Y 2 8 G

��6 » �"2 T5+�p*O?+�� 0 $9T{$%�F�w|}$ � T{+3p 0 � $9p 2 YZ$ ��¡��w$�)ZOS& 2 8 $�x�+3& 2 r3��+�8±'*)�+�x�$ZYZ+3& �12 T 8F+�&�)ZO � 7 � @ 2 Y 2 8���'*)�+]8?+�&S+�YZ)��@3+�T O 8�$9x�+�& 2 r���+�8 0 7"�98?8 �"0�2 8 0 $9T $98 »�2 7=$9&?+y8��I+J �!�ºX8FO 2 8%$�x�+3& 2 r���+y8�Y�+ » +�T $�6�+�Y�+ 0 +�& 2 )�T 0 $�p��R)�p*OS$JY�+&S+3/�& 2 8X'9)U+ 2 8X/9+3p�+�& 2 7 � @ 2 T +J8?s�$ 0 ¡ 2 T 2 Y 2 8]YZ+I�V��p�$9&?T�8�+ 2 8VQ*��p�$�&ST�83! �V��p�$�&ST�8 : ��& �"2 p�/�)�7 2 &?��p�$�&ST�8 E/�+�p�+3& 2 7 � @ 2 T 2 $�x�+3& 2 r�sz$�YZ+Áoqp*O?+�&S8?+ 0 r3sz$ : £ 4 � �

∩E +3p�'9) 2 p*O?$ '*)�+ Q���p�$9&?T�8 : ��& �"2 p�/�)U7 2 &F� 0 $9p�$�&ST�8 E

| 2 @�+3T¢$�T{+y8FT{$ 0 $9T 2 $�x�+3& 2 r3s�$5Y�+J«\p � sz$ : �Iv �∪E !

�V��p�$�&ST�8�T 2 x�+ �12 T[0, 1]× [0, 1] → [0, 1]

+wYZ+ » +�T 8 2 O � 8R| 2 @3+�& 0 � p 0 $ 2�¿ � $9T 2 8�!±# 2 & 2�2 x�&?+y8F+�p*O 2 &�+�8FO?+y82�¿ � $�T 2 8 »�2 T{$*8])�8 2 & 2 p�$�O 2 r3sz$

T (x, y)x 2 & 2 &S+3xU&?+y8F+�p9O 2 &��V��pU$�&ST{8V+H8?+R� 2 T

x, y, z+

w$*8�$�x�+3& 2 YZ$*83!

��+3T{$98�+�p9ODsz$NI 9!

T (0, 0) = 0K�!T (x, y) = T (y, x)

PU!T (x, 1) = x

��!T [T (x, y), z] = T [x, T (y, z)]

u�!T (z, w) ≤ T (x, y) se z ≤ x e w ≤ y£\8tx�& � T{+ � & 2 8H+BO?+3& 0 + � & 2 8J&S+3/9& 2 8J/ 2 & 2 p9OS+3T '9)U+ 2 $�x�+3& 2 r�sz$;x�$�Y�+³8?+3&-&S+�YZ)U@ � Y 2Á2 )�T 2 $9x�+�& 2 r3s�$;Y�+

)�p � sz$ 0 71��8S8 �"0�2 !�£\7=¤�T Y � 8?8?$ � 2 O?+�& 0 + � & 2 &?+�/�& 2 / 2 & 2 p9OS+ 2 + ¿ � 8ROS¼3p 0 �"2 YZ+V)�T +37"+3T{+�p9OS$�pU+3)ZOS&?$�!�£ 8F+�/�)�p�Y 2&S+3/�& 2 / 2 & 2 p9OS+-'*)�+ 2 $9&SYZ+�T¾YZ+ 2�»�2 7 �"2 r3sz$�Y 2 $�x�+3& 2 r3s�${pUsz$ � p��U)�+�p 0 �12 p�$B&?+y8F)�7=O 2 YZ$U!w£ '9) 2 &?O 2 &S+3/�& 22 8S8?+3/�)U& 2 '*)�+�x�$ZYZ+�T5$*8�| 2 @3+�& 2 )Up � s�$�YZ+I'*) 2 71'*)�+3&]p DUT5+�&?$�YZ+ 0 $�p��R)�p*OS$98]+3TÀ'*) 2 71'*)�+3&]$�&DYZ+3TÀ'*) 2 p�YZ$| 2 @�+3T{$98 2 $9x�+�& 2 r3s�$�YZ$ � 8 2 YZ$ � 8�! > � p 2 7"T{+3p*O?+ 2 '*) � p*O 2 &?+�/�& 2�2 8?8?+3/9)�& 2 '*)�+H)�T 2 Y � T � p�) � r3s�$�p�${/�& 2 )YZ+ � p 0 7")U8?s�$�p�$³/�& 2 )ÁYZ$*8 0 $�p��R)�p*OS$98�pUsz$�x�$ZYZ+5x�&?$ZYZ)U@ � &�)�T 2 )�T{+3p*O?$Bp�$B/9& 2 ) YZ+ � p 0 7=)U8Ssz$cY 2 )Up � s�$YZ$*8 0 $�pz�R)�p*O?$98�!£ $�x�+3& 2 r�sz$JYZ+wx�&S$ZYZ)ZO?$ 2 7=/9¤36�& �10 $�O 2 T�6�¤�T ¤w)UT 2 �V��p�$9&?T +wx�$ZYZ+\8?+3& )U8 2 Y 2 x 2 & 2 YZ+��Up � & 2 $9x�+�& 2 �

r3s�$tYZ+ � p*O?+�&S8?+ 0 r3s�$-YZ+ 0 $�p��R)�p*OS$98�pU+36�)�7"$98?$98�! 4 $t+�p*O 2 p*OS$ ��+ ¿ � 8FO?+�Tª$9)ZO?& 2 8X$�x�+3& 2 r3�*+y8�'*)�+\8 2 O � 8F| 2 @�+3T 2+�8FO?+y8 2�¿ � $�T 2 8%+�'9)U+(8F+�&Ss�$ 2 x�&S+�8?+3p*O 2 Y 2 8�T 2 � 8 2 Y �"2 p*O?+�!�#%$�Y�+\8F+�&�x�&S$ »�2 YZ$H'*)�+wx 2 & 2 '*) 2 7"'*)�+�&��V��p�$9&?T

T [µA(x), µB(y)] ≤ min[µA(x), µB(x)]: KZ!" �� E

+Hx�$�&?O 2 p*OS$ 2 $�x�+3& 2 r�sz${YZ+JT �=p � T{$�¤H)�T¢7 � T � O?+t8?)�x�+3& � $9&�x 2 & 2 +y8RO 2 $�x�+3& 2 r3s�$³!Q*��p�$�&ST�8\/9+3p�+�& 2 7 � @ 2 T¾$ 0 $�p 0 + � O?$³Y�+5$9x�+�& 2 r3s�$BYZ+5)�p � sz$U!tº�7 2 8JT 2 x�+ �12 T

[0, 1] × [0, 1] → [0, 1]+

O 2 Tt6�¤3T 8 2 O � 8R| 2 @3+�T 0 � p 0 $ 2�¿ � $9T 2 8�! # 2 & 2w2 x�&S+�8?+3p*O 2 &�+y8ROS+�8 2z¿ � $9T 2 8 »�2 T{$98 )�8 2 &±$(8 �"Tt6�$�7"$S(x, y)

x 2 & 2&S+3x�&S+�8?+3p*O 2 &�Q���p�$9&?T�8�+H8?+R� 2 T

x, y, z+

w$98�$9x�+�& 2 pUYZ$98�!%��+�T5$*8�+3p*OSs�$JI

Page 32: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

Kz� � ���`� ��� � ���R� � � � ���� � � � ���� � ��� 9!

S(1, 1) = 1

K�!S(x, y) = S(y, x)

PU!S(x, 0) = x

��!S[S(x, y), z] = S[x, S(y, z)]

u�!S(z, w) ≤ S(x, y) se z ≤ x e w ≤ y

£\8tx�& � T{+ � & 2 8H+BO?+3& 0 + � & 2 8J&S+3/9& 2 8J/ 2 & 2 p9OS+3T '9)U+ 2 $�x�+3& 2 r�sz$;x�$�Y�+³8?+3&-&S+�YZ)U@ � Y 2Á2 )�T 2 $9x�+�& 2 r3s�$;Y�+� p9OS+3&D8F+ 0 r�sz$ 0 71��8S8 �"0�2 !(£(7"¤3T Y � 8S8F$ � 2 OS+3& 0 + � & 2 &?+�/�& 2 / 2 & 2 p9OS+ 2 + ¿ � 8ROS¼3p 0 �"2 YZ+t)�TÂ+�7=+�T5+�p*O?$�p�+�)ZO?&S$U!�£8?+3/�)UpUY 2 &?+�/�& 2 / 2 & 2 p*O?+V'9)U+ 2 $9&SYZ+�T YZ+ 2�»�2 7 �"2 r3s�$\Y 2 $�x�+3& 2 r�sz$\p�sz$ � p �U)�+3p 0 �12 pU$I&S+�8?)�7=O 2 YZ$U!�£ '*) 2 &FO 2&S+3/�& 2Á2 8?8?+3/9)�& 2 '*)�+³x�$ZYZ+�T{$98t| 2 @3+�& 2 � p*OS+3&D8F+ 0 r3s�$ YZ+;'*) 2 71'*)�+3&�p DUT5+�&?$ YZ+ 0 $�pz�R)�p*O?$98t+3T~'*) 2 7"'*)�+�&$�&DYZ+�T '*) 2 pUYZ$�| 2 @�+3T{$98 2 $�x�+3& 2 r�sz$ YZ$ � 8 2 YZ$ � 83! > � p 2 7"T{+3p*O?+ 2 '9) � p*O 2 &?+�/�& 2 2 8?8?+3/9)�& 2 '*)�+ )�T 2Y � T � p�) � r3s�$�p�$5/�& 2 )³YZ+ � p 0 7=)U8Ssz$�p�$5/�& 2 )BYZ$*8 0 $�pz�R)�p*O?$98]pUs�$�x�$ZYZ+�xU&?$ZYZ)�@ � &�)�T 2 )�T{+3p*O?$5p�$5/�& 2 )BY�+� p 0 7")U8Ssz$�Y 2 � p*OS+3&D8F+ 0 r3s�$5Y�$98 0 $9p��R)�p*O?$*83!£ $9x�+�& 2 r3s�${Y�+�8?$�T 2 pUs�$B¤-)�T 2 Q*��pU$�&ST¾+�pUsz$�x�$ZYZ+�8?+3&I)U8 2 Y 2 x�$9& 0 & � O?¤3& � $98wT 2 OS+3T���O �"0 $98\x 2 & 2YZ+��Up � & 2 $9x�+�& 2 r3sz$wYZ+V«(p � sz$(Y�+ 0 $9p��R)�p*O?$*8±p�+�6�)�7"$98?$98�!�«(T 2 $9x�+�& 2 r3s�$ 2 7 OS+3&Sp 2 O �=»�2 ¤ 2 YZ+X8?$�T 2 7 � T � O 2 Y 2

'*)�+J¤-YZ+���p � Y�$ 0 $�T{$NIµA(x) + µB(x) − µA(x) × µB(x)#%$ZYZ+-8F+�&�x�&?$ »�2 YZ$5'*)�+Hx 2 & 2 '*) 2 7"'*)�+�&wQ���p�$9&?T

max[µA(x), µB(x)] ≤ S[µA(x), µB(y): KZ!" �� E

+{x�$9&FO 2 p*O?$ 2 $�x�+3& 2 r�sz$cYZ+�T�� ¿ � T{$;¤{)�T 7 � T � O?+ � pZ|}+�& � $�&Hx 2 & 2 +�8FO 2 $9x�+�& 2 r3sz$�!{�I8H8?+3/9) � p*OS+�8J7 � T � O?+�8x�$�Y�+3T¾8?+3&w+y8RO 2 6�+37"+ 0 � YZ$*8Vx 2 & 2 Q*��pU$�&ST{8V+-�V��p�$�&ST�8

T [µA(x), µB(y)] ≤ min[µA(x), µB(x)] ≤ max[µA(x), µB(x)] ≤ S[µA(x), µB(y): KZ!LKz� E

��x�+3& 2 r���+y8�6 2 8?+ 2 Y 2 8�p�+y8RO 2 8\p�$�&ST 2 8wx�$ZYZ+3T 8?+3&wOS$�&Sp 2 &(&S+�8FO?& � O �"»�2 8w+-$*8 0 ¡ 2 T 2 YZ$98�$9x�+�& 2 YZ$�&S+�8wY�+2 /9&?+�/ 2 r3sz$³8?s�$c+�T{x�&?+�/ 2 YZ$98Ix 2 & 2 p�+�6�)�7 � @ 2 & 2 &?+�/ � sz$c+3p*O?&S+ 2 8J$9x�+�& 2 r���+�8JYZ+�T �"p � T5$¨+�T�� ¿ � T{$¨'9)U+pUs�${8Ssz$ 0 $96�+�&FO 2 8�x�+37 2 8�p�$9&?T 2 83!�#%$�&?O 2 p*OS$�OS+3T{$98w'*)�+

min(x1, x2, . . . , xn) ≤ A(x1, x2, . . . , xn) ≤ max(x1, x2, . . . , xn): KZ!LKZ E

$�p�YZ+A&S+3x�&S+�8?+3p*O 2 )�T $9x�+�& 2 YZ$�&JYZ+ 2 /9&?+�/ 2 r3sz$�!I«\T $9x�+�& 2 YZ$�&IYZ+�8FO?+5O � x�$c6 2 8FO 2 p*OS+�+�T5xU&?+�/ 2 Y�$B¤ 2

T{¤�Y �12{2 & � OST{¤�O �1032 IA(x1, x2, . . . , xn) =

1

n

n∑

i=1

xi

�"��� �� #3� -��� # 6BA �2> � #�46G- �7'�� *,-C6 # / ����� - @4#��M\��)�TÂ+3p�$9&?T{+-+y8R|}$9&Sr3$�YZ+tx�+�8S'*) � 8 2 x 2 & 2 Y�+��Up � &�|}$�&ST 2 8wp�+�6�)�7"$98 2 8 2 7 OS+3&Sp 2 O �=»�2 8(YZ+t$9x�+�& 2 r���+�8(+3p*O?&S+0 $9p��R)�p*O?$*83!{£ + ¿ x�+3& � ¼�p 0 �12 x�&S�zO �1032 O?+�T YZ+�T5$9pU8FO?& 2 YZ$;'*)�+5O 2 � 8I|}$�&ST 2 8 2 7=O?+�&?p 2 O �"»�2 8J8Ssz$cp�+ 0 +�8S8?��& �12 8�!oR8FO?$c$ 0 $�&S&?+tx�& � p 0 � x 2 7=T{+�p9OS+�pU$ 0�2 8?$³Y 2 $�x�+3& 2 r�sz$BYZ+ � p*O?+3&D8?+ 0 r�sz$ ��+{+3T )UT /�& 2 )ÁT{+�p�$�&Ip�$ 032 8?$³Y�$0 $9T{x�7=+�T{+3p*O?$�!ºX8FO 2 8w$�x�+3& 2 r3�*+y8w8?s�$ 0 ¡ 2 T 2 Y 2 8 0 $9T{x�+�pU8 2 O?.9& �12 8�x�$�&D'*)�+H+37 2 8wxU&?$ 0 )U& 2 T 0 $9T5x�+3p�8 2 & 2 8(YZ+��Up � r���+�8

+�8FO?& � O 2 8�Y 2 Y 2 8wx�$�&-  2 Y�+3¡¨x 2 & 2 $*8($�x�+3& 2 Y�$�&S+�8w£ 4 � ���Iv + 4 �I�J!UºX7 2 8\x�&S$ 0 )�& 2 TW| 2 @�+3& 0 $�TÂ'*)�+ 2 8&S+37 2 r���+y8]+�p*O?&S+IxU&?$9x�$*8 � r���+�8 0 )��R$98 »�2 7"$�&S+�8V+y8ROS+R� 2 T¾Tt) � O?$�8?+3x 2 & 2 YZ$98V8?+R� 2 T¾T{+3p�$*8w8F+�pU8 � » + � 8�!� 2 T{$*8 2 p 2 7 � 8 2 &�+y8ROS+�8%x�&S$�6U7=+�T 2 8 » +3& � � 032 pUYZ$I$J'*)�+w$ 0 $�&S&?+ 0 $9T 2 $�x�+3& 2 r3s�$�YZ+ � p*O?+�&S8?+ 0 r3sz$5!�[]$�T{$

)�T 2 $�x�+3& 2 r�sz$cYZ+ � p*OS+3&D8F+ 0 r3s�$ 0 $�T�6 � p 2 Y �"» +�&S8?$98 0 $�p��R)�p*OS$98 ��)ZO � 7 � @ 2 pUY�$ 2 $9x�+�& 2 r3sz$;YZ+�T �"p � T{$ ��)�TD�p �10 $ »�2 7"$�&%6 2 � ¿ $\x�$ZYZ+�T 2 8 032 & 2 &�O?$ZYZ$*8�$*8�$�)ZOS&?$*8 0 $�T{x�$9p�+3p*OS+�8�!�oR8FO?$IO?& 2 @]x�&?$96�7"+3T 2 8%YZ+�YZ+�8?+3T{x�+3p�¡U$p�$*8 0 $�p*O?&S$�7 2 YZ$9&?+y8�p�+36U)�7=$*8F$*83! []$�p*O?&S$�7 2 Y�$�&S+�8�p�+�6�)�7"$98?$98 OS& 2 6 2 7"¡ 2 T 0 $�T )�T 2 8?¤3& � +]YZ+V&?+�/�& 2 8±Y 2 |}$�&ST 2o > 2 p9OS+ 0 +�YZ+�p9OS+�8w�wM\º 4¦0 $9pU8F+y'LK�+3p*O?+9!X[ 2 Y 2 )�T 2 YZ+�8FO 2 8]&S+3/9& 2 8X¤ 2�»�2 7 �12 Y 2 +J$5x�+�8?$�YZ+ 032 Y 2 )UT¾YZ$*80 $9pU8?+�'LK�+3p*OS+�8 ��p 2 8?$�7")Ur3s�$ ��p 2 7 ��¤tY 2 YZ${x�+�7=$B&?+y8F)�7=O 2 YZ$�Y 2 $�x�+3& 2 r�sz$�YZ+ � p*O?+3&D8?+ 0 r�sz${+ ¿ + 0 )ZO 2 Y 2 8?$�6�&S+

Page 33: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R���U� ��� � � �s��� � � � � � � ���� � ��� � ��� � * ��� � ����� K*�$98 2 p*O?+ 0 +yYZ+3p*O?+y83! #�$9&FO 2 p*O?$ �%)�T 0 $�T{x�$�p�+�p*O?+BYZ+³)�T 2 &S+3/9& 2 '*)�+BO?+3pU¡ 2 )�T »�2 7"$�&5YZ+�|})UpUr3s�$Á6 2 �=¿ $x�$�Y�+J| 2 @3+�& 0 $9TW'*)�+ 2 '*)�+37 2 &?+�/�& 2 p�sz$�O?+3pU¡ 2 x�+y8F${&S+37"+ »�2 p*OS+Ip 2 8?$�7")Ur�sz${YZ${x�&?$96�7"+3T 2 !� 2 T{$*8 0 $�pU8 � YZ+3& 2 &])�T 2 &S+3/9& 2 + ¿ +3T{x�7"$5+ » +3& � � 032 & 0 $9T5$5+�8FO?+JT 2 8 032 & 2 T{+3p*O?$-$ 0 $�&S&?+9! []$9pU8 � YZ+�&?+ 2

8?+3/�) � p*O?+J&S+3/9& 2o > ,!/ aG2{oFQ µ�m�e � � £ 4 � V � H�/c.A:_8;OU8B1�.�1 oFQ m�°}�ym £ 4 � �� 1A8;O48B1�. >l #� :f2�oFQ k0.A8�352��wM(º 4 H�2 > ,o/

� ��e}���>U2 @3+�pUYZ$ 0 $�T�'*)�+ 2 8 »�2 & � � » + � 8 2 8?8?)�T 2 T »�2 7=$9&?+y8V+ ¿ +3T{x�7"$ �Zx�$ZYZ+3T{$98�+y8 0 &?+ » +�&

µmuito(compre) = min(µbaixo(preco), µalta(lucratividade), µbaixo(endividamento): K�! K9K E

µmuito(compre) = min(0.25, 0.98, 0.85) = 0.25: KZ!LKzP E

��6U8?+3& » +X'*)�+V$Ix�+y8F$ �Up 2 7�Y 2 &S+3/9& 2 � 0 $9) � /�) 2 7��U! K9u�� 2 x�+�8 2 &%YZ+wYZ$ � 8%YZ$*8 »�2 7"$�&S+�8�+3T{x�&S+3/ 2 YZ$*8±|}$*8?8?+3T6�+3TWx�+3&?O?$�YZ+� 9! �U!M\� YZ$ � 8�O � x�$*8t6���8 �"0 $985YZ+³$9x�+�& 2 r���+�8 2 7=O?+�&?p 2 O �"»�2 8 � 2 '*)�+37 2 8t6 2 8?+ 2 Y 2 8t+3T $9x�+�& 2 r���+�8 2 & � OST5¤3O �1032 8

8 � T{x�7"+�8{+ 2 856 2 8F+ 2 Y 2 8t+�T |})�pUr3�*+y8{T 2 � 8 0 $�T{x�7"+ ¿�2 8�! º ¿ +�T5xU7=$*8�YZ+;$�x�+3& 2 r���+y8�8 � T{x�7"+�8{8Ssz$NI³8?$�T 27 � T � O 2 Y 2 �Vx�&?$ZYZ)�O?$ +¨T{¤�Y �12 ! £ 0 7 2 8S8F+¨YZ+;|})�p�r���+�8�YZ+Á© 2 /�+�&{¤¨)�T�+ ¿ +�T{x�7=$ Y�$ 8F+�/�)�pUY�$ O � x�$U!£+�8 0 $97=¡ 2 Y�$ $�x�+3& 2 YZ$9& 0 $�T{x�+3pU8 2 OS.�& � $ x 2 & 2 )�T¸8 � 8FO?+3T 2 x 2 &FO �"0 )�7 2 &�&S+�'*)�+3& 2 7=/9)�T 2 + ¿ x�+�& � ¼3p 0 �12 +¨¤/�+�& 2 7=T{+�p9OS+J|}+ � O?$³¡�+3)�& � 8RO �"0�2 T{+3p*O?+ �U$96U8F+�& »�2 pUYZ$z�q8?+I$B+�|}+ � O?$³Y 2 x�&S$�x�$98FO 2 !�«\T 2 &S+3/�& 2 x�&D��O �1032 Y � @�'9)U+YZ+ » +��q8?+5xU&?$ 0 )U& 2 &F�q8F+{8?+3T{x�&S+ 0 $�T{+�r 2 &Hx�+37 2 8F$97=)�r3sz$;T 2 � 8-8 � T{x�7"+�8�!B#X& � T{+ � &?$cO?+y8RO 2 �q8F+�$*8J$�x�+3& 2 YZ$9&?+y8YZ+   2 YZ+3¡ �]YZ+�x�$ � 8{$*8{$�x�+3& 2 YZ$9&?+y8tx�&S$ZYZ)ZOS$ �VT5¤yY �12 +¨8?$�T 2 7 � T � O 2 Y 2 ! #�$9&�D�7=O � T5$ 2 85|})�pUr���+y8�T 2 � 80 $9T{x�7=+ ¿�2 8]O � x�$ 2 8wYZ+J© 2 /9+3&y!���� ��� �����������������12 �����-��! #�� �£ 0 7 2 8S8?+JY�+H$�x�+3& 2 r���+y8 2 7=O?+3&Sp 2 O �"»�2 8 � 7")U8FO?& 2 Y 2 8�p 2 � 2 6�+�7 2 K�! u ��Tt)UY 2 T 2 «\p � s�$�+ 2 oqp9OS+3&D8F+ 0 r�sz$5x 2 & 2$�)�O?& 2 8w$9x�+�& 2 r���+�8 2 7=/9¤36�& �1032 8\8 � T{x�7=+y83!(£�8\$9x�+�& 2 r���+�8\YZ+{  2 YZ+3¡¨+y8RODsz$�T{$98FO?& 2 Y 2 8wp 2 � 2 6�+�7 2 KZ!Lu5x 2 & 2&S+�|}+3&S¼3p 0 �12 !

� 2 6�+37 2 K�! u IX��x�+�& 2 r���+�8 2 7"/�¤�6�& �"0�2 8 0 $9T5x�+3p�8 2 O?.�& �12 8VYZ+Hoqp*O?+3&D8?+ 0 r�sz$�+-«(p � sz$

������ ���������������! !"#� $%�&�'"���)(+*�,�-

min(µA(x), µB(x)) max(µA(x), µB(x))

Product1 µA(x) × µB(x) µA(x) + µB(x) − (µA(x) × µB(x))

Media (µA(x) + µB(x))/2 (2 × min(µA(x), µB(x)) + 4 × (max(µA(x), µB(x)))/6

Media2 Media2 Media2

Media1

2 Media1

2 Media1

2+/.+01( �324052768(!*�(max(0, µA(x) + µB(x) − 1) min(1, µA(x) + µB(x))

� 2 6�+�7 2 K�! �JI%� � x�$98wYZ+H$9x�+�& 2 r���+�8 2 7"/�¤�6�& �1032 8 0 $�T{x�+3pU8 2 OS.�& �"2 8]Y�+Hoqp9OS+3&D8F+ 0 r�sz$t+-«\p � sz$� � x�$ oqp*O?+�&S8?+ 0 r3s�$ «(p � sz$  2 YZ+3¡

min(µA(x), µB(y)) max(µA(x), µB(y))#�&?$ZYZ)�O?$µA(x) × µB(y) µA(x) + µB(y) − µA(x) × µB(y)§¨¤yY �"2

(µA(x) + µB(y))/2 (2 × min(µA(x), µB(y)) + 4 × max(µA(x), µB(y)))/6Q�$9T 2 , � T � O 2 Y 2max(0, µA(x) + µB(y) − 1) min(1, µA(x) + µB(y)

# 2 & 2 | 2�0 � 7 � O 2 & 2 8 0 $9T5x 2 & 2 r���+y8�+3p*O?&S+H+�8FO?+y8w$�x�+3& 2 pUY�$98 »�2 T{$98(T5$*8ROS& 2 &w/�&D��� 0 $98�+HO 2 6�+37 2 8(Y�+ 032 Y 2)�T 2 Y 2 8H|})UpUr���+�8 0 $9T{x�+�pU8 2 O?.9& �12 8�! []$�T{$Á$ x 2 YZ&Dsz$ÁYZ+ 0 $9T5x 2 & 2 r3s�$c8Ssz$ $98-$9x�+�& 2 YZ$�&S+�8tYZ+;  2 YZ+3¡ �

Page 34: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

Kz� � ���`� ��� � ���R� � � � ���� � � � ���� � ���T{$98FO?& 2 T{$98Ip 2 8 > � /9)�& 2 8HKZ!LK��;+�KZ!LK�uB$98J/�&D��� 0 $98JYZ+y8ROS+�8J$9x�+�& 2 YZ$�&S+�8�!5£�8JO 2 6�+�7 2 8-KZ!��B+�K�! �cT{$98FO?& 2 T2 7"/�)UpU8 »�2 7"$�&S+�8V&?+y8F)U7 O 2 p*O?+y8]x 2 & 250�2 Y 2 )�T 2 YZ+y8RO 2 8�$�x�+3& 2 r3��+�8�!

f(x,y)

0

0.5

1 0

0.5

1

0

0.5

1

and = min(x,y)

> � /9)�& 2 K�! Kz�JI���x�+�& 2 YZ$�&w£ 4 � YZ+t  2 YZ+3¡�!

f(x,y)

0

0.5

1 0

0.5

1

0

0.5

1

or = max(x,y)

> � /9)�& 2 K�! K9u I���x�+�& 2 YZ$�&\�Iv YZ+�  2 YZ+3¡±!

_ # � #f�$#f� ��"Xb*��mU����� ! ���±�����y��¢$�x�+3& 2 YZ$9&�x�&S$ZYZ)ZOS$;O?+3T 2 x�&S$�x�& � +yY 2 Y�+5YZ+�&S+�8?x�+ � O 2 & 2 8 032 & 290 O?+�&&�18FO �1032 8�T � p ��T 2�¿ Y 2 $�x�+3& 2 r3s�$cY�+� p9OS+3&D8F+ 0 r�sz${YZ+��Up � Y 2 x�$�&�  2 YZ+3¡�!

0 AND ∀(µA(x)) = 0

1 AND ∀(µA(x)) = µA(x)

«(T 2 $�)ZOS& 2�032 & 290 O?+�&&�18FO �1032 � p*O?+3&S+�8S8 2 p*O?+�¤I'*)�+I$�xU&?$ZYZ)ZOS$�T�)UY 2 x 2 & 2t0�2 Y 2 x 2 &wYZ+ :µA(x), µB(x)

E !oR8FO?${pUsz${$ 0 $9&?&S+Ip�+ 0 +�8S8 2 & �12 T{+3p*OS+�'*) 2 p�YZ$�8F+J)U8 2 $�$�x�+3& 2 YZ$9&]T �"p � T{$�!£\8 > � /�)U& 2 8\KZ!LKz�5+�K�! K*�-T{$*8ROS& 2 TW$98(/�&D��� 0 $98wYZ+y8ROS+�8w$9x�+�& 2 YZ$�&S+�8�!X£\8(O 2 6�+�7 2 8\KZ! �{+�KZ!" ��5T{$98FO?& 2 T

2 7"/�)UpU8 »�2 7"$�&S+�8V&?+y8F)U7 O 2 p*O?+y8]x 2 & 250�2 Y 2 )�T 2 YZ+y8RO 2 8�$�x�+3& 2 r3��+�8�!

Page 35: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R���U� ��� � � �s��� � � � � � � ���� � ��� � ��� � * ��� � ����� K����! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

��! �����9� ��! ���9��� ��! ���9��� �U! �9����� �U! �9����� �U! �9�������!LK�uz�9� ��! ���9��� ��!LK�u���� �U! K9uz��� �U! K9uz��� �U! K9uz�����!Luz���9� ��! ���9��� ��!LK�u���� �U! u������ �U! u������ �U! u��������!��zuz�9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� �U!L��uz��� �! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

� 2 6�+�7 2 K�!L� I%£ 4 � 0 $�T¢$9x�+�& 2 YZ$�&�T �"p � T{$ :min(µ(x), µ(y))

E !��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

��! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!LK�uz�9� ��!LK�u���� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!Luz���9� ��!Luz�9��� ��!Luz�9��� �U! u������ �U!L��uz��� 9! �9�������!��zuz�9� ��!��zu���� ��!��zu���� �U!L��uz��� �U!L��uz��� 9! �9����� �! �����9� �! ���9��� �! ���9��� 9! �9����� 9! �9����� 9! �9�����

� 2 6�+37 2 K�! �JIX�Iv 0 $�T¢$9x�+�& 2 YZ$�&VT�� ¿ � T5$ :max(µ(x), µ(y))

E !

_ # � #f�$#1_ ��"Xb*��mU������b*h �����efm�ª$9x�+�& 2 YZ$�&IT{¤�Y �12 pUsz$c¤58?+3p�8 � » +�7�+�T + ¿�0 +�8S8F$³x 2 & 2 $*8 »�2 7=$9&?+y8\T �"p � T5$ ��+{T�� ¿ � T{$�!t#%$�&J+ ¿ +�T5xU7=$ �2 8S8?)�T � pUY�$5$*8w8F+�/�) � p*O?+y8 »�2 7=$9&?+y8MI

µA(x) = 0.94, µB(x) = 0.58, µC(x) = 0.23

��+3T{$98�p 2 $9x�+�& 2 r3sz$5YZ+���p � Y 2 x�$9&�  2 YZ+�¡µA(x)AND µB(x)AND µC(x) = 0.23

+Hp 2 $�x�+3& 2 r�sz$5YZ+HT{¤�Y �12(µA(x)AND µB(x))AND µC(x) ≡ ((0.94 + 0.58)/2 + 0.23)/2 = 0.495

«(T­x�&S$�6U7=+�T 2 ¤c'*)�+ 2 $�&DYZ+�T YZ+ 0 ��7 0 )�7=$ ¤ � T{x�$9&FO 2 p*O?+9!�[ 2 8?$ 2 $�&DYZ+3T­+3T '*)�+³$98�$�x�+3& 2 pUYZ$*82 x 2 &S+ 0 +�T�8?+R� 2 T{$ZY � � 032 Y 2 $ &S+�8?)�7 O 2 YZ$ Y 2 + ¿ x�&S+�8S8?s�$ ¤ 2 7 OS+3& 2 YZ$�! �I8BT5+y8FT{$*8 »�2 7"$�&S+�8 �]+�T�$9)ZO?& 2$�&DYZ+�T ��|}$9&?pU+ 0 +�T¢$�8F+�/�) � p*O?+J&S+�8?)�7 O 2 YZ$

(µC(x)AND µB(x))AND µA(x) ≡ ((0.23 + 0.58)/2 + 0.94)/2 = 0.6725

£\8 > � /�)�& 2 8VKZ!LKz�-+HKZ!LKz�tT{$98FO?& 2 T $98]/�&D��� 0 $98]YZ+�8FO?+y8]$9x�+�& 2 YZ$�&S+�8�!�£�8]O 2 6�+37 2 8wK�!= 9 (+JK�!= �KHT{$98FO?& 2 T2 7"/�)UpU8 »�2 7"$�&S+�8V&?+y8F)U7 O 2 p*O?+y8]x 2 & 250�2 Y 2 )�T 2 YZ+y8RO 2 8�$�x�+3& 2 r3��+�8�!

��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������! �����9� ��! ���9��� ��! ���9��� �U! �9����� �U! �9����� �U! �9�������!LK�uz�9� ��! ���9��� ��! ���*K�u �U!= �K�uz� �U!= y�*�zu �U! K9uz�����!Luz���9� ��! ���9��� ��!" yK9uz� �U! K9uz��� �U! P��zuz� �U! u��������!��zuz�9� ��! ���9��� ��!" ����zu �U! P��zuz� �U! u��9K�u �U!L��uz��� �! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

� 2 6�+�7 2 KZ! � I £ 4 � 0 $�T¢$9x�+�& 2 YZ$�&�x�&S$ZYZ)ZO?$ :µ(x) ∗ µ(y)

E !

Page 36: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

P�� � ���`� ��� � ���R� � � � ���� � � � ���� � �����! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

��! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!LK�uz�9� ��!LK�u���� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!Luz���9� ��!Luz�9��� ��!Luz�9��� �U! u������ �U!L��uz��� 9! �9�������!��zuz�9� ��!��zu���� ��!��zu���� �U!L��uz��� �U!L��uz��� 9! �9����� �! �����9� �! ���9��� �! ���9��� 9! �9����� 9! �9����� 9! �9�����

� 2 6�+37 2 K�!= y� IX�Ivx 2 & 2 ${$9x�+�& 2 YZ$�&�x�&S$ZYZ)ZO?$ :(µ(x) + µ(y)) − (µ(x) ∗ µ(y))

E !

f(x,y)

0

0.5

1 0

0.5

1

0

0.5

1

and = x*y

> � /9)�& 2 KZ!LKz�JIX��x�+�& 2 YZ$�&�£ 4 � 0 $9T¢x�&S$ZYZ)ZO?$�!

_ # � #f�$#"® ��� � m � e%� e}�ym���m£\8 > � /9)�& 2 8-KZ! P��c+³K�! PU 5T{$*8ROS& 2 T $98-/�&D��� 0 $98JYZ+�8FO?+y8-$9x�+�& 2 YZ$�&S+�8�!�£\8-O 2 6�+37 2 8�KZ!" �Pc+³KZ!" 3�;T{$98FO?& 2 T2 7"/�)UpU8 »�2 7"$�&S+�8V&?+y8F)U7 O 2 p*O?+y8]x 2 & 250�2 Y 2 )�T 2 YZ+y8RO 2 8�$�x�+3& 2 r3��+�8�!

� � p9OS+3&S+�8S8 2 p*O?+{p�$zO 2 &-'*)�+�$¨$9x�+�& 2 YZ$�&-8?$�T 2 7 � T � O 2 Y 2 $�6�+�YZ+ 0 + 2 7"+ � Y 2 + ¿�0 7")U8Ssz$¨YZ$ÁT{+ � $ 0 $�T{$0 $9pU8?+�'LK�¼3p 0 �12 Y 2 |}$�&ST 2Á0 $9T{$Á$Á$�x�+3& 2 Y�$�&J|}$ � YZ+��Up � YZ$U!#� 2 T5$*8 0 $�pU8 � Y�+3& 2 &H)�T »�2 7"$�&µA(x)

�%$ 8F+�)0 $9T{x�7=+�T{+3p*O?${¤HY 2 Y�$5x�+37"$ »�2 7"$�&

µA(x) = 1 − µA(x)

£ $�x�+3& 2 r�sz$5YZ+ � p*OS+3&D8F+ 0 r3s�$ 0 $�T¢${$9x�+�& 2 YZ$�&�T �=p � T{$ 2 x�7 �1032 Y 252 +y8ROS+�8 »�2 7"$�&S+�8]|}$�&Sp�+ 0 +

µA(x) AND µA(x) = min(µA(x), µA(x)),

� 2 6�+37 2 K�!= 9 I £ 4 � 0 $9T¢$�x�+3& 2 Y�$�&�T{¤�Y �12;:(µ(x) + µ(y))/2.0

E !

��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������! �����9� ��! ���9��� ��!" yK9uz� �U! K9uz��� �U! P��zuz� �U! u��������!LK�uz�9� ��!" yK9uz� ��!LK�u���� �U! P��zuz� �U! u������ �U! �*K�uz���!Luz���9� ��!LK�u���� ��! P*��uz� �U! u������ �U! �*K�uz� �U!L��uz�����!��zuz�9� ��! P*��uz� ��!Luz�9��� �U! �*K�uz� �U!L��uz��� �U! ���zuz� �! �����9� ��!Luz�9��� ��! �9K9uz� �U!L��uz��� �U! ���zuz� 9! �9�����

Page 37: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R���U� ��� � � �s��� � � � � � � ���� � ��� � ��� � * ��� � ����� PU

f(x,y)

0

0.5

1 0

0.5

1

0

0.5

1

or = (x+y)-(x*y)

> � /�)�& 2 KZ!LK9� I ��x�+3& 2 YZ$9&\�Iv x�&S$ZYZ)ZOS$U!

� 2 6�+�7 2 K�!= �K I��Iv x 2 & 2 ${$9x�+�& 2 YZ$�&VT{¤�Y �12¨:(2 ∗ min(µ(x) + µ(y)) + 4 ∗ max(µ(x) ∗ µ(y)))/6

E !

��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������! �����9� ��! ���9��� ��!" ��9�*� �U! P9P�P�P �U! u������ �U! �9���*���!LK�uz�9� ��!" ��9�*� ��!LK�u���� �U! �� ��*� �U! u���P�P �U!L��uz�����!Luz���9� ��! P�P9P�P ��! �U y�*� �U! u������ �U! �9���*� �U! �9P�P�P��!��zuz�9� ��!Luz�9��� ��!Luz�9P�P �U! �9���*� �U!L�z����� �U! �U ��*� �! �����9� ��! ���9�*� ��!��zu���� �U! �9P�P�P �U! �U ��*� 9! �9�����

+3p�'9) 2 p*O?$�'*)�+J+�8FO 2 $�x�+3& 2 r�sz$ 0 $9T¢${$�x�+3& 2 YZ$9&�8?$�T 2 7 � T � O 2 Y 2 |}$9&?pU+ 0 +

µA(x) AND µA(x) = max(0, µA(x) + µA(x) − 1) = 0

oR8FO?$�8 � /�p � � 032 '*)�+�pU+�8FO?+ 2 8?x�+ 0 O?$�+�8FO 2 $�x�+3& 2 r�sz$-+y8ROD�-T 2 � 8Vx�&?. ¿ � T 2 Y 2 7=.9/ �1032 �V$�$�7"+ 2 p 2 YZ$5'*)�+IY 2p�+�6�)�7"$98 2 !

���� �� �������� �-' � ������� � ����� � � � % � ,-)��������

£ 0 7 2 8S8F+\Y�+\$�x�+3& 2 r3��+�8 2 7 OS+3&Sp 2 O �=»�2 8 � 7")U8FO?& 2 Y 2 8�p 2 � 2 6�+�7 2 KZ!" yu ��Tt)�Y 2 T 2 «\p � sz$t+ 2 oqp*O?+�&S8?+ 0 r3s�$Hx 2 & 2|})�pUr3��+�8wT 2 � 8 0 $9T{x�7=+ ¿�2 8�!�£�8w$�x�+3& 2 r3�*+y8�YZ+t  2 YZ+�¡³+y8RODsz${T{$98FO?& 2 Y 2 8Vp 2 � 2 6�+�7 2 x 2 & 2 &?+3|}+3&S¼3p 0 �"2 !

��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������! �����9� ��! ���9��� ��! ���9��� �U! �9����� �U! �9����� �U! �9�������!LK�uz�9� ��! ���9��� ��! ���9��� �U! �9����� �U! �9����� �U! K9uz�����!Luz���9� ��! ���9��� ��! ���9��� �U! �9����� �U! K9uz��� �U! u��������!��zuz�9� ��! ���9��� ��! ���9��� �U! K9uz��� �U! u������ �U!L��uz��� �! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

� 2 6�+37 2 K�!= yP I%£ 4 � 0 $9TW8?$�T 2 7 � T � O 2 Y 2max(0, µ(x) + µ(y) − 1)

!

Page 38: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

P9K � ���`� ��� � ���R� � � � ���� � � � ���� � ���

f(x,y)

0

0.5

1 0

0.5

1

0

0.5

1

and = (x+y)/2

> � /9)�& 2 K�! K�� I���x�+�& 2 YZ$�&w£ 4 � 0 $�T¢T{¤�Y �"2 !f(x,y)

0

0.5

1 0

0.5

1

0

0.5

1

or media

> � /9)�& 2 K�! K�� IX��x�+3& 2 YZ$9&w�Iv 0 $9T¢T{¤�Y �12 !

_ # � #1_ #f� ��"Xb*��mU������b*h�²���� "Xb���hymZ� �U��ef��ht��b��Hm $�b9�#�&S$�x�$98FO?$*8Bx�$�&¨vw$9p © 2 /9+3&¨YZ$ oqpU8FO � O?)ZOS$�x 2 & 2 oqp*O?+�7 � /�¼�p 0 �12 Y 2 §Á�9'9) � p 2 YZ$ oq$�p 2 []$�7"7"+3/�+9!ÀºX8FO?+�8$�x�+3& 2 Y�$�&S+�8�O?+�T )UT x 2 &S½�T{+�O?&S$

k��'9)U+ »�2 & �12 +3p*OS&?+\�J+

∞�*&?+�x�&?+y8F+�p*O 2 p�YZ$ 2 |}$�&Dr 2 Y 2H0 $�p�+ ¿ s�$J+3p*O?&S+($98

0 $9p��R)�p*O?$*8%+w8?s�$ � T{x�$9&FO 2 p*O?+y8�p 2I0 $�pU8FO?&S)Ur�sz$�Y�+VT{$ZYZ+�7=$*8%p�+36�)U7=$*8F$*8 0 $�&S&?+3O?$*8�+ » +�& � � 0 � » + � 8�! � 2 T{+�8?T 2T 2 p�+ � & 2 '*)�+ 2 7".�/ �1032 �V$�$�7"+ 2 p 2 ��YZ+ » � YZ$ 2 8?+3) 0�2 &D��O?+�&�p � O � Y�$ 2 x�&S+�8?+3p*O 2 &wY � � 0 )�71Y 2 YZ+�8w+�T¾T{$ZYZ+�7 2 &w$Tt)UpUYZ$�&S+ 2 7 ��$B+�8S'9)U+3T 2 T �=p � T{$ ��T{� ¿ � T{${|}&S+�'LK�+3p*OS+3T{+3p*O?+t&?+y8F)�7=O 2 +�T¾)UT�T{$ZYZ+�7=$�'*)�+t8?+ 0 $�T{x�$9&FO 2+3&S& 2 O �1032 T{+3p*O?+('*) 2 pUYZ$H$98]Y 2 Y�$98�+y8RODsz$Hp�$*8�+ ¿ OS&?+�T{$98�YZ$ 0 $9p��R)�p*O?$t8?)�x�$�&?O?+9! 4 +�8FO?+ 0�2 8?$Hx�$ZYZ+�T{$98�)�8 2 &${x 2 &D½zT{+3O?&S$

kx 2 & 25»�2 & �"2 & 2 p�+�6�)�7"$98 � Y 2 YZ+HYZ$�8 � 8FO?+�T 2 !

�I8%$�x�+3& 2 Y�$�&S+�8%YZ+w)�p � sz$H+ � p*O?+�&S8?+ 0 r3sz$JYZ+�© 2 /�+3& 0 $�p » +�&?/9+3T¦x 2 & 2 $*8%$�x�+3& 2 YZ$9&?+y8�T 2z¿ ��T � p�'*) 2 p�YZ$$tx 2 &D½zT{+�OS&?$

kO?$�&Sp 2 ��8?+wTt) � O?$5/�& 2 p�YZ+�!��I8�$9x�+�& 2 YZ$�&S+�8��Iv +�£ 4 � YZ+\© 2 /�+�&�|}$�&Sp�+ 0 +3T )�T 2 T 2 p�+ � & 2

|f� 0 � 7�+ �U+ ¿ � » +�7�Y�+I8 � p*OS$�p � @ 2 & 2 8 � p*O?+3p�8 � Y 2 Y�+�8�Y 2 8]$�x�+3& 2 r3��+�8�!�ºX8FO 2 8 � p*O?$9p �12 x�$ZYZ+I8?+3&]|}+ � O 2 O 2 p*O?$�x 2 & 2032 Y 2 T{$�Y�+37"$ 0 $�T{${x 2 & 25032 Y 2 $9x�+�& 2 r3s�$�YZ${T{$ZYZ+37"$U!�5"]b9��m������ ��¯ ��b��Hm $�b*�£|}.�&STt)�7 2 x 2 & 2 $ 0 �z7 0 )�7"$�YZ${$�x�+3& 2 pUY�$��Iv¤

µA∪B = min{

1,[

µA(x)k + µB(x)k]1/k

} : KZ!LK�� E

Page 39: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R���U� ��� � � �s��� � � � � � � ���� � ��� � ��� � * ��� � ����� P9P��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

��! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!LK�uz�9� ��!LK�u���� ��!Luz�9��� �U!L��uz��� 9! �9����� 9! �9�������!Luz���9� ��!Luz�9��� ��!��zu���� 9! �9����� 9! �9����� 9! �9�������!��zuz�9� ��!��zu���� �! ���9��� 9! �9����� 9! �9����� 9! �9����� �! �����9� �! ���9��� �! ���9��� 9! �9����� 9! �9����� 9! �9�����

� 2 6�+�7 2 KZ!" 3�NIX�Iv 0 $�TW$9x�+�& 2 YZ$�&�8?$�T 2 7 � T � O 2 Y 2(min(1, µ(x) + µ(y))

!f(x,y)

0

0.5

1 0

0.5

1

0

0.5

1

and = max(0, x+y-1)

> � /�)�& 2 KZ! P�� IX��x�+3& 2 YZ$9&V£ 4 � 0 $�T¾8F$9T 2 7 � T � O 2 Y 2 !

3�) 2 pUYZ$k¤Bx�+�'*)�+�p�$ � 2 )Up � s�$Á&S+�8?x�$9pUYZ+�x 2 & 2 x�+�'*)�+�p�$98 »�2 7"$�&S+�8-Y 2 8H|})�pUr3��+�8 � � 8FO?$ � T{x���+B)UT 2

|}$�&ST 2 |}& 2�0�2 Y�+�)�p � sz$U!{£ªT{+�Y � Y 2 '*)�+k0 &?+y8 0 + 2 )�p � s�$³&S+�8?x�$9pUYZ+{x 2 & 2 T 2 � $9&?+y8 »�2 7=$9&?+y8 2 O?¤�'*)�+�8F+

+�8FO 2 6 � 7 � @ 2 +3TW)�TWT�� ¿ � T{$U! #%$�&�+ ¿ +�T5xU7=$ ��x 2 & 2k = 1

2 |})UpUr3s�${8?+IO?$9&?p 2µA∪B = min[1, µA(x) + µB(x)]

3�) 2 pUYZ$k → ∞

2 |})UpUr3s�$ � 0�2 � /�) 2 7 2max(a, b)

��pU$³+�p9O 2 p*O?$ � 0 $�T{$�+y8ROS+�&S+�8?)�7=O 2 YZ$BpUs�$B¤�.�6 » � $2 xU&?+y8F+�p9O 2 &S+3T{$98 2 /9$�& 2 )�T 2 x�&S$ »�2 !� b*����b�� m _ #f�

limk→∞ min{

1,[

µA(x)k + µB(x)k]1/k

}

= max(µA(x), µB(x))

# 2 & 2µA(x) = 0

�µB(x) = 0

¤�$�6 » � $�'9)U+�$HOS+3$9&?+�T 2 ¤ » �z7 � Y�$U!�3�) 2 p�YZ$µA(x) = µB(x)

$tO?+3$9&?+�T 2O 2 Tt6�¤3T¾¤ » ��7 � YZ$�x�$�&D'*)�+J$�7 � T � OS+tYZ+

21/w '*) 2 p�YZ$ w → ∞¤ � /�) 2 7 2 �!63�) 2 pUY�$

a 6= bOS+3T{$98('*)�+J$

T �=p � T{$�8?+I&S+�Y�)�@ 2(µA(x)k + µB(x)k)

! #�&S+ 0 � 8 2 T{$98�+�p*OSsz$5x�&S$ »�2 &w'*)�+lim

k→∞(µA(x)k + µA(x)k)1/k = max(µA(x), µB(x))

� 2 T{$*8 2 8S8F)UT � & ��8F+�T¾x�+3&DY 2 YZ+-/9+3p�+�& 2 7 � Y 2 YZ+H'9)U+µA(x) < µB(x)

+ »�2 T{$*8�| 2 @�+3&Q = (µA(x)k +

������ ���������+���/� +"#� $%�&�'"���)(+*�,�-

min(µA(x), µB(x)) max(µA(x), µB(x))F (��!,��1 − min(1, ((1 − µA(x))k + (1 − µB(x))k)1/k) min(1, (µA(x)k + µB(y)k)1/k)

� 2 6�+37 2 KZ!" yu I ��x�+3& 2 r3�*+y8 0 $�T{x�+3pU8 2 OS.�& �"2 8V6 2 8?+ 2 Y 2 8V+3T¢|})�pUr���+y8(YZ+Hoqp*O?+�&S8?+ 0 r3sz$�+-«(p � sz$

Page 40: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

Pz� � ���`� ��� � ���R� � � � ���� � � � ���� � ���

f(x,y)

0

0.5

1 0

0.5

1

0

0.5

1

or soma limitada

> � /�)U& 2 KZ! P� I���x�+3& 2 Y�$�&(�Iv 0 $�TW8?$�T 2 7 � T � O 2 Y 2 !

µB(x)k)1/k ! ��+�8FO?+HT{$ZYZ$5O?+3T{$*8('*)�+

limk→∞

ln Q =ln (µA(x)k + µB(x)k)

k«\8 2 p�YZ$ 2 &S+3/9& 2 YZ+H7�� M\$�x � O 2 7 $�6ZOS+3T{$98

limk→∞

ln Q = limµA(x)k ln µA(x) + µk

B(x) ln µB(x)

µA(x)k + µB(x)k

limk→∞

ln Q = lim

µA(x)k

µkB

(x)ln µA(x) + ln µB(x)

µA(x)k

µB(x)k + 1= ln µB(x)

#%$�&?O 2 p9OS$tOS+3T{$98w'*)�+lim

k→∞Q = lim

k→∞(µA(x)k + µA(x)k)1/k = µB(x) = max(µA(x), µB(x))

vw+y8RO 2 xU&?$ »�2 &�'*)�+H$5O?+�$�&S+3T 2�2 � p�Y 2 ¤ » ��7 � YZ$�'*) 2 p�YZ${$5T �"p � T5$ »�2 7=+{ �! 4 +�8FO?+ 032 8F$5OS+3T{$98w'*)�+(µA(x)k + µA(x)k)1/k ≥ 1

µA(x)k + µA(x)k ≥ 1

x 2 & 2 OS$�Y�$w!�3�) 2 pUYZ$

w → ∞� 2 D�7=O � T 2 YZ+y8 � /�) 2 7"Y 2 YZ+ »�2 7"+t8?+

a$9)

b|}$9&?+�T � /�) 2 � 8 2 ���x�$9&FO 2 p*O?${$

O?+�$�&S+3T 2 � 032 x�&S$ »�2 YZ$�!£\8t� 2 6�+�7 2 8-KZ!" �� �%KZ!" �����KZ!" ��cT{$98FO?& 2 T $*8 »�2 7"$�&S+�8JY 2 $9x�+�& 2 r3sz$;YZ+ � p*O?+�&S8?+ 0 r3sz$cx 2 & 2 O?&S¼�8 »�2 7=$9&?+y8

YZ$³x 2 &D½zT{+�OS&?$k!H��6U8?+3& »�2 &\'*)�+ � YZ$�T{+�8?T{$BT{$ZYZ$³'*)�+�p 2 $�x�+3& 2 r�sz$�YZ+ � p*OS+3&D8F+ 0 r3s�$ � 2 $�x�+3& 2 r�sz$�YZ+

)�p � sz$�YZ+J© 2 /�+�&]OS+3pUY�+Hx 2 & 2�2 $9x�+�& 2 r3sz${YZ+JT�� ¿ � T5$ : � 2 6�+�7 2 K�! � E]2 T{+�Y � Y 2 '*)�+k0 &S+�8 0 +�!

�5"]b9��m������ �B´¨` ��b��Hm $�b*�£|}.�&STt)�7 2 x 2 & 2 $ 0 �z7 0 )�7"$�YZ${$�x�+3& 2 pUY�${£ 4 �¦¤

µA∩B(x) = 1 − min{

1,[

(1 − µA(x))k + (1 − µB(x))k]1/k

} : KZ!LK�u E3�) 2 pUYZ$

k¤ x�+y'9)U+3p�$ 2 � p*OS+3&D8F+ 0 r3s�$ 8F.�&?+y8Fx�$�pUY�+Áx 2 & 2�»�2 7"$�&S+�8 2 7 OS$98BY 2 8�|})UpUr���+�8cYZ+ � p 0 7=)�8?s�$

8 � /9p � � 032 p�YZ$�)�T 2 |}$9&?T 2 |}$9&FOS+tYZ+ � p*O?+�&S8?+ 0 r3s�$U!(£ T{+yY � Y 2 '*)�+k0 &?+y8 0 + 2 |})�p�r3sz$³&?+y8Fx�$�p�YZ+ 2�»�2 7=$9&?+y8

032 Y 25» +�@IT{+�p�$�&S+�8�Y 2 8V|})�pUr���+y8 2 OS¤I+y8RO 2 6 � 7 � @ 2 &�+3T¾)�TWT �"p � T{$�!

Page 41: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R���?� ��� � � � �%� � �����s�� � � ��� ������� � � ��� � � �� � ���N �8�!� � �� �� P*u��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

��! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!LK�uz�9� ��!LK�u���� ��!LKz�U � �U! u������ �U!L��uz��� 9! �9�������!Luz���9� ��!Luz�9��� ��!Luz�9��� �U! u9K�KZ �U!L��uz�� 9! �9�������!��zuz�9� ��!��zu���� ��!��zu���� �U!L��uz�� �U!L�z��P9K 9! �9����� �! �����9� �! ���9��� �! ���9��� �U!= y����� �U!= y����� 9! �9�����

� 2 6�+37 2 KZ!" ��JI��Iv YZ+J© 2 /�+3&�x 2 & 2k = 16

!��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

��! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!LK�uz�9� ��!LK�u���� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!Luz���9� ��!Luz�9��� ��!Luz�9��� �U! u������ �U!L��uz��� 9! �9�������!��zuz�9� ��!��zu���� ��!��zu���� �U!L��uz�� �U!L��uz�9K 9! �9����� �! �����9� �! ���9��� �! ���9��� �U!= y����� �U!= y����� 9! �9�����

� 2 6�+37 2 KZ!" �� I��Iv YZ+J© 2 /�+3&�x 2 & 2k = 64

!

4 $ »�2 T5+�p*O?+IpUsz$5¤H.�6 » � ${'*)�+Jx 2 & 2k → ∞

2 |})�p�r3sz$�8?+J&?+yYZ)�@ 2min (µA(x), µB(x))

!�# 2 & 2 x�&?$ »�2 &+�8FO 2{2 �U&?T 2 r3sz$�p�.98�x�$ZYZ+3T{$*8 0 $9pU8 � YZ+�& 2 &V'*)�+ 2 x 2 &FO � &£\8V� 2 6�+37 2 8]K�!= y� �ZKZ!LKz� �*KZ!LKZ \T5$*8ROS& 2 T $98 »�2 7=$9&?+y8XY 2 $�x�+3& 2 r3s�$-YZ+ � p*O?+�&S8?+ 0 r3sz$tx 2 & 2 O?&S¼�8 »�2 7"$�&S+�8�Y�$

x 2 &S½�T5+3O?&S$k!X��6U8F+�& »�2 &X'*)�+ 2 T{+�Y � Y 2 '9)U+

k0 &S+�8 0 + 2 $9x�+�& 2 r3s�$JY�+ � p*OS+3&D8F+ 0 r3s�$-YZ+w© 2 /�+3&]8F+ 2 x�&S$ ¿ � T 2

Y 2 $�x�+3& 2 r�sz${YZ+ � p*OS+3&D8F+ 0 r3s�$5Y�+�  2 YZ+�¡ : � 2 6�+�7 2 K�!L� E !�5"]b9��m�������´¨b $�m��9��� ��b �Jm�$�b*�© 2 /9+3&�O 2 Tt6�¤3T YZ+��Up � )H)�T 2 $�x�+3& 2 r3s�$VY�+�p�+3/ 2 r�sz$ 0 $9T )�T 2 |}$�&ST 2 8 � T � 7 2 & 2 8 )U8 2 Y 2 8�p 2 8 $�x�+3& 2 r3�*+y8

YZ+H)�p � s�$5+ � p*O?+3&D8?+ 0 r�sz$c!�£ +�'*) 2 r3sz${'*)�+HYZ+���p�+H+�8FO 2 $�x�+3& 2 r�sz$�¤ 2 8?+3/9) � p9OS+ I

µA(x) = (1 − µA(x)k)1

k: KZ!LKz� E

$�p�YZ+\$ »�2 7=$9&VY�+kY�+ » +�+�8FO 2 &]p�$ � p*OS+3& »�2 7=$

[0, 5]!�# 2 & 2

k = 12 |})�pUr3s�$�8?+\&S+�YZ)U@ 2 |})�pUr3s�$�YZ+��Up � Y 2

x�$�&]  2 YZ+3¡ :µA(x) = 1−µA(x)

E ! £ > � /9)�& 2 K�! P*K�K�! P*KwT{$98FO?& 2�0 $�T{$H$ »�2 7"$�&�YZ+k

� p��U)�+�p 0 �12H2 $�x�+3& 2 r3s�$YZ+ 4 +3/ 2 r3s�$�YZ+J© 2 /�+3&y!

�"��� � > - �i�7#3� �$� #��7-�������� # �C>��G� *,+.- @4# A � �<! i�7��6 /1#324!45,��67��6,�!±£t!�  2 YZ+3¡ |}$ � $;x�& � T{+ � &S$ 2 � p*O?&S$�Y�)�@ � & � Y�¤ �12 '*)�+{)�T 0 $9p��R)�p*O?$;p�+�6�)�7"$98Jx�$�Y�+�8F+�& » � 8ROS$ 0 $�T{$;)�Tx�$�p*O?$�+3TÀ)�TÀ¡ � x�+3& 0 )�6�$ : �M�8�6 � �� ����� E !%� 2 &FO �\$*8 �9$ :('*)%+�'*) �U y���9K E + ¿ OS+3pUYZ+�) 2 � YZ¤ �"2 +(|}+�@�p�$ »�2 80 $9p*O?& � 6�) � r3�*+y8(x 2 & 2B2 O?+�$�& �"2 !�«�8 2 pUYZ$ 0 $�p 0 + � O?$*8(/�+�$�T{¤�OS& �10 $*8w+37"+5+3p 0 $�p*O?&S$�)¨&S+�8?x�$*8RO 2 8wx 2 & 2�2 7"/�)�T 2 8Y 2 8{x�+�&?/9)�p*O 2 8�|})�p�Y 2 T{+3p*O 2 � 8�x 2 & 2 2 ,±.9/ �1032 4 +�6�)�7"$98 2 I;'9)�sz$ p�+36�)U7=$*8F$ ¤;)�T 0 $�p��R)�p*OS$ pU+36�)�7"$98?$ G'*) 2 p*OS$5)UT 0 $9p��R)�p*O?${p�+�6�)�7"$98?${¤I)UT¾8?)�6 0 $�p��R)Up9OS$�YZ+J$�)ZOS&?$ G

��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!LK�uz�9� ��!LK�u���� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������!Luz���9� ��!Luz�9��� ��!Luz�9��� �U! u������ �U!L��uz��� 9! �9�������!��zuz�9� ��!��zu���� ��!��zu���� �U!L��uz��� �U!L��uz��� 9! �9����� �! �����9� �! ���9��� �! ���9��� �U!= y����� �U!= y����� 9! �9�����

� 2 6�+37 2 KZ!" �� IX�Iv YZ+J© 2 /9+3&�x 2 & 2k = 128

!

Page 42: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

P�� � ���`� ��� � ���R� � � � ���� � � � ���� � �����! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

��! �����9� ��! ���9��� ��! ���9��� �U! �9����� �U! �9����� �U! �9�������!LK�uz�9� ��! ���9��� ��!LKZ y��� �U! Kz�9��� �U! K9uz��� �U! K9uz�����!Luz���9� ��! ���9��� ��!LK��*��� �U! �Z����� �U! u������ �U! u��������!��zuz�9� ��! ���9��� ��!LK�u���� �U! u������ �U!L�zP���� �U!L��uz��� �! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

� 2 6�+37 2 KZ!" �� I £ 4 � YZ+J© 2 /�+�&Vx 2 & 2k = 16

!��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

��! �����9� ��! ���9��� ��! ���9��� �U! �9����� �U! �9����� �U! �9�������!LK�uz�9� ��! ���9��� ��!LK��� �� �U! K9uz��� �U! K9uz��� �U! K9uz�����!Luz���9� ��! ���9��� ��!LK�u���� �U! �*�z�9� �U! u������ �U! u��������!��zuz�9� ��! ���9��� ��!LK�u���� �U! u������ �U!L������P �U!L��uz��� �! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

� 2 6�+37 2 KZ!LKz� I £ 4 � YZ+J© 2 /�+�&Vx 2 & 2k = 64

!

�I8{x�&S$ ��8S8 � $9p 2 � 8{Y 2 �z&S+ 2�0 $*8ROS)�T 2 T~)U8 2 &{|})UpUr���+�8B'9)U+¨T 2 x�+ �"2 T~)�T¸YZ$�T �"p � $Xp�$ � p*O?+3& »�2 7"$

[0, 1]x 2 & 2 &S+3xU&?+y8F+�p9O 2 & 0 $�p��R)Up9OS$98³p�+36U)�7=$*8F$*83! ºX8RO 2 8B|})�pUr3�*+y8Bx�$ZYZ+3T &S+3xU&?+y8F+�p9O 2 & 2 8F+�T{½�p*O �1032 YZ$

0 $9p��R)�p*O?$�!B# 2 & 2 &?+�x�&S+�8?+3p*O 2 & 0 $�p��R)Up9OS$98HpU+36�)�7"$98?$98J)�8 2 pUYZ$ 2 /�+�$�T{+�OS& �12c0 $�p*O � p�) 2 &?+�T5$*8J)U8 2 pUYZ$;+y8ROS+T{+�8?T{$BYZ$�T �"p � $�+ � p*O?+3& »�2 7"$U!�# 2 & 2 8 � T{x�7 � � 032 & »�2 T{$98 0 $�pU8 � YZ+3& 2 &('*)�+�$BYZ$�T �"p � $

X¤tYZ+��Up � YZ$ 0 $�T{$

X = {x1, x2, . . . , xn}!��¦'9)U+�¤I+3p*OSs�$t)�T 0 $�p��R)Up9OS$�pU+36�)�7"$98?$

A��YZ+��Up � YZ$5p�+y8ROS+IYZ$9T �=p � $ �Z'*) 2 pUYZ$t+�7=+

¤�&S+3x�&S+�8?+3p*O 2 YZ$B/9+3$9T5+3O?& �"0�2 T{+3p*OS+ G £ &S+3x�$98FO 2 ¤�)�T x�$�p*OS$;Y�+3p*O?&S$¨YZ+�)�T ¡ � x�+3& 0 )�6�$¨)�p � OS��& � $¨YZ+nY � T{+3pU8?��+�8�!

[]$�p�8 � YZ+3&S+ 0 $9T{$�+ ¿ +3T{x�7"$�)�T 0 $�p��R)�p*OS$ 0 $9TÂY�$ � 8(+37"+3T{+�p9OS$98X = {x1, x2}

!�� ¡ � x�+3& 0 )�6�$Bp�+y8ROS+032 8F$�O?+�T�Y�) 2 8wY � T{+3pU8?��+�8 �Z$�)¨8?+R� 2 )�T¾'*) 2 YZ& 2 YZ${YZ+H7 2 YZ$ � /�) 2 7 2 )�T;!]� 0 $�p��R)�p*OS$5x�$zOS¼3p 0 �"2�0 7"�98?8 �10 $YZ+

X0 $�p*OS¤3T¢$*8V8?+3/9) � p9OS+�8 0 $9p��R)�p*O?$*8

2X = {∅, {x1}, {x2}, X}!�ºX8FO?+�8 0 $�pz�R)�p*O?$98�YZ+��Up�+�TW$98 » ¤3&?O �10 +�8

YZ$t'*) 2 YZ& 2 YZ$J+ 0 $�&S&?+y8Fx�$�p�YZ+3T 2 $98 x�$�p*OS$98 : � � � E � : �� � E � : � �= E + : �= E �*$�pUY�+($98X)�pU8�+\@3+3&S$98 � pUY �"0�2 T '9)U+)�T¹+37"+3T{+3p*O?$cx�+3&?O?+3p 0 +{$�) pUsz$ 2 $ 0 $�p��R)�p*OS$U!{£ > � /�)�& 2 KZ! P�PBT5$*8ROS& 2 +�8FO?+y8 0 $9p��R)�p*O?$*8J&?+�x�&?+y8F+�p*O 2 Y�$980 $9T{$5x�$�p*OS$98�![]$�p�8 � YZ+3&S+3T{$*8 2 /�$9& 2 $ 0 $9p��R)�p*O?$�pU+36�)�7"$98?$

A = {(x1, 3/8), (x2, 6/8)}�UT5$*8ROS& 2 YZ${p 2�> � /9)�& 2 K�! P9P�!

4 +y8ROS+ 0 $�pz�R)�p*O?$ ��$c+�7=+�T{+3p*O?$x1x�+3&?O?+�p 0 + 2 $ 0 $�p��R)�p*OS$

A0 $�T /9& 2 ) YZ+ � p 0 7")U8Ssz$

µA(x1) = 3/8� +�pZ�

'*) 2 p*OS$ '*)�+x2x�+3&?O?+3p 0 + 0 $�T¸/�& 2 )

µA(x2) = 6/8!ª�¹'*)�+ 8?+3& �12 +3p*OSs�$ $ 0 $�p��R)�p*OS$ YZ+ÁOS$ZYZ$98³$98

8?)�6 0 $�p��R)Up9OS$98�p�+36U)�7=$*8F$*8�� � 8FO?$B¤���$ 0 $9p��R)�p*O?$³x�$�O?¼�p 0 �12 p�+�6�)�7"$98?$F (2X)

G ºX8FO?+ 0 $9p��R)�p*O?$B¤t)�T¡ � x�+�&F�0 )U6�$�)�p � OD�z& � $

In = [0, 1]n!���$�Y�$5${+y8Fx 2 r�${p�+ 0 +�8S8S�z& � $�x 2 & 2 &S+3xU&?+y8F+�p9O 2 &�/�+�$�T{+�OS& �1032 T{+3p*O?+ 0 $9p��R)�p*O?$*8

p�+�6�)�7"$98?$98Ix�$ZYZ+�8?+3&-Y�+��Up � YZ$;x�$�&{X, In}

!B� +�8?x 2 r�$ � p*OS+3& � $9&JY�$ 0 )�6�$c¤�xU&?+�+3p 0 ¡ � YZ$cx�+�7=$ 0 $�p��R)�p*OS$x�$zO?¼�p 0 �12 p�+36U)�7=$*8F$�YZ+

X!

� x�$�p*O?$ 0 +3p*OS& 2 7 :CE YZ$ 0 )�6�$B¤5$ 0 $�p��R)Up9OS$ 0 $�TÂT 2 � $�&I/�& 2 )ÁYZ+�p�+36�)U7=$*8 � Y 2 Y�+���x�$9&S'*)�+tO?$ZYZ$*8\$*8

8?+3)U8t+37"+3T{+3p*O?$*8HOS+3T­/�& 2 ) YZ+ � p 0 7=)�8?s�$ � /9) 2 7 21/2!¨ºX8FO?+Bx�$�p*OS$Á¤�O?$�O 2 7=T{+3p*OS+ � p 0 $9T5x 2 O&� » +�7 0 $9T 2

��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�������! �����9� ��! ���9��� ��! ���9��� �U! �9����� �U! �9����� �U! �9�������!LK�uz�9� ��! ���9��� ��!LK���uz� �U! K9uz��� �U! K9uz��� �U! K9uz�����!Luz���9� ��! ���9��� ��!LK�u���� �U! �*�*��P �U! u������ �U! u��������!��zuz�9� ��! ���9��� ��!LK�u���� �U! u������ �U!L���9��� �U!L��uz��� �! �����9� ��! ���9��� ��!LK�u���� �U! u������ �U!L��uz��� 9! �9�����

� 2 6�+37 2 K�! K� I £ 4 � YZ+J© 2 /�+3&�x 2 & 2k = 128

!

Page 43: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R���?� ��� � � � �%� � �����s�� � � ��� ������� � � ��� � � �� � ���N �8�!� � �� �� P��

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

Negacao de Yager

f(x)=x

k=0.5

k=0.75

k=1.0

k=2.0

k=5.0

> � /�)U& 2 KZ! P9K I%¬I&D��� 0 $98VT{$98FO?& 2 p�YZ$ 2 � p��U)U¼3p 0 �"2 YZ${x 2 &S½�T5+3O?&S$kp 2 |})�p�r3sz$�YZ+Hp�+�/ 2 r3sz$5YZ+J© 2 /9+3&y!

O?+�$�& �"2 YZ$*8 0 $9p��R)�p*O?$*8�O?& 2 Y �"0 � $9p 2 � 8��Ux�$9&S'*)�+H+�7=+ » � $�7 2 $98wx�& � p 0 �"x � $98wY 2 p�sz$ 0 $9p9OS& 2 Y � r3sz${+-Y 2 + ¿�0 7")U8Ssz$YZ${T{+ � $�YZ+H£(& � 8FO?.�O?+�7=+y83! 4 +y8ROS+Hx�$9p*O?$5O?+�T5$*8w'*)�+

C = C ∩ C = C ∪ C = C: K�! K*� E

«(T � p*O?+�&?+y8?8 2 p*OS+�| 2 OS$J¤V$Ix�$98 �"0 � $9p 2 T{+3p*OS$�&S+37 2 O �=» $\p�$J+�8?x 2 r�$�Y�$98 0 $9p��R)�p*O?$*8�&?+y8F)�7=O 2 p*O?+�8%Y 2I2 xU7 �1032 �r3s�$HY 2 8%$9x�+�& 2 r���+�8%YZ+w)�p � s�$ � � p*O?+3& 0 +y8?8Ssz$ 2 )�T 0 $�p��R)Up9OS$J+(8F+�) 0 $�T{x�7"+3T{+3p*O?$�!�#%$9& + ¿ +3T{x�7"$ � 0 $�pU8 � YZ+3&S+$ 0 $9p��R)�p*O?$

A290 � T 2 T5+�p 0 � $�p 2 YZ$�!X«\8 2 pUY�$ 2 8w$�x�+3& 2 r3��+�8�YZ+

max�min

x 2 & 2 � T{x�7"+3T{+�p9O 2 & 2 8w$�x�+3& 2 �r���+y8�6U��8 �"0�2 8�8?$�6�&S+ 0 $�pz�R)�p*O?$98VOS+3T{$98w'*)�+J$*8 0 $�pz�R)�p*O?$98�&S+�8?)�7=O 2 p9OS+�8w8Ssz$

A = (3/8, 6/8)

A = (5/8, 2/8)

A ∪ A = (5/8, 6/8)

A ∩ A = (3/8, 2/8)

£ > � /9)�& 2 K�! P��{T5$*8ROS& 2{0 $�T{${+�8FO?+�8(x�$9p9OS$98(8?+HY � 8ROS& � 6�)�+�T¾p�$�+y8Fx 2 r�$�! ºX8FO?+�8(x�$9p9OS$98(8?+HT{$ » +�T¢+3T8 � p 0 &?$9p �12 8F+ 2 | 2 8RO 2 pUYZ$ +3T Y � &?+yr3sz$ 2 $98 0 $9p��R)�p*O?$*8 0 7"�98?8 �"0 $98H$9) +3T~Y � &S+�r�sz$ 2 $Áx�$9p*O?$ 0 +3p*O?& 2 7]$�pUY�+$ 0 $9&?&S+ 2 p�+36U)�7=$*8 � Y 2 YZ+ÁT�� ¿ � T 2 ! Q�+3/9)�pUYZ$ ��$98<��$�$ x�$�p*O?$ 0 +3p*OS& 2 7 0 $9&?&S+�8?x�$9pUYZ+ 2 $ 6�)�& 2�0 $�p�+�/�&S$Y 2 7".�/ �"0�2 p�+�6�)�7"$98 2 �±$9pUYZ+�p 2 Y 2 ¤�Y � 8RO � p�/9) � » +37 :B'*)%+A'*) � y���*K E ! 4 $¨x�$9p9OS$ 0 +3p*OS& 2 7%O?+3T{$*8H'9)U+�)�T 22 ��&?T 2 r�sz$c¤{+�'*) �"»�2 7=+�p*O?+ 2 8F) 2 p�+3/ 2 r3s�$U!5ºX8FO?+�x�$�p*O?$¨¤ 2 8?$�7")Ur3s�$³x 2 & 2 $*8Jx 2 & 2 YZ$ ¿ $*8�YZ$¨T{+�p9O � &S$98?$YZ+³[]&S+�O 2 ��YZ$*8 0 $9p��R)�p*O?$*8HYZ+{O?$ZYZ$*8J$98 0 $9p��R)�p*O?$*8H'*)�+{pUsz$¨8Ssz$cT5+�Tt6�&S$98HY�+{8 � T{+�8?T{$98 �±+�O 0 ! 4 +�8FO?+�8x 2 & 2 YZ$ ¿ $98HOS+3T{$98{'*)�+c)�T 2�2 �U&ST 2 r3s�$

A� T{x�7 �1032

A+

A� T{x�7 �1032

A! ºX7 2 8tOS+3T~$�T{+�8?T{$ /9& 2 ) YZ+

Page 44: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

P�� � ���`� ��� � ���R� � � � ���� � � � ���� � ���

ø=(0,0)

X=(1,1){x2}=(0,1)

{x1}=(1,0)

x1

x2

3/8

6/8A=(3/8; 6/8)

C=(1/2, 1/2)

> � /�)U& 2 KZ! P�PJI�[]$9p��R)�p*O?$*8wvw+3xU&?+y8F+�p9O 2 YZ$*8 0 $�T{${x�$9p*O?$98�!

ø=(0,0)

X=(1,1){x2}=(0,1)

{x1}=(1,0)

x1

x2

3/8

6/8A

5/8

2/8AA A

A A

> � /�)�& 2 KZ! Pz�JI%#�& � p 0 �"x � $98wYZ+-£\& � 8ROS.zOS+37"+�8�p�${ºX8?x 2 r3$U!

» +�&SY 2 YZ+I+Hx�$9&FO 2 p*O?$5x�$�Y�+3T¾8?+3&w+�p*O?+3p�Y � Y 2 8 0 $9T{$V (A) = V (A)

V (A) = 1 − V (A)

V (A) = 1/2

£ 0�2 &DY � p 2 7 � Y 2 YZ+HYZ+t)UT 0 $�p��R)�p*OS$ 0 71��8S8 �10 $ 0 $�p*O 2 '*) 2 p9OS$98w+�7=+�T{+3p*O?$*8(x�+3&?O?+3p 0 +3T 2 +�8FO?+ 0 $�pz�R)�p*O?$U!# 2 & 2{0 $�pz�R)�p*O?$98�pU+36�)�7"$98?$98 2 YZ+��Up � r3s�$�K�!= �u-T{$*8ROS& 2 '*)�+ 2�032 &SY � p 2 7 � Y 2 Y�+IYZ+H)UT 0 $9p��R)�p*O?${p�+�6�)�7"$98?${¤2 8?$�T 2 YZ$*8�/�& 2 )�8 YZ+ � p 0 7")U8Ssz$-YZ+ 032 Y 2 )�TÀYZ$98X+37"+3T{+3p*O?$*8�YZ$ 0 $�p��R)Up9OS$U!%£ 0�2 &DY � p 2 7 � Y 2 YZ+(YZ$ 0 $�p��R)�p*OS$A = (3/8, 6/8)

¤ � /9) 2 7 2|A| = 3/8 + 6/8 = 9/8

!5£ T{+yY � Y 2 Y 2;032 &DY � p 2 7 � Y 2 YZ+�O 2 Tt6�¤3T OS+3T¹)�T 2� p9OS+3&Sx�&S+�O 2 r3s�$B/�+�$�T{¤�OS& �1032c0 $9pU8 � 8FO?+�p9OS+�!5º�7 2 ¤ � /�) 2 7 2 $³O 2 T 2 p�¡�$¨YZ$ » +3O?$9&J'*)�+ 0 $�T{+�r 2 p 2 $9& � /�+�T +O?+�&?T � p 2 p�$ 0 $9p��R)�p*O?$

A� 0 $9T5$5+�8FOS� � 7=)�8ROS& 2 YZ${p 2{> � /�)U& 2 KZ! P9u�!

# 2 & 2{0 $�T{x�&S$ »�2 &�+�8FO 2 � p*O?+�&?x�&S+�O 2 r�sz$ »�2 T{$98 0 $�pU8 � YZ+3& 2 & 2 |}.9&?T�)�7 2 YZ+�§ � p���$.�(8<�*ABx 2 & 2{032 7 0 )U7 2 & 2Y � 8ROD½zp 0 �"2 +�p9OS&?+HYZ$ � 8�x�$�p*OS$98

P = (p1, p2, . . . , pn)+

Q = (q1, q2, . . . , qn)¤-Y 2 Y 2 x�$�&

d(P, Q) = (

n∑

i=1

|pi − qi|t)

1

t: KZ!LKz� E

Page 45: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R���?� ��� � � � �%� � �����s�� � � ��� ������� � � ��� � � �� � ���N �8�!� � �� �� P9�

ø=(0,0)

X=(1,1){x2}=(0,1)

{x1}=(1,0)

x1

x2

3/8

6/8A=(3/8; 6/8)

|A|

> � /9)�& 2 KZ! P9u IX[ 2 &DY � p 2 7 � Y 2 YZ+ 0 $�T{$ � +�OS$�&y!

$�p�YZ+n¤�$¨pQD�T{+3&S$¨YZ+ 0 $�$9&SYZ+�p 2 Y 2 8IYZ$¨x�$9p*O?$;+

t¤�$;O � x�$ÁYZ+�Y � 8FOS½zp 0 �12c2 8?+3& 0�2 7 0 )�7 2 Y 2 ! 3�) 2 p�YZ$

t = 1OS+3T{$98 2 Y � 8FOS½zp 0 �12 Y�+�§ 2 p�¡ 2 OFO 2 p�+{x 2 & 2

t = 2O?+�T{$98 2 Y � 8ROD½zp 0 �"2 º�) 0 7 � YZ+ 2 p 2 !B£ÀY � 8ROD½zp 0 �"2

YZ+;§ 2 pU¡ 2 O?O 2 p�x�$ZYZ+cO 2 Tt6�¤3T~8?+3& 0 ¡ 2 T 2 Y 2 Y�+cY � 8FOS½zp 0 �12 YZ+;M 2 T{T � p�/ p�+36U)�7=$*8 2 ! £¾Y � 8FOS½�p 0 �12 Y�+M 2 T{T � p�/{p�+�6�)�7"$98 2 +�p*O?&S+�)�T 0 $9p��R)�p*O?$

A = (x1, µ(x1)), (x2, µ(x2)), . . . , (xn, µ(xn))+ 2 $9& � /�+�T �*$9)

$ 0 $9p��R)�p*O?$ »�2 @ � $ :O = 0, 0, . . . , 0

+-Y 2 Y 2 x�$9&

d(A, O) =n

i

|µA(xi) − 0|

d(A, O) =

n∑

i

|µA(xi)|

d(A, O) = |A|: K�! K�� E

Page 46: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�9� � ���`� ��� � ���R� � � � ���� � � � ���� � ���

Page 47: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� ��������� � �

� ����� ��� � � � �w� � � ��� � � � � ����� ��� � �

� �������� � � �

; ��� �i�G� ��@4! �����«(T 2 &S+37 2 r�sz$ � T5xU7 �1032 +�T¹)�T 2Á2 8?8?$ 0 �12 r�sz$c+3p*O?&S+�+37"+3T{+3p*OS$98tY�+BY � |}+3&S+3p*O?+y8 0 $9p��R)�p*O?$*83!¨Q�+�$¨/9& 2 )�Y�+2 8S8?$ 0 �"2 r3s�$B+3p*OS&?+�$*8J+37"+3T{+3p*O?$*8H¤��;$�) {+3p*ODsz$cO?+�T5$*8H)�T 2 &S+37 2 r3s�$ 0 7"�98?8 �1032 !³[ 2 8F$;+y8ROS+�/�& 2 ) x�$*8?8 2»�2 & �"2 &�+3p*O?&S+H��+� ��O?+�T5$*8�)�T 2 &?+�7 2 r3s�$tp�+�6�)�7"$98 2 !£ &?+�7 2 r3sz$B+3p*O?&S+�$*8\+�7=+�T{+3p*O?$*8�x�$ZYZ+{8F+{YZ+ » +3& 2 )UT 2 x�&S$�x�& � +�Y 2 YZ+ 0 $�Tt)UT � )UT 2 '*) 2 7 � Y 2 YZ+ ��)�T 2

&S+3/�& 2 �V+�O 0 ! «\T 2 8F+�p*O?+3p�r 2 O 2 7 0 $�T{$ <x¤ T 2 � $9&B'*)�+

yC � pUY �"0�2 )�T 2 &S+37 2 r�sz$�+3p*O?&S+ YZ$ � 8 »�2 7"$�&S+�8�!

��6U8?+3& »�2 &�'*)�+ 2 $�&DYZ+3T�+3p*O?&S+;$98{+37"+3T{+�p9OS$98�¤ � T{x�$9&FO 2 p*O?+9! #%$�&�+ ¿ +�T{x�7=$ � < +�8FO 2 |})�p�r3sz$�¤¨)�T¸YZ$*80 $9T{x�$9p�+3p*O?+y8wYZ${x�&S$�/�& 2 T 2�C ��8F$9T{+3p*O?+ »�2 7"+Jp�+�8FO 2 $�&DYZ+�Tc!«(T 2 &?+�7 2 r3s�$�YZ$�O � x�$ < $�&S+�8 � 8ROS$�&�¤V)UT YZ$98 0 $�T{x�$9p�+3p*OS+�8%YZ$I&S�9Y � $ C x�$ZYZ+w8?+3& 8?)�6U8FO � O?) �1Y 2 x�+�7 2 &S+3/�& 2

< 8F+5$;$�6Z�R+3O?$c¤5)UT &?+y8 � 8FO?$�&I+3p*ODsz$c+37"+5¤ 0 $�T{x�$9p�+3p*OS+5YZ$;&D��Y � $ C !73�) 2 pUYZ$cYZ$ � 8J+37"+3T{+3p*OS$98Ix�+�&FOS+3p 0 +3T2 )�T 2 &S+37 2 r�sz$R�]p�.985p�$*85&?+3|}+3&S+3p 0 �"2 T5$*8 2 +�7=+y8 0 $�T{$ $ x 2 &{$�&DYZ+3p 2 YZ$

(a, b) ∈ R�X$�)

aRb� 0 $�T

$�+37"+3T{+3p*OS$a8F+�pUYZ$�$ xU& � T5+ � &S$�+;$ +�7=+�T{+3p*O?$

b8?+3p�YZ$�$�8?+3/9)�pUYZ$�! «\T 2 &S+37 2 r�sz$ YZ+

n+�7=+�T5+�p*O?$98

¤ 0 ¡ 2 T 2 Y�$ YZ+c)�T 2 &S+37 2 r3s�$n�q�z& �"2 ��+ 2 2 8?8?$ 0 �12 r�sz$ÁYZ+y8ROS+�8

n+37"+3T{+3p*OS$98�p 2 &S+37 2 r�sz$Á¤c)�T 2 pZ��)�x�7 2 !

«(T 2 &S+37 2 r3s�$�¤�'*) 2 71'*)�+3& 0 $9p��R)�p*O?$;YZ+{pZ��)�x�7 2 8I$�&DYZ+3p 2 Y 2 83!-vw+�7 2 r���+�8I8Ssz$�|}$9&?T 2 Y 2 8IYZ+ 0 $9p��R)�p*O?$*8IY�++37"+3T{+�p9OS$98�+J+37 2 8w8Ssz${+�TW8 � T5+y8FT 2 8 0 $�pz�R)�p*O?$98�!vw+�7 2 r���+�8Ix�$�Y�+3T¹8?+3&Hx�+3p�8 2 Y 2 8 0 $�T{$;T 2 x�+ 2 T{+3p*OS$98�! > )�pUr3�*+y8H8?s�$cO 2 T�6�¤�T T 2 x�+ 2 T{+3p*O?$*8��±T 2 8

&S+37 2 r���+y8-8?s�$ÁT 2 x�+ 2 T5+�p*O?$98tYZ+ 0�2 & 2�0 OS+3&-T 2 � 8t/�+3& 2 7�!Á«(T 2 |})�pUr3s�$¨| 2 @³)�T­T 2 x�+ 2 T{+�p9OS$ YZ+�T�) � O?$98x 2 & 2 )UT � � 8FO?$c¤�T 2 � 8IYZ+�)UT +37"+3T{+3p*O?$cYZ$ 0 $�pz�R)�p*O?$³Y�+�$9& � /�+�T¾x�$ZYZ+3T 8?+3&IT 2 x�+ 2 YZ$98\+3T )�T D�p �"0 $+37"+3T{+�p9OS$tYZ$ 0 $�pz�R)�p*O?$�Y�+�8FO � p�$U!%«(T 2 &S+37 2 r�sz$H¤�)�TªO � x�${YZ+\T 2 x�+ 2 T{+3p*O?$tT 2 � 8X/�+�& 2 7�YZ$HO � x�${'9)U+(7=+ »�2Tt) � OS$98Vx 2 & 2 T�) � OS$98 � � 8FO?$5¤�x�$ZYZ+3T{$*8XOS+3&�T 2 � 8]Y�+I)UTÀ+37"+3T{+3p*O?${YZ$ 0 $�pz�R)�p*O?${YZ+�$9& � /�+�T 7=+ »�2 pUYZ$�x 2 & 2)�T�+37"+3T{+3p*OS$�YZ$ 0 $9p��R)�p*O?$BY�+�8FO � p�$�+-T 2 � 8(YZ+H)�T�+37"+3T{+�p9OS$BYZ$ 0 $�p��R)�p*OS$�YZ+�8FO � p�$�x�$�Y�+t8F+�& 2 8S8?$ 0 �"2 YZ$0 $9T¢)�T¾+37"+3T{+3p*O?$�YZ$ 0 $�p��R)Up9OS${$�& � /9+3T;!

; ��� � � ��@4!P�7� A -�� �7# 6G*,- $�«(T 2 8F+y'LK�¼3p 0 �"2 $�&DYZ+3p 2 Y 2 Y�+

r+�7=+�T5+�p*O?$98 ��+y8 0 & � O?$*8-p 2 |}$�&ST 2

(a1, a2, . . . , ar)��¤ 0 ¡ 2 T 2 Y 2 )�T 2 pZ�

)�x�7 2 $9&SYZ+�p 2 Y 2 !Á[]$�pU8 � Y�+3&S+�$98 0 $�p��R)�p*OS$98 0 7"�98?8 �10 $98A1, A2, . . . , Ar

��$ 0 $�p��R)�p*OS$ YZ+�OS$�Y 2 8 2 8H&?��)�xU7 2 8(a1, a2, . . . , ar)

��$9pUYZ+a1 ∈ A1

�a2 ∈ A2

�. . .�ar ∈ Ar

��+�¤�YZ+3p�$�O 2 YZ$¨x�$9&A1 × A2 × . . . × Ar

!³QZ+O?$ZYZ$*8I$98 0 $�pz�R)�p*O?$98Ai8Ssz$ � /9) 2 � 8\$cx�&S$ZYZ)ZO?$¨[ 2 &?O?+y8 �12 p�$Bx�$ZYZ+{8F+�& 2 6�&S+ » �12 YZ$ 0 $�T{$

Ar !��ªxU&?$ZYZ)ZOS$[ 2 &?O?+y8 �12 pU$�pUs�$5O?+3T¾&?+�7 2 r3s�$ 0 $�T¢$�x�&S$ZYZ)ZOS$5+�p*O?&S+I+�7=+�T{+3p*O?$*8wYZ$98 0 $�p��R)�p*OS$98�!]�¦x�&S$ZYZ)ZOS$�[ 2 &FOS+�8 �"2 p�$YZ+-YZ$ � 8�)Up �=» +3&D8F$*8

X+

Y¤-YZ+�OS+3&ST � p 2 YZ$�Y 2 8F+�/�) � p*O?+JT 2 p�+ � & 2

X × Y = {(x, y) | x ∈ X, y ∈ Y }: PU!= E

��

Page 48: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�*K � ����� ��� � � �?� � � � �s��� � � � � ��� �� � ��� � � � � �s��� � �D ��� � � �� ������ b � "�°l�{® #f����� Q�+F� 2 T¦$*8 0 $9p��R)�p*O?$*8

A = {1, 2, 3, 4}+

B = {0, 1}! #%$�Y�+3T{$98�|}$�&ST 2 & » ��& � $98�x�&?$ZYZ)�O?$98[ 2 &?O?+y8 �12 pU$98 2 x 2 &?O � &wYZ+�8FO?+y8 0 $9p��R)�p*O?$*83!%#%$�&w+ ¿ +�T{x�7=$NI

A × B = {{1, 0}, {2, 0}, {3, 0}, {4, 0}, {1, 1}, {2, 1}, {3, 1}, {4, 1}}B × B = {{0, 0}, {0, 1}, {1, 0}, {1, 1}}

; �<; � # 5M-��� # 6 A 5���6G67*,+.-�6«(T 2 &?+�7 2 r3s�$-¤�)UT¢8F)�6 0 $9p��R)�p*O?${YZ$5x�&S$�Y�)ZO?$�[ 2 &?O?+�8 �12 p�$

A1 ×A2 × . . .×Ar!]�xU&?$ZYZ)ZOS$�[ 2 &FOS+�8 �"2 p�$

x�$�Y�+c8?+3& 0 $9pU8 � YZ+�& 2 YZ$¨)�T 2 &S+37 2 r�sz$Á8?+3T &S+�8FO?& � r3��+�8 � � 8FO?$ ¤ � O?$ZYZ$*8t$*8t+�7=+�T{+3p*O?$*8t+y8RODsz$ &S+37 2�0 � $9p 2 Y�$98+3p*OS&?+{8 � 8?+3T pU+3p�¡�)�T 2 &?+y8ROS& � r3s�$U!H«(T 2 &?+�7 2 r3s�$�+3p*O?&S+{YZ$ � 8 0 $�pz�R)�p*O?$98 :

r = 2E �±$�) 8F+F� 2 )UT 2 &S+37 2 r3s�$

8?)�6 0 $�p��R)Up9OS$cY�$³x�&S$ZYZ)ZOS$¨[ 2 &FOS+�8 �"2 p�$A1 × A2

¤ 0 ¡ 2 T 2 Y 2 YZ+�&S+37 2 r�sz$�6 � pU��& �12 !53�) 2 pUY�$A1 6= A2

2&S+37 2 r3s�$³6 � pU��& �12 ¤�)�T /�& 2 |}$;6 � x 2 &?O � YZ$ ��'*) 2 pUYZ$

A1 = A22 &S+37 2 r�sz$;¤�)�T /�& 2 |}$¨Y � &?+ 0 � $�p 2 YZ$U!�«\T 2

&S+37 2 r3s�$�x�$ZYZ+H8F+�&(YZ+3pU$zO 2 Y 2 x�$9&R(A1, A2, . . . , Ar)

��$9)¨8F+F� 2

R(A1, A2, . . . , Ar) ⊆ A1 × A2 × . . . × Ar

£ &?+y8ROS& � r3s�$ 2 $5&S+37 2�0 � $�p 2 T{+3p*O?$t+3p*O?&S+�$*8Vx 2 &S+�8 ��$�)³8?+R� 2 � 2 |}$9&Sr 2 Y 2 &S+37 2 r3s�$t+�p9OS&?+I+�8FO?+y8Vx 2 &S+�8VYZ+ 032 Y 2«(p �=» +3&D8F$¨x�$ZYZ+c8?+3&�T{+�Y � Y 2 x�+37 2 |})�p�r3sz$ 032 & 2�0 O?+�&&�18RO �"0�2

µ�%$�pUYZ+³$ »�2 7"$�&³ � p�Y �"0�2 '*)�+³$98-+�7=+�T{+3p*O?$*8

+�8FOSs�${&?+�7 290 � $�p 2 YZ$98]+H��T{$*8ROS& 2 '*)�+JpUsz${¡U�5&S+37 2 r�sz$ � � 8ROS${¤

µX×Y (x, y) =

{

1, (x, y) ∈ X × Y0, (x, y) /∈ X × Y

: PU! K E

��� b � "�°l� ® #f��_ � � 2 T{$98 0 $9pU8 � YZ+�& 2 & 2 &S+37 2 r3s�$cY�+{Y �=» � 8 � 6 � 7 � Y 2 Y�+��Rd�±p�$ 0 $9p��R)�p*O?$

S = {1, 2, 3, 4, 6}YZ+��Up � Y 2 x�+�7 2{2 ��&?T 2 r�sz$ <xY �=» � YZ+

yC !

Rd ⊂ S × S�Z¤I)UT 2 &S+37 2 r�sz$t6 � pU��& �1250 71��8S8 �1032 '*)�+J+3p » $97 » +IYZ$ � 8

+37"+3T{+�p9OS$98x+

yO � & 2 YZ$*8wYZ$�x�&S$�Y�)ZO?$³[ 2 &FO?+y8 �12 pU$5YZ+

S0 $�T¾+37"+HT5+y8FT{$�! � |f� 0 � 7�7 � 8RO 2 &wOS$ZYZ$98w$*8�x 2 &S+�8Y 2 &?+�7 2 r3s�$-+ » +3& � � 032 &V'*)�+ 2 &S+37 2 r3s�$-+3T¢8 � ¤J)�T 0 $9p��R)�p*O?$ �*$ 0 $9p��R)�p*O?${YZ+�O?$ZYZ$*8]$*8Xx 2 &S+�8VYZ+�pQD�T{+�&?$*8'*)�+HY �"» � YZ+�TW$�)�O?&S$U!

Rd = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}: P�! P E

«(T 2 &?+�7 2 r3s�$tx�$�Y�+J8?+3&�&S+3x�&S+�8?+3p*O 2 Y 2 Y�+I$9)ZO?&S$98VT{$ZYZ$98 ��x�$9&�+ ¿ +3T{x�7"${)�T¢/�& 2 |}$5Y � &?+ 0 � $�p 2 YZ$ 0 $�T{$+�8FOS� � pUY �"0�2 YZ$-p 2-> � /9)�& 2 PU!= 9! � 0 $�p��R)�p*OS$

x+y8ROD� O Y � &?+ � O 2 +\$

yO +�8S'*)�+3&DY 2 !�#%$�&]+ ¿ +3T{x�7"$ � 2 8F+3O 2 '*)�+

7 � / 2 $98 0 �=& 0 )�7"$98 0 $�p*OS+3pUYZ$H$*8 »�2 7"$�&S+�8w w+IK � p�Y �"0�2 '*)�+($ »�2 7"$�&w �Y �=» � YZ+�K�!�£ 8?+�O 2 x�$�Y�+\8F+�&�6 � Y � &?+ 0 � $�p 2 7x 2 & 2 $98 0�2 8?$98V+3T�'9)U+ 2 &S+37 2 r�sz$ »�2 7=+Jp�$*8wYZ$ � 8(8?+3p*O � YZ$*83!��)ZOS&?$*8]T{$ZYZ$*8wYZ+J&?+�x�&?+y8F+�p*O 2 &V&S+37 2 r���+y8V8Ssz$5O 2 6�+37 2 8�+JT 2 O?& � @3+�8�! £¦� 2 6�+�7 2 PU!= IT{$98FO?& 2 )UT 2 $9)ZO?& 2

&S+3x�&S+�8?+3p*O 2 r3s�$�Y 2 T{+�8?T 2 &S+37 2 r�sz$�!

yx

K P � � K � � P � � � � � � � �� � � � �

� 2 6�+�7 2 P�!" I � 2 6�+37 2�0 $9&?&S+�8?x�$9pUYZ+�p9OS+ 2 vw+�7 2 r3sz$�PU! P£¢&S+3xU&?+y8F+�p9O 2 r�sz$;x�$9&�T{+ � $ YZ+³T 2 O?& � @�+�8-¤³)�T $�)�O?&S$ÁT{$�Y�$ YZ+³T5$*8ROS& 2 &t&S+37 2 r3�*+y83! ºX8RO 2 O � x�$ Y�+

T 2 OS& � @t¤ 0 ¡ 2 T 2 Y 2 YZ+�T 2 O?& � @�&?+�7 2 r3s�$U!�£ T 2 OS& � @t&S+37 2 r�sz$ 0 $9&?&S+�8?x�$�pUYZ+�p*O?+ 2 &?+�7 2 r3sz$�Y 2 ºX'*) 2 r3s�$BP�! P|}$ � $�6�O � Y 2 O � & 2 pUYZ$���8?+ 2 xU& � T{+ � & 2 7 � p�¡ 2 + 2 x�& � T{+ � & 2�0 $97=)�p 2 +t+y8ROD��T{$98FO?& 2 Y 2 p 2 T 2 O?& � @���$�pUYZ+t)�T­

Page 49: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�?���?� � � � �s��� � � � � ������ � ��� �*P

1

2

3

4

6

1

2

3

4

6

> � /�)U& 2 P�!" IX¬I& 2 |}${Y 2 &?+�7 2 r3s�$5PU! P

� pUY �1032 '*)�+x: Y 2 7 � p�¡ 29E Y �"» � Y�+

y: Y 2�0 $�7")�p 29E !

R =

1 1 1 1 10 1 0 1 10 0 1 0 10 0 0 1 00 0 0 0 1

: PU! � E

vw+�7 2 r���+�8wx�$ZYZ+3T 8?+3&IYZ+���p � Y 2 8(x 2 & 2 )�p �"» +�&S8?$98 0 $�p*O&�"p�)�$98�!w#%$�&�+ ¿ +3T{x�7"$ ��8F+F� 2B2 &S+37 2 r�sz$ 0 $�p*O&�"p�) 2YZ+��Up � Y 2 x�+�7 2 + ¿ xU&?+y8?8Ssz$ 2 6 2 � ¿ $

R = {(x, y) | y ≥ x/2, x ∈ X, y ∈ Y }

'*)�+Jx�$ZYZ+-8?+3&(Y�+��Up � Y 2 x�+37 2 8F) 2 |})�p�r3sz$ 0�2 & 2�0 OS+3& �"8FO �103250 $9T{$

µR(x, y) =

{

1, y ≥ x/20, y < x/2

+-'*)�+J/�& 2 � 0�2 T{+3p*OS+\OS+3T 2 |}$9&?T 2 T{$98FO?& 2 Y 2 p 2{> � /9)�& 2 P�!LKZ!

������ &(� � �-!�)-� 2�! �-��� � � ��� ��23� ��������&(2�� � �"! #����Q�)�x�$�pU¡ 2 '*)�+

n+37"+3T{+3p*O?$*8XYZ$�)�p �"» +�&S8?$

X+�8FOSs�$-&?+�7 290 � $�p 2 YZ$98 0 $�T

m+37"+3T{+3p*O?$*8]Y�$-)�p �"» +�&S8?$

Y!�Q�+ 2032 &SY � p 2 7 � Y 2 YZ+HYZ+

nX+ 2�032 &DY � p 2 7 � Y 2 YZ+HYZ+

nY��+3p*OSs�$ 2�032 &SY � p 2 7 � Y 2 Y�+-Y 2 &?+�7 2 r3s�$

R��+3p*O?&S+

+�8FO?+y8(YZ$ � 8\)�p �"» +3&D8?$98�¤nX×Y

�nX × nY

!�£ 032 &SY � p 2 7 � Y 2 Y�+-YZ$ 0 $9p��R)�p*O?$�x�$zOS¼3p 0 �"2 '*)�+-Y�+�8 0 &S+ » +H+y8RO 2&S+37 2 r3s�$ �

P (X × Y )��¤

nP (X×Y ) = 2(nXnY ) !�#%$�&J+ ¿ +�T{x�7=$ � 0 $9pU8 � YZ+�&?+5$ 0 $�pz�R)�p*O?$ A = {0, 1, 2}+5$0 $9p��R)�p*O?$

B = {x, y}� 2c032 &DY � p 2 7 � Y 2 YZ+�Y 2 &S+37 2 r�sz$�'*)�+ 0 $�p�8 � 8ROS+�YZ+�x�&S$�Y�)ZO?$¨[ 2 &?O?+�8 �12 p�$BY�+

A × B¤

� /�) 2 7 23 · 2 = 6

�Z+ 2{032 &SY � p 2 7 � Y 2 YZ+IY�$ 0 $�p��R)Up9OS${x�$�O?¼3p 0 �12 ¤23·2 = 64

!

Page 50: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

��� � ����� ��� � � �?� � � � �s��� � � � � ��� �� � ��� � � � � �s��� � �D ��� � � �� ���

1 2 4 5 60 3

0

1

2

3

4

5

> � /9)�& 2 PU! K I ¬I&D��� 0 ${Y 2 + ¿ x�&S+�8S8?s�$y = x/2

����� � � ' ����!3���-��� ����� � ��� 23� ���������$!�)-����!3� ��&(2�� � �"! #����Q�+F� 2

S)�T x�&S$ZYZ)ZOS$¨[ 2 &FOS+�8 �"2 p�$BYZ+�YZ$ � 8 0 $9p��R)�p*O?$*8

X+

Y:S = X × Y

E +x)�T +37"+3T{+3p*O?$;YZ+

X+

y)�T +37"+3T{+3p*O?$;YZ+

Y!�[]$�pU8 � Y�+3&S+3T{$98

R)�T 2 &?+�7 2 r3s�$B+3T

S!t#%$ZYZ+3T{$*8�)�8 2 &I$¨8&�=T�6�$97=$

RYZ+{$�)ZOS&?$

T{$ZYZ$;x 2 & 2 &S+3xU&?+y8F+�p9O 2 &I&?+�7 2 r���+�8�!-#%$�&H+ ¿ +3T{x�7"$ �±x�$ZYZ+3T{$98J+y8 0 &?+ » +�&xRy

'*) 2 pUYZ$(x, y) ∈ R(X, Y )

!ºX8FO 2 &S+37 2 r�sz$tx�$ZYZ+JO?+�& 2 8w8?+3/9) � p*OS+�8�x�&S$�x�& � +�Y 2 YZ+�8�I¯Bb � b � e ��m � «(T 2 &S+37 2 r�sz$

R¤I&?+��U+ ¿ �"»�2 8F+�x 2 & 2 '*) 2 71'*)�+3&]+37"+3T{+3p*OS$

xY�+

S2 &?+�7 2 r3sz$

xRx|}$9& » ��7 � Y 2 �

� 8FO?${¤J$�x 2 & :x, x

E O 2 Tt6�¤3T¾x�+�&FOS+3p 0 +3& 2 &?+�7 2 r3s�$ ��$�);8F+F� 2µR(x, x) = 1

��B���ye��S��b � b � e ��m � «(T 2 &S+37 2 r�sz$I¤ 2 p*O � ��&?+��U+ ¿ �"»�2 8?+wpUsz$H¡U$�) » +3& )�Tª+37"+3T{+3p*OS$

x+3T

Sx 2 & 2 $-'9) 2 7

xRx|}$�& » �z7 � Y�$ �Z$�);8?+R� 2µR(x, x) = 0

���e � �*�3��elk�m � «(T 2 &S+37 2 r3s�$�¤�8 � T{¤�O?& �1032 8?+-x 2 & 2 O?$ZYZ$

x+

y+�T

S|}$9&?+�T » �z7 � Y 2 8 2 8w+ ¿ x�&S+�8S8F��+y8

xRy+

yRx�Z$9)c8?+R� 2

µR(x, y) = 1+

µR(y, x) = 1�

�Bhyhye � �*�3��elk�m � «\T 2 &S+37 2 r�sz$c¤ 2 8S8 � T{¤�O?& �1032 8F+{pUs�$³¡�$9) » +�&J+37"+3T{+3p*O?$*8x+

y+3T

Sx 2 & 2 $98J'*) 2 � 8 2 8

YZ) 2 8�&S+37 2 r3��+�8xRy

+yRx

|}$9&?+�T¢+ ¿ x�&S+�8S8F��+y8 » ��7 � Y 2 84��B���ye��Dh�e%� �*�3��efk9m � «(T 2 &?+�7 2 r3s�$�¤ 2 p*O � �q8 � T{¤�OS& �1032 8?+tx 2 & 2 O?$ZYZ$

x+

y+�T

S2�0 $�p*O?+ 0 +�&\'*)�+58?+

xRy+

yRx|}$�&S+3T » ��7 � YZ$98�+�p*OSsz$

x = y�

� ��mU��hye}�ye ��m � «(T 2 &S+37 2 r�sz$I¤wO?& 2 pU8 � O �"»�2 8?+wx 2 & 2 O?$ZYZ$x�y��+

z+3T

SO �=» +3&ST5$*8�'*)�+(8?+

xRy|}$�& » ��7 � YZ$H+

yRzO 2 T�6�¤�T » �z7 � Y�$�+�p9ODsz$

xRz¤ » ��7 � YZ$�O 2 Tt6�¤3T � � 8FO?${¤J8F+

µR(x, y) = 1+

µR(y, z) = 1+�p*OSsz$

µR(x, z) = 1�

²�����b�kz�ym���m � «\T 2 &S+37 2 r3s�$�¤ 0 $�p�+ 0 O 2 Y 2 '*) 2 p�YZ$Bx 2 & 2 OS$ZYZ$x+

y+3T

S$³8?+3/9) � p9OS+H|}$�& » +�&SY 2 YZ+ I�Q�+

x 6= y+�p9ODsz${$�)

xRy¤ » �z7 � YZ${$�)

yRx¤��

� ��elk�m�� b�h�� ��b*����m � «\T 2 &S+37 2 r�sz$�¤{YZ+�8FO?+�O � x�$;'*) 2 p�YZ$Bx 2 & 2 O?$ZYZ$x, y

+z+3T

SO �=» +3&ST5$*8�'*)�+58F+

xRz¤ » ��7 � YZ${+

yRzO 2 Tt6�¤3T¾+3p*OSs�$

x = y!

� ��elk�m�� ��e}��b�e}�ym � «\T 2 &S+37 2 r�sz$H¤�Y�+�8FO?+�O � x�$5'9) 2 pUYZ$�x 2 & 2 OS$�Y�$x, y

+z+3T

SO �=» +3&ST{$98X'*)�+I8F+

xRy¤ » ��7 � YZ${+xRz

O 2 Tt6�¤3TW+�p*OSsz$y = z

!i5e�����efk�m � «\T 2 &S+37 2 r�sz${'9)U+I¤KDUp �"0�2"O +y8?'*)�+�&SY 2 + O Y � &?+ � O 2 ¤ 0 ¡ 2 T 2 Y 2 YZ+H6 � � D�p �1032 !

Page 51: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�?���?� � � � �s��� � � � � ������ � ��� ��u��� b � "�°l� ® #l��® � Q�+R� 2

R)UT 2 &?+�7 2 r3s�$¨6 � p��z& �"2 Y�+��Up � Y 2 +3p*O?&S+B+�7=+�T5+�p*O?$985YZ+³)�T T{+�8?T5$ 0 $�pz�R)�p*O?$ �$�) 8?+R� 2

R(X, X)! []$�p�8 � YZ+3&S+¨'*)�+;$ 0 $�p��R)Up9OS$

X8?+R� 2 $ 0 $9p��R)�p*O?$ Y�+;Y � 8 0 � x�7 � p 2 8�YZ+;)�T 0 )�&D8F$ YZ+

/�& 2 Y�) 2 r�sz$ ��+ 2 &?+�7 2 r3s�$R8?+R� 2 YZ+��Up � Y 2Á0 $9T{$¨¤d,!/ Y�1 / e � 8 " 8;:&2 1 !;#�$ZYZ+�T{$98H+�p9ODsz$ÁY � @3+�&�'*)�+�+y8RO 2

&S+37 2 r3s�$-¤ 2 p*O � ��&S+���+ ¿ �=»�2 x�$�&D'*)�+\pU+3p�¡�)�T 2 Y � 8 0 � x�7 � p 2 ¤Ix�&?¤3��&S+�'*) � 8 � OS$tYZ+37 2 T5+y8FT 2 !%£&S+37 2 r3s�$HO 2 Tt6�¤3T¤(OS& 2 pU8 � O �=»�2 x�$�&V'9)U+�8?+\)�T 2 Y � 8 0 � x�7 � p 2 ¤\x�&S¤���&?+y'*) � 8 � O?$-YZ+�)�T 2 $�)�O?& 2 '9)U+\x�$�&�8F) 2-» +3@\¤\x�&S¤���&?+y'*) � 8 � O?$YZ+�)�T 2 O?+�& 0 + � & 2 +3p*OSs�$ 2 x�& � T5+ � & 2 Y � 8 0 � x�7 � p 2 ¤\$96 » �12 T{+3p*OS+()�T x�&S¤���&?+y'9) � 8 � OS$tY 2 O?+3& 0 + � & 2 !%��+3T{$98]'9)U+8?+B)�T 2 Y � 8 0 � x�7 � p 2 ¤Bx�&S¤���&?+y'*) � 8 � O?$ YZ+³)�T 2 $9)ZO?& 2 �%$Á&S+ » +3&D8F$¨p�)�p 0�2 $ 0 $9&?&S+3&D� ��x�$9&FO 2 p*O?$ 2 &?+�7 2 r3s�$¨¤2 8S8 � T{¤�OS& �1032 !® #"® #1_ #f� ¯³b*°lm���g�b*h5��b � � ��e ��mU° ����k9efm b � ��°fb*� ����k�elm«(T 2 &S+37 2 r3s�$ 0 71��8S8 �"0�2 6 � pU�z& �12

R(X, X)'9)U+�¤J&S+��U+ ¿ �"»�2 �Z8 � T{¤�OS& �1032 +IO?& 2 pU8 � O �"»�2 ¤ 0 ¡ 2 T 2 Y 2 YZ+J&S+37 2 r3s�$

YZ+V+�'*) �"»�2 7"¼3p 0 �"2 !�£ > � /�)�& 2 PU! P\T{$98FO?& 2 +3T |}$9&?T 2 YZ+V/�& 2 |}$\&?+�7 2 r���+�8�&?+��U+ ¿ �"»�2 8���8 � T{¤3O?& �"0�2 8±+]O?& 2 p�8 � O �"»�2 8�!vw+�7 2 r���+�8-Y�+�+�'*) �"»�2 7"¼3p 0 �"2 8Ssz$ � T5x�$�&?O 2 p*O?+�8-+3T &?+ 0 $�pU¡�+ 0 � T{+3p*OS$;Y�+�x 2 Y�&?��+�8-+ 0 $9p9OS&?$97=+9!�# 2 & 2¨032 Y 2

1 2

3

(a) Reflexiva

1 2

3

(b) Simétrica

1 2

3

(b) Transitiva

> � /�)U& 2 P�! P I�¬I& 2 |}$*8 � 7")U8ROS& 2 pUYZ$ 2 8�xU&?$9x�& � +�Y 2 YZ+y8]YZ+H&S+���+ ¿ �=» � Y 2 Y�+��Z8 � T{+�OS& �12 +JO?& 2 pU8 � O �=» � Y 2 Y�+�!+37"+3T{+�p9OS$

x ∈ X�*x�$ZYZ+�T5$*8XYZ+��Up � &X)�T 0 $�p��R)�p*OS$

Ax�*'*)�+ 0 $9p9OS¤3T O?$ZYZ$*8�$98X+37"+3T{+3p*OS$98

yYZ+

X'9)U+\8?s�$

&S+37 2�0 � $�p 2 YZ$*8 2xx�+37 2 &S+37 2 r�sz$-YZ+�+�'*) �"»�2 7=¼�p 0 �12 !�� 0 $9p��R)�p*O?$

Axx�$ZYZ+�8?+3&VYZ+��Up � YZ$t|}$�&ST 2 7"T5+�p*O?+ 0 $�T{$

Ax = {y | (x, y) ∈ R(X, X)}.

� +37"+3T{+3p*OS$x¤�x 2 &?O?+XYZ+

AxY�+ » � Y�$ 2 x�&S$�x�& � +�Y 2 YZ+ YZ+X&S+��U+ ¿ �=» � Y 2 YZ+9! �*�w'9)U+

R¤ O?& 2 pU8 � O �"»�2 +X8 � T{¤�OS& �1032 �

032 Y 2 +�7=+�T{+3p*O?${YZ+Ax¤�&S+37 2�0 � $9p 2 YZ$ 0 $�T 0�2 Y 2 )�TWYZ$*8X$9)ZO?&S$98]+37"+3T{+�p9OS$98�! 4 +3pU¡*)UTÀ+37"+3T{+3p*O?${YZ+

Ax¤&S+37 2�0 � $�p 2 YZ$ 0 $�TÀ$9)ZO?&S$5+37"+3T{+3p*O?${'*)�+JpUsz$5+�8FO?+F� 2 +�T

Ax! ºX8FO?+ 0 $9p��R)�p*O?${¤HYZ+���p � Y�$ 0 $�T{$5)�T 2�0 7 2 8?8?+

YZ+H+y'9) �=»�2 7"¼3p 0 �12 YZ+R(X, X)

0 $9T¢&S+�8?x�+ � OS$ 2x!

��� b � "�°l�;® #f��¶ � � 2 T{$98 0 $�pU8 � YZ+3& 2 & 2 &S+37 2 r�sz$R(X, X)

p�$ 0 $�p��R)�p*OS$X = {1, 2, 3, 4, 5, 6, 7}

��YZ+��Up � Y 20 $9T{$

x+

yO?¼�TÀ$tT{+�8?T{$t&S+�8FO?$t'*) 2 p�YZ$�Y �"» � Y � YZ$*8Xx�$�&�P�!�#%$ZYZ+3T{$98]7 � 8FO 2 &XOS$�Y�$98]$98]x 2 &S+�8]YZ+\pQD�T{+3&S$98

'*)�+ 0 $9T5x���+3T 2 &S+37 2 r�sz$ ��'*)�+H8Ssz$NIR = {(1, 1), (1, 4), (1, 7), (2, 2), (2, 5), (3, 3), (3, 6), (4, 1), (4, 4), (5, 2), (5, 5), (6, 3), (6, 6), (7, 1), (7, 4), (7, 7)}

[]7 2 & 2 T{+3p*O?+]+�8FO 2 &?+�7 2 r3sz$�¤]&S+���+ ¿ �=»�2 ��8 � T5¤3O?& �"0�2 +]O?& 2 p�8 � O �"»�2 +Vx�$9&FO 2 p*O?$�¤V)UT 2 &S+37 2 r3s�$\YZ+V+�'*) �"»�2 7"¼3p 0 �"2 !£\8wO?&S¼�8 0 7 2 8S8?+�8�YZ+H+�'*) �"»�2 7"¼3p 0 �"2 YZ+��Up � Y 2 8wx 2 & 2 +y8RO 2 &S+37 2 r�sz${8?s�$ 2 8w8F+�/�) � p*O?+y8MI

A1 = A4 = A7 = {1, 4, 7}A2 = A5 = {2, 5}A3 = A6 = {3, 6}

£ > � /9)�& 2 T{$98FO?& 2 $�/�& 2 |}${'9)U+I&S+3xU&?+y8F+�p9O 2 +y8RO 2 &?+�7 2 r3sz$�!«(T 2 &?+�7 2 r3sz$ YZ+³+y'*) �=»�2 7=¼�p 0 �12 Y �"» � Y�+³+�T /�&S)�x�$*8�8?+3x 2 & 2 YZ$*8-+37"+3T{+�p9OS$985'*)�+³8Ssz$ +�'*) �"»�2 7=+�p*O?+�858?+

7"+ »�2 &ST{$98V+3T 0 $�p*O 252 YZ+��Up � r3s�${Y 2 &S+37 2 r�sz$�!vw+�7 2 r���+�8 ��+3TÂ)�TÂ)�p �=» +3&D8F$

X��'*)�+ 2 xU&?+y8F+�p9O 2 T 2 8(xU&?$9x�& � +�Y 2 YZ+y8wYZ+t&S+��U+ ¿ �=» � Y 2 YZ+H+58 � T{+�OS& �12 8?s�$

0 ¡ 2 T 2 Y 2 8�YZ+H&S+37 2 r�sz$5YZ+JO?$97=+�&S½�p 0 �12 $9)cY�+Hx�&?$ ¿ � T � Y 2 YZ+�!

Page 52: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�9� � ����� ��� � � �?� � � � �s��� � � � � ��� �� � ��� � � � � �s��� � �D ��� � � �� ���

1 2 3

4 5 6

7

> � /9)�& 2 PU! �NI�¬I& 2 |}$5YZ+H)UT 2 &S+37 2 r�sz$5YZ+H+�'*) �"»�2 7"¼3p 0 �"2 !

��� b � "�°l� ® #l�Zn � Q�)�x�$�p�¡ 2 '9)U+Á+3T )�T 0 � & 0 ) � OS$ +37"+�OS&<�9p �"0 $ ¤ 0 $9T5x�$98FO?$ Y�+ 0 � p 0 $ T{.ZYZ)�7"$98 � p*OS+3&?�0 $9p�+ 0 O 2 YZ$98�!�«\T +3pU/�+3pU¡�+ � &?$c+�8FOS�cx�&S$ 0 )�& 2 p�YZ$³Y � 8?x�$9&H+�8FO?+�8IT{.ZYZ)�7"$98J+�T )�T 2 x 2 8FO � 7"¡ 2 7"+ »�2 pUYZ$;+3T0 $9p*O 2 | 2 OS$�&S+�8{O 2 � 8 0 $�T{$JI p DUT5+�&?$ Y�+ � p9OS+3& 0 $9p�+ ¿ ��+�8{+�p9OS&?+ $98�T{.ZYZ)�7"$98B+ 0 $9T�$ T�)�pUYZ$ + ¿ O?+�& � $�& �O 2 T 2 p�¡�$¨YZ$*8HT{.ZYZ)�7"$98 ��+�O 0 !³ºX8ROS+�8HT{.ZYZ)�7"$98-x�$ZYZ+�T 8?+3&H+�p*)UT5+�& 2 YZ$98 0 $�T{$;+37"+3T{+3p*OS$98-YZ$ 0 $�p��R)�p*OS$M = {m1, m2, m3, m4, m5}

!�£ &S+37 2 r�sz$5+3p*O?&S+J+�8FO?+�8�T{.ZYZ)�7"$98�x�$ZYZ+-8?+3&w+ ¿ x�&S+�8S8 2 x�+37 2 &?+�7 2 r3sz$

R =

1 1 0 0 01 1 0 0 10 0 1 0 00 0 0 1 00 1 0 0 1

ºX8FO 2 &S+37 2 r�sz$¨¤�&S+���+ ¿ �=»�2 +³8 � T{¤�O?& �1032 +³x�$9&FO 2 p*O?$¨)�T 2 &?+�7 2 r3sz$¨YZ+BO?$97=+�&S½�p 0 �12 $�) x�&S$ ¿ � T � Y 2 Y�+���+x�$�Y�+I8?+3&V+ ¿ x�&S+�8S8 2 p�$t/9& 2 |}$-T{$98FO?& 2 YZ$-p 2�> � /�)�& 2 !�£ > � /�)U& 2 � pUY �1032 '*)�+\$*8XT{.ZYZ)�7"$98�Pt+\�5x�$ZYZ+�T¢8F+�&0 $97=$ 0�2 YZ$*8\7 �=» &S+3T{+3p*OS+�p 2 x 2 8RO � 7"¡ 2 x�$9&J'*)�+{$cp � » +37 YZ+ � p*O?+3& 0 $9p�+ ¿ s�$�+3p*O?&S+5+37"+�8J+5$98I$�)ZOS&?$*8IT5.ZYZ)U7=$*8pUs�$5¤ � T{x�$9&FO 2 p*O?+J+HpUs�${'*)�+J+�7=+y8�pUsz${+�8FOSs�$ � p*O?+�&F� 0 $9p�+ 0 O 2 YZ$98 0 $9T¢$98�$9)ZO?&S$98�!���� �����������������#�'*% � ��2���������� &(23� �"�"!�#�� �Q�+F� 2 T

R+

SYZ) 2 85&?+�7 2 r���+�8�p�$ )�p �"» +�&S8?$�[ 2 &FOS+�8 �"2 p�$

X × Y,+;YZ+��Up 2�2 &S+37 2 r�sz$Áp�)�7 2 + 2 &S+37 2 r3s�$0 $9T{x�7=+3O 2{0 $9T{$ 2 8�T 2 O?& � @3+�8V&S+37 2 r�sz$

O+

ET{$98FO?& 2 Y 2 8 2 6 2 �=¿ $U!

O =

0 0 · · · 00 0 · · · 0!!!0 0 · · · 0

E =

1 1 · · · 11 1 · · · 1!!!1 1 · · · 1

: PU! u E

«(T O � x�$¨+y8Fx�+ 0 �"2 7�YZ+{&?+�7 2 r3sz$c¤ 2 &?+�7 2 r3sz$ � Y�+3p*O � Y 2 Y�+ :IE ��'*)�+{&S+37 2�0 � $9p 2³0�2 Y 2 +37"+3T{+�p9OS$ 0 $�T +37"+

T{+�8?T{$U! # 2 & 2 )�T )Up �=» +3&D8F$ [ 2 &?O?+y8 �12 p�$X × X

2 &?+�7 2 r3s�$ � Y�+3p*O � Y 2 Y�+³x�$ZYZ+c8?+3&{&S+3x�&S+�8?+3p*O 2 YZ$ ��x�$�&+ ¿ +�T5xU7=$ �Zx�+�7 2 T 2 OS& � @

I =

1 0 · · · 00 1 · · · 0!!!0 0 · · · 1

: PU! � E

£¾O?& 2 pU8?x�$98 � r�sz$ Y�+c)UT 2 T 2 OS& � @c&?+�7 2 r3sz$ :R−1 E |}$�&Sp�+ 0 + 2 T 2 OS& � @;Y 2 &?+�7 2 r3sz$ � p » +3&D8 2 YZ+

R+;¤

YZ+��Up � Y 2�0 $9T5$

Page 53: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�?���?� � � � �s��� � � � � ������ � ��� �Z�

1

2 5

3 4

> � /�)U& 2 P�!Lu I�¬I& 2 |}${YZ+H)�T 2 &?+�7 2 r3s�$5YZ+JOS$�7"+3&D½zp 0 �"2

µR−1(y, x) == µR(x, y): P�!�� E

$�p�YZ+µR+

µR−1

x�$ZYZ+�T »�2 7"+3&(�5$9) 9!(R−1)−1 = R

: P�! � Eº�p*OSs�$�8F+ 2 &?+�7 2 r3s�$

R¤-Y�+��Up � Y 2 x�+37 2 T 2 O?& � @J&S+37 2 r�sz$�PU! �

R =

µR(x1, y1) µR(x1, y2) · · · µR(x1, yn)µR(x2, y1) µR(x2, y2) · · · µR(x2, yn)!!!µR(xm, y1) µR(xm, y2) · · · µR(xm, yn)

: PU! � E

2 &?+�7 2 r3s�$ � p » +3&D8 2 YZ+R��'*)�+J¤

R−1 OS+3T 2 T 2 O?& � @J&S+37 2 r3s�$9PU!= y�

R−1 =

µR(x1, y1) µR(x2, y1) · · · µR(xm, y1)µR(x1, y2) µR(x2, y2) · · · µR(xm, y2)!!!µR(x1, yn) µR(x2, yn) · · · µR(xm, yn)

: P�!" �� E

#%$ZYZ+3T{$98wY�+��Up � & 2 8w8?+3/9) � p9OS+�8�$9x�+�& 2 r���+�8]+3p*O?&S+J&?+�7 2 r���+�8 0 7"�98?8 �"0�2 8�I�c��el��� �

R ∪ S → µR∪S = µR(x, y) ∨ µS(x, y)

����yb9��hyb�k����U� �R ∩ S → µR∩S = µR(x, y) ∧ µS(x, y)]

²�� � "�°lb � b*���y� �R → µR(x, y) = 1 − µR(x, y)

²������ ��� �R ⊂ S → µR(x, y) ≤ µS(x, y)

���b�����ef��m���b��∅ → O

+X → E

# 2 & 2 +y8RO 2 8�$9x�+�& 2 r���+�8 0 $9T¢&S+37 2 r���+y8V'*)�+ 29032 6 2 & 2 TÀYZ+��Up � Y 2 8 »�2 7"+3T 2 8w8?+3/9) � p*OS+�8Vx�&S$�x�& � +yY 2 Y�+�8�I

Page 54: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�9� � ����� ��� � � �?� � � � �s��� � � � � ��� �� � ��� � � � � �s��� � �D ��� � � �� ���²�� � ����mZ�ye �±ef��m���b��

R ∪ S = S ∪ R�R ∩ S = S ∩ R

�Bhyhy�±k�elmZ��e �±ef��m���b��R ∪ (S ∪ T ) = (R ∪ S) ∪ T

�R ∩ (S ∩ T ) = (R ∩ S) ∩ T

`cefh��3��e}µ%���ye �±el��m���b��R ∪ (S ∩ T ) = (R ∪ S) ∩ (R ∪ T )

�R ∩ (S ∪ T ) = (R ∩ S) ∪ (R ∩ T )

�� ����°l���9��� �R = R

���b�� "X��� ����k9efm �A ∪ A = A, A ∩ A = A

���b�����ef��m���b��A ∪ ∅ = A, A ∩ X = A, A ∩ ∅ = ∅, A ∪ X = X

������ &('*% ��'*�"!�� ��' �-����� 23� � ��' &(2�� � �"! #����

Q�)�x�$�pU¡ 2 '9)U+¨+ ¿ � 8FO 2 )�T 2 &?+�7 2 r3s�$R'*)�+;T 2 x�+ �12 +37"+3T{+3p*OS$98�YZ+;)�T�)�p �"» +3&D8?$

X+�T�)�T�)�p �"» +�&S8?$

Y� +{)�T 2 $�)ZOS& 2 &?+�7 2 r3s�$

S'*)�+5T 2 x�+ �12 +�7=+�T{+3p*O?$*8IY�$³)�p �=» +3&D8F$

Yp�$c)�p �"» +�&S8?$

Z..[]$9T5x�$98 � r3sz$³¤ 2

&S+37 2 r3s�$�'*)�+tT 2 x�+ �12 $*8(T{+�8?T{$98�+37"+3T{+3p*OS$98\Y�+Xp�$*8\T{+y8FT{$98\+37"+3T{+3p*OS$98IYZ+

Z� 0 $�T{$�+y8ROD� � pUY �1032 YZ$

p 2-> � /�)�& 2 P�! ��!%M\�tY �=» +3&D8 2 8%T 2 p�+ � & 2 8�Y�+($�6ZOS+3& 2t0 $�T{x�$98 � r�sz$HYZ+IYZ) 2 8�&?+�7 2 r���+�8 �98?+3pUY�$ 2 T 2 � 8 0 $9Tt)�T

X Y Z

R S

T

> � /�)U& 2 P�! � IX[]$�T{x�$98 � r�sz${YZ+-vw+�7 2 r���+�82t0 ¡ 2 T 2 Y 2H0 $�T{x�$98 � r�sz$HT�� ¿ � T5$���T �"p � T5$�! ��)�O?& 2 8�T 2 p�+ � & 2 8�8Ssz$-+�p 0 $9p9OS& 2 Y 2 8X8?s�$-T�� ¿ � T{$���x�&S$ZYZ)ZOS$H+T�� ¿ � T5$���T{¤yY �"2 !® #"® #=¶ #f� ²���� "X��h�el����� m � � ef�£ $�x�+3& 2 r�sz$5YZ+ 0 $9T5x�$98 � r3sz${YZ+-YZ) 2 8�&S+37 2 r���+y8]$96ZO � Y 2{2 x 2 &?O � &(Y 2 8�$�x�+3& 2 r���+y8VY�+HT{� ¿ � T{${+JT �"p � T{$�¤T 2 p�+ � & 2 T 2 � 8 0 $�Tt)UT YZ+�8F+{$96ZO?+�& 0 $�T{x�$98 � r�sz$U!�£ 0 $�T{x�$98 � r�sz$cY 2 8I&S+37 2 r���+y8

R+

S:R ◦ S

E ¤{)�T 2p�$ »�2 &?+�7 2 r3s�$tp�${x�&S$ZYZ)ZO?$³[ 2 &FOS+�8 �"2 p�$

X × ZYZ+��Up � Y 2 x�+37 2 |}.�&STt)�7 2

R ◦ S =

X×Z

maxy

[min(µR(x, y), µS(y, z))]/(x, z): PU!= 9 E

$�p�YZ+;$�8&�"Tt6�$�7"$◦8 � /9p � � 032 2 0 $�T{x�$98 � r�sz$ Y 2 8{&S+37 2 r���+y8

R+

S! 3�) 2 p�YZ$ $�x�&S$ZYZ)ZOS$ [ 2 &FOS+�8 �"2 p�$ ¤

Y � 8 0 &?+3O?${$�8 �"Tt6�$�7"$�YZ+J)�p � sz${x�$�Y�+-8F+�&(8F)�6�8RO � OS) �"Y�$�x�+�7 2 8?$�T 2 + 2 |}.9&?T�)�7 2 � 0�2

R ◦ S =∑

X×Z

maxy

[min(µR(x, y), µS(y, z))]/(x, z): PU!= �K E

Y 2 |}.�&STt)�7 2 290 � T 2 x�$�Y�+3T{$98 » +�& � � 032 &�'*)�+c$ /9& 2 ) YZ+ � p 0 7=)�8?s�$ YZ+ 032 Y 2 x 2 &5Y�+³+�7=+�T5+�p*O?$98�p 2 p�$ »�2&S+37 2 r3s�$ »�2 7=+

µR◦S(x, z) = maxy

[min(µR(x, y), µS(y, z))].: P�!" �P E

Page 55: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�?���?� � � � �s��� � � � � ������ � ��� �*�£À$9x�+�& 2 r3sz$¨YZ+ 0 �z7 0 )U7=$ÁYZ$*8H/�& 2 )�8HY 2 � p 0 7")U8Ssz$ÁYZ$*8-+37"+3T{+�p9OS$98Hp 2 &?+�7 2 r3sz$;¤ 0�2 7 0 )�7 2 Y�$ÁYZ+�|}$�&ST 28 � T � 7 2 & 2 )�T 2 Tt)�7=O � x�7 �"0�2 r3s�$�YZ+HT 2 OS& � @3+y8�$�pUYZ+ 2 $9x�+�& 2 r3s�$5YZ+HT �"p � T5$ 0 $�&S&S+�8?x�$9pUYZ+ 2 Tt)�7=O � x�7 �"0�2 r3s�$

+ 2 YZ+JT�� ¿ � T{$ 2 8F$9T 2 !

��� b � "�°l� ® #f� � �[]$�T{$c+ ¿ +3T{x�7"$;YZ+�8FO?+ 0 ��7 0 )�7"$ »�2 T{$*8 0 $�pU8 � YZ+3& 2 & 2 8J&S+37 2 r���+y8

R+

S'*)�+{7 � / 2 T $*8 »�2 7=$9&?+y8

xi2 $*8»�2 7"$�&S+�8

yi+-$98 »�2 7=$9&?+y8

yi2 $98 »�2 7=$9&?+y8

zi�UT{$*8ROS& 2 Y 2 8�p 2B> � /�)U& 2 PU!L��� 0 )�� 2 8(T 2 OS& � @3+y8(YZ+-&S+37 2 r�sz${+�8FOSs�$

T{$98FO?& 2 Y 2 8V+3TWPU!= ��5+HPU!= �u-&S+�8?x�+ 0 O �=»�2 T{+�p9OS+�!

X1

X2

X3

X4

y1

y2

y3

z1

z2

z3

> � /9)�& 2 P�!�� I vw+�7 2 r���+�8R+

S!

R =

µR(x1, y1) µR(x1, y2) µR(x1, y3)µR(x2, y1) µR(x2, y2) µR(x3, y3)µR(x3, y1) µR(x3, y2) µR(x3, y3)µR(x4, y1) µR(x4, y2) µR(x4, y3)

=

1 1 00 1 00 0 10 0 1

: P�!" 3� E

+S =

µS(y1, z1) µS(y1, z2) µS(y1, z3)µS(y2, z1) µS(y2, z2) µS(y2, z3)µS(y3, z1) µS(y3, z2) µS(y3, z3)

=

1 0 00 0 11 0 0

: P�!" yu E£ 0 $9T{x�$*8 � r3s�$�+�p9OS&?+J+y8RO 2 8wYZ) 2 8�&?+�7 2 r���+�8V)ZO � 7 � @ 2 p�YZ$z�q8F+ 2 |}.�&STt)�7 2 P�!" �Pt|}$�&Sp�+ 0 + 2 8?+3/9) � p9OS+JT 2 O?& � @

R ◦ S =

1 0 10 0 11 0 01 0 0

'*)�+ 0 $9&?&S+�8?x�$�pUYZ+ 2 $�Y �12 /9& 2 T 2 Y 2{> � /9)�& 2 PU! �U!]��6U8F+�& » +\$*8 032 T � p�¡�$*8�+3p*O?&S+J$98�x�$�p*O?$*8wYZ+H$�& � /9+3TÀ+H$*8YZ+IYZ+�8FO � p�$tp 2 8XY�) 2 8 �U/�)U& 2 8 : PU!L�H+�P�! � E + » +�& � ��'*)�+ 0 $�T{$-+�7=+y8X+y8RODsz$-&S+37 2�0 � $9p 2 YZ$*8 Y 2 T{+y8FT 2 T 2 pU+ � & 2 !

® #"® #=¶ #1_ ²���� "X��h�el����� j � e � � � ! �����������4 +y8ROS+�O � x�$ YZ+ 0 $�T{x�$98 � r�sz$ 2 $�x�+3& 2 r�sz$¨YZ+�x�&S$ZYZ)ZOS$ 8F)U6U8RO � OS) �]2 |})�pUr3s�$ÁYZ+�T �"p � T{$�! �\+y8ROS+�T{$�Y�$ 2+�'*) 2 r3s�$5YZ+ 0 $�T{x�$98 � r�sz$ � 032

Page 56: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

uz� � ����� ��� � � �?� � � � �s��� � � � � ��� �� � ��� � � � � �s��� � �D ��� � � �� ���

X1

X2

X3

X4

z1

z2

z3

> � /9)�& 2 PU! �JI � �12 /�& 2 T 2 Y 2 &?+�7 2 r3s�$R ◦ S

R · S =

X×Z

maxy

[µR(x, y) · µS(y, z)]/(x, z): PU!= y� E

# 2 & 2 )Up �=» +3&D8F$*8�Y � 8 0 &S+�OS$98I$c8 � p 2 7�YZ+ � p*O?+3/9& 2 r3s�$�¤�8F)�6�8RO � OS) �"Y�$cx�+37"$¨8 � p 2 7%Y�+{8?$�T 2 O?.�& � $U!5� »�2 7=$9&IY 2|})�pUr�sz$�YZ+ � p 0 7")U8Ssz${Y 2 &S+37 2 r3s�$�YZ+ 0 $�T{x�$98 � r�sz$ »�2 7=+

µR◦S(x, z) = maxy

[µR(x, y) · µS(y, z)]: P�!" �� E

® #"® #=¶ #"® ²���� "X��h�el����� j � e � � � �����elm4\2¨0 $�T{x�$98 � r�sz$cT{� ¿ � T{$z��T{¤�Y �12c2 $9x�+�& 2 r3s�$cYZ+�T �"p � T5$¨¤�8?)�6U8FO � O?)��"YZ$¨x�+37 2 T{¤�Y �12¨2 & � OST{¤�O �1032;0 $�T{$+�8FOS� � pUY �1032 YZ${p 2 |}.�&STt)�7 2 P�!" �� 2 6 2 � ¿ $

R < + > S =

X×Z

maxy

{1

2[µR(x, y) + µS(y, z)]}/(x, z)

: PU!= y� E+ 2 |})�pUr�sz$�YZ+ � p 0 7")U8Ssz$ »�2 7=+

µR◦S(x, z) = maxy

{1

2[µR(x, y) + µS(y, z)]}

: P�!" �� E

������ � ��2�� ��� � � � � ��,-2�'*� � �º�T &S+37 2 r3�*+y8wp�+�6�)�7"$98 2 8($*8\+�7=+�T{+3p*O?$*8(x�$�Y�+3TÂ+�8FO 2 &�&S+37 2�0 � $9p 2 YZ$*8�x 2 & 0 �12 7"T{+3p*O?+ � 2 8?8 � T 0 $�T{$�)�TÂ+37"+��T{+3p*O?$�x�$ZYZ+]x�+�&FOS+3p 0 +3& 2 )�T 0 $9p��R)�p*O?$I8F$9T{+3p*O?+ 2 O?¤])�T 0 +3&?O?$ »�2 7=$9&�!�vw+37 2 r3�*+y8±p�+�6�)�7"$98 2 8�8?s�$ 0 $9p��R)�p*O?$*8p�+�6�)�7"$98?$98-YZ+��Up � YZ$98t+3T x�&S$�Y�)ZO?$*8�[ 2 &?O?+y8 �12 p�$*8HYZ+ 0 $�p��R)Up9OS$98 0 7"�98?8 �10 $98 ��$9) 8F+F� 2 $Á«(p �=» +3&D8F$ÁYZ+ � � 8R�0 )U&S8?$¨YZ+ �=¿ $�)�YZ+5O?+3&�8F$9T5+�p*O?+{)�T 2 Y � T{+�pU8?s�$ :

XE x 2 & 2 8?+3&HTt)U7 O � �qY � T5+�pU8 � $9p 2 7 :

X × Y × Z × · · ·E !

��$ T{+�8?T{$ÁT{$�Y�$ '*)�+ ��@3+3T{$*8 0 $9T 0 $�pz�R)�p*O?$98tp�+36U)�7=$*8F$*8-x�$ZYZ+3T{$98t&?+�x�&S+�8?+3p*O 2 &-&?+�7 2 r���+�8-p�+36�)U7=$*8 2 8T{$98FO?& 2 pUY�$c$*8 »�2 7"$�&S+�8J&S+37 2�0 � $�p 2 YZ$*8�+�$Á8?+3)�/9& 2 ) YZ+B&?+�7 2 r3sz$ 0 $9T{$c+y8ROD� � p�Y �"0�2 YZ$¨p 2 +�'*) 2 r�sz$;P�!LKz� �+ ¿ +�T5xU7=$�YZ+H&S+37 2 r�sz$t6 � pU��& �12 !

R = {(x, y), µR(x, y)}: P�!LKz� E

Page 57: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�?���?� � � � �s��� � � � � ������ � ��� u� «(TªT{$ZYZ$HYZ+w&?+�x�&S+�8?+3p*O 2 & 2 &S+37 2 r�sz$I¤ � p�Y �"0�2 & 0�2 Y 2 )�T 2 Y 2 8�OF��)�x�7 2 8�Y 2 &S+37 2 r�sz$�+(8F+�)5/9& 2 )5Y�+w&?+�7 2 r3sz$ �� 8ROS$5¤

R =∑

(xi,yi)∈X×X

µR(xi, yi)/(xi, yi): P�!LKZ E

$�p�YZ+ $(8F$9T 2 O?.9& � $ � pUY �"0�2 '9)U+ 2 &?+�7 2 r3s�$R¤%|}$�&ST 2 Y 2 x�+�7 2 )�p � sz$wYZ+ OS$�Y�$98 $*8�+37"+3T{+3p*O?$*8

µR(xi, yi)/(xi, yi)!

# 2 & 2 )UTWx�&S$ZYZ)ZO?$B[ 2 &FOS+�8 �"2 p�$ 0 $�p*O&�"p�)�$5O?+�&&� 2 T{$98

R =

X×Y

µR(x, y)/(x, y): P�!LK�K E

#%$ZYZ+3T{$98�O 2 Tt6�¤3T &?+�x�&?+y8F+�p*O 2 &�)UT 2 &?+�7 2 r3s�$ 6 � pU��& �12m × n

x�$9&BT{+ � $ YZ+ )�T 2 T 2 OS& � @ 0 $9T 28?+3/�) � p*O?+I|}$9&?T 2

R =

µR(x1, y1) µ(x1, y2) · · · µR(x1, yn)µR(x2, y1) µR(x2, y2) · · · µR(x2, yn)!!! !!! !!! !!!µR(xm, y1) µR(xm, y2) · · · µR(xm, yn)

: P�!LKzP E

£\8I&S+37 2 r3�*+y8(p�+�6�)�7"$98 2 8�p�)�7 2 + 0 $�T{x�7"+�O 2O�E+

IO?¼�T 2 T{+y8FT 2 Y�+��Up � r3s�$cY 2 8�&?+�7 2 r3sz$ 0 71��8S8 �1032 8

� pUY �1032 Y 2 8(p 2 8(T 2 O?& � @�+�8\Y 2 8(+y'*) 2 r3�*+y8(+�TÂPU! u5+5P�! �z&S+�8?x�+ 0 O �=»�2 T{+�p9OS+�!(£�8?8 � T 0 $�T{$�&S+37 2 r���+y8 0 71��8S8 �1032 8O?¼�TÂ&?+�7 2 r���+�8 � p » +�&S8 2 8 � 2 8\&S+37 2 r3�*+y8wp�+�6�)�7"$98 2 O?¼�T 2 8\8?) 2 8 � p » +3&D8 2 8(YZ+��Up � Y 2 8IYZ$BT{+y8FT{$�T{$ZYZ$U!�#%$�&?�O 2 p*O?$ 2 OS& 2 pU8?x�$*8 � r3s�$;Y�+�)�T 2 T 2 OS& � @�&S+37 2 r3s�$ :

RT E |}$�&Sp�+ 0 + 2 T 2 OS& � @BY 2 &S+37 2 r�sz$ � p » +3&D8 2 YZ+R+B¤

YZ+��Up � Y 2�0 $9T5$

µRT (x, y) == µR(x, y): P�!LK�� E

+(RT )T = R

: P�!LK�u Eº�p*OSs�$�8F+ 2 &?+�7 2 r3s�$

R¤-Y�+��Up � Y 2 x�+37 2 T 2 O?& � @J&S+37 2 r�sz$�PU! K��

R =

µR(x1, y1) µR(x1, y2) · · · µR(x1, yn)µR(x2, y1 µR(x2, y2 · · · µR(x2, yn!!!µR(xm, y1 µR(xm, y2 · · · µR(xm, yn

: P�!LKz� E

2 &?+�7 2 r3s�$ � p » +3&D8 2 YZ+R��'*)�+J¤

RT O?+�T 2 T 2 OS& � @J&?+�7 2 r3sz$*P�!LK9�

RT =

µR(x1, y1) µR(x2, y1) · · · µR(xm, y1)µR(x1, y2) µR(x2, y2) · · · µR(xm, y2)!!!µR(x1, yn) µR(x2, yn) · · · µR(xm, yn)

: P�!LK9� E

������ � � ' ����!3���-��� ����� � ��� 23� ������� � ����,�23'*� � �Q�+F� 2

S)�T­x�&S$ZYZ)ZO?$�[ 2 &?O?+�8 �12 p�$ÁYZ+cYZ$ � 8 0 $�p��R)Up9OS$98

X+

Y:S = X × Y

E +x)�T­+37"+3T{+�p9OS$ YZ+

X+y)�T¹+37"+3T{+3p*OS$;Y�+

Y!³[]$�pU8 � YZ+3&S+3T{$98

R)UT 2 &S+37 2 r�sz$³p�+36�)U7=$*8 2 +3T

S!�ºX8FO 2 &S+37 2 r�sz$Bx�$ZYZ+{O?+�& 2 8

8?+3/�) � p*O?+y8�x�&S$�x�& � +yY 2 Y�+�8�I¯Bb � b � e ��m � «(T 2 &S+37 2 r3s�$

Rp�+�6�)�7"$98 2 ¤H&?+��U+ ¿ �"»�2 8?+Hx 2 & 2 '*) 2 71'*)�+3&\+37"+3T{+3p*O?$

xY�+

S2 &?+�7 2 r3sz$

xRx|}$�& » �z7 � Y 2 � � 8ROS$5¤H${x 2 & :x, x

E O 2 T�6�¤�T¢x�+3&?O?+3p 0 +3& 2 &S+37 2 r�sz$ �9$9)c8?+R� 2µR(x, x) = 1

��B���ye��S��b � b � e ��m � «(T 2 &S+37 2 r�sz$I¤ 2 p*O � ��&?+��U+ ¿ �"»�2 8?+wpUsz$H¡U$�) » +3& )�Tª+37"+3T{+3p*OS$

x+3T

Sx 2 & 2 $-'9) 2 7

xRx|}$�& » �z7 � Y�$����e � �*�3��elk�m � «(T 2 &S+37 2 r3s�$�¤�8 � T{¤�O?& �1032 8?+-x 2 & 2 O?$ZYZ$

x+

y+�T

S|}$9&?+�T » �z7 � Y 2 8 2 8w+ ¿ x�&S+�8S8F��+y8

xRy+

yRx�

Page 58: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

u�K � ����� ��� � � �?� � � � �s��� � � � � ��� �� � ��� � � � � �s��� � �D ��� � � �� ����Bhyhye � �*�3��elk�m � «\T 2 &S+37 2 r�sz$5¤ 2 8S8 � T{¤�O?& �1032 8?+JpUsz${¡�$9) » +�&�+37"+3T{+3p*O?$*8

x+

y+3T

Sx 2 & 2 $98w'*) 2 � 8

xRy+yRx

|}$�&S+3T¢+ ¿ x�&?+y8?8?��+�8 » �z7 � Y 2 8���B���ye��Dh�e%� �*�3��efk9m � «(T 2 &?+�7 2 r3s�$�¤ 2 p*O � �q8 � T{¤�OS& �1032 8?+tx 2 & 2 O?$ZYZ$

x+

y+�T

S2�0 $�p*O?+ 0 +�&\'*)�+58?+

xRy+

yRx|}$�&S+3T » ��7 � YZ$98�+�p*OSsz$

x = y�

� ��mU��hye}�ye ��m � «(T 2 &?+�7 2 r3s�$t¤IO?& 2 pU8 � O �"»�2 8?+Ix 2 & 2 O?$ZYZ$x�y�Z+

z+3T

SO �"» +�&?T{$*8]'*)�+H8?+

xRy|}$9& » �z7 � YZ$

+yRz

O 2 T�6�¤�T » �z7 � YZ${+3p*OSs�$xRz

¤ » ��7 � YZ$5O 2 Tt6�¤3TC�²�����b�kz�ym���m � «\T 2 &S+37 2 r3s�$�¤ 0 $�p�+ 0 O 2 Y 2 '*) 2 p�YZ$Bx 2 & 2 OS$ZYZ$

x+

y+3T

S$³8?+3/9) � p9OS+H|}$�& » +�&SY 2 YZ+ I�Q�+

x 6= y+�p9ODsz${$�)

xRy¤ » �z7 � YZ${$�)

yRx¤��

� ��elk�m�� b�h�� ��b*����m � «\T 2 &S+37 2 r�sz$�¤{YZ+�8FO?+�O � x�$;'*) 2 p�YZ$Bx 2 & 2 O?$ZYZ$x, y

+z+3T

SO �=» +3&ST5$*8�'*)�+58F+

xRz¤ » ��7 � YZ${+

yRzO 2 Tt6�¤3T¾+3p*OSs�$

x = y!

� ��elk�m�� ��e}��b�e}�ym � «\T 2 &S+37 2 r�sz$H¤�Y�+�8FO?+�O � x�$5'9) 2 pUYZ$�x 2 & 2 OS$�Y�$x, y

+z+3T

SO �=» +3&ST{$98X'*)�+I8F+

xRy¤ » ��7 � YZ${+xRz

O 2 Tt6�¤3TW+�p*OSsz$y = z

!����� �����������������#�'*% � ��2���������� � ��� ,-2�'*� ���Q�+F� 2 T

R+

SY�) 2 8]&?+�7 2 r���+�8Xp�+36U)�7=$*8 2 8�p�$�)�p �"» +�&S8?$5[ 2 &?O?+y8 �12 pU$U!�#�$ZYZ+�T{$98VYZ+��Up � & 2 8V8F+�/�) � p*O?+y8X$9x�+�& 2 �

r���+y8�+3p*O?&S+J&?+�7 2 r���+�8Vp�+�6�)�7"$98 2 8�I�c��el��� �

R ∪ S → µR∪S = µR(x, y) ∨ µS(x, y)

����yb9��hyb������ �R ∩ S → µR∩S = µR(x, y) ∧ µS(x, y)

²�� � "�°lb � b*���y� �R → µR(x, y) = 1 − µR(x, y)

²������ ��� �R ⊂ S → µR(x, y) ≤ µS(x, y)

���b�����ef��m���b��∅ → O

+X → E

# 2 & 2 +y8RO 2 8�$9x�+�& 2 r���+�8 0 $9T¢&S+37 2 r���+y8V'*)�+ 29032 6 2 & 2 TÀYZ+��Up � Y 2 8 »�2 7"+3T 2 8w8?+3/9) � p*OS+�8Vx�&S$�x�& � +yY 2 Y�+�8�I²�� � ����mZ�ye �±ef��m���b��

R ∪ S = S ∪ R�R ∩ S = S ∩ R

�Bhyhy�±k�elmZ��e �±ef��m���b��R ∪ (S ∪ T ) = (R ∪ S) ∪ T

�R ∩ (S ∩ T ) = (R ∩ S) ∩ T

`cefh��3��e}µ%���ye �±el��m���b��R ∪ (S ∩ T ) = (R ∪ S) ∩ (R ∪ T )

�R ∩ (S ∪ T ) = (R ∩ S) ∪ (R ∩ T )

�� ����°l���9��� �R = R

���b�� "X��� ����k9efm �A ∪ A = A, A ∩ A = A

���b�����ef��m���b��A ∪ ∅ = A, A ∩ X = A, A ∩ ∅ = ∅, A ∪ X = X

[]$�T{$ 2�0 $9p*O?+ 0 + 0 $9TÂ$98 0 $�p��R)Up9OS$98�p�+36U)�7=$*8F$*8�� 2 8�&S+37 2 r���+y8wp�+36U)�7=$*8 2 8wO 2 Tt6�¤3T p�sz$�$96�+yYZ+ 0 +3T 2 8\7=+ � 8Y 2 + ¿Z0 7=)�8?s�$ÁYZ$¨T{+ � $Á+ 0 $9p9OS& 2 Y � r3sz$ �Z�F�Á'*)�+�x�$ZYZ+3T{$*8JO?+�&�8?$�6�&S+3x�$98 � r�sz$c+3p*O?&S+�)�T 2 &S+37 2 r�sz$c+�$Á8F+�)0 $9T{x�7=+�T{+3p*O?$�! ��+�8FO?+HT{$ZYZ$5O?+�T{$98('*)�+

R ∩ R 6= E: P�!LKz� E

+R ∪ R 6= O

: P�!LKz� E��� b � "�°l� ® #f� � �Q�)�x�$�pU¡ 2 '*)�+�p�.*8�O?+�T5$*8 2 8 YZ) 2 8�&?+�7 2 r���+�8 :

R+

SE � pUY �1032 Y 2 8�p 2 8�� 2 6�+37 2 8%PU! P9�\+wP�! P� 9!%� xU&?$96�7=+�T 2

¤ 032 7 0 )�7 2 & 2 «\p � s�${+Ioqp*OS+3&D8F+yr3sz$�+3p*O?&S+I+�8FO 2 8�&S+37 2 r3��+�8�!�oR8FO?$5¤���8?+R0 $9&?&S+�8?x�$9pUYZ+ O &S+37 2 r�sz$

x¤ � /�) 2 7 2

y+

SO &?+�7 2 r3s�$

x¤�T 2 � $9&I'*)�+

y��'*)�+�&?+�T{$98($96ZO?+�&

R⋃

S,'9)U+�8 � /9p � � 032

x¤ � /�) 2 7 2

y$�)

x¤tT 2 � $�&

'*)�+y+

R⋂

S�*'*)�+\8 � /�p � � 032

x¤ � /�) 2 7 2

y+

x¤(T 2 � $�&V'9)U+

y! ��+ » +3T{$98�O?$9T 2 & 0 ) � Y 2 YZ$ 0 $�T 2 7"/�)�T 2 8

Page 59: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�?���?� � � � �s��� � � � � ������ � ��� u�PYZ+y8RO 2 8-$9x�+�& 2 r���+�8-x�$ � 8�+37 2 8tx�$ZYZ+�T &S+�8?)�7=O 2 &-+3T Y 2 YZ$98�'9)U+BpUs�$¨O?+�T­8 � /�p � � 032 YZ$Á8?+3T�½zp*O �10 $ YZ+B&S+ 2 78 � /9p � � 032 Y�$U! 4 $ 032 8F$tO?+�T5$*8w'*)�+H$�&S+�8?)�7=O 2 YZ$5Y 2 $9x�+�& 2 r3s�$�YZ+ � p*O?+3&D8?+�r3s�$tx�$ZYZ+-8F+�& � p*O?+�&?xU&?+3O 2 Y�$ 0 $�T{$x¤ � /�) 2 7 2

y+3T 0 +3&?O 2 T{+�Y � Y 2 +JO 2 Tt6�¤3T¢T 2 � $�&('*)�+

y+3T 0 +3&?O?${/�& 2 )±!

R (x e igual a y) =

1 2 3 4 5 61 1.0 0.8 0.6 0.4 0.2 0.02 0.8 1.0 0.8 0.6 0.4 0.23 0.6 0.8 1.0 0.8 0.6 0.44 0.4 0.6 0.8 1.0 0.8 0.65 0.2 0.4 0.6 0.8 1.0 0.86 0.0 0.2 0.4 0.6 0.8 1.0

: P�! P�� E

S (x e maior que y) =

1 2 3 4 5 61 0.0 0.0 0.0 0.0 0.0 0.02 0.2 0.0 0.0 0.0 0.0 0.03 0.4 0.2 0.0 0.0 0.0 0.04 0.6 0.4 0.2 0.0 0.0 0.05 0.8 0.6 0.4 0.2 0.0 0.06 1.0 0.8 0.6 0.4 0.2 0.0

: P�! P� E

£¦)�p � s�$BY 2 8\&?+�7 2 r���+�8R+

S¤H|}$9&?T 2 Y 2 +�8 0 $97=¡�+�pUYZ$���8?+H$�T 2 � $�& »�2 7=$9&\YZ$98�YZ$ � 8(/9& 2 )U8(YZ+ � p 0 7=)�8?s�$

p 2 8\&S+37 2 r3�*+y8(+ 2 � p*O?+3&D8?+�r3s�$ 2 x 2 &?O � &�YZ$*8\T{+3pU$�&S+�8 »�2 7=$9&?+y8���$�6ZOS+3pUY�$z�q8F+�$98�&?+y8F)�7=O 2 YZ$98\T{$98FO?& 2 YZ$*8(p 2 8T 2 OS& � @3+y8wP�! P9Kt+JPU! P9P�!

R ∪ S [(x igual y) OU (x maior y)] =

1 2 3 4 5 61 1.0 0.8 0.6 0.4 0.2 0.02 0.8 1.0 0.8 0.6 0.4 0.23 0.6 0.8 1.0 0.8 0.6 0.44 0.6 0.6 0.8 1.0 0.8 0.65 0.8 0.6 0.6 0.8 1.0 0.86 1.0 0.8 0.6 0.6 0.8 1.0

: P�! P9K E

R ∩ S [(x igual y) E (x maior y)] =

1 2 3 4 5 61 0.0 0.0 0.0 0.0 0.0 0.02 0.2 0.0 0.0 0.0 0.0 0.03 0.4 0.2 0.0 0.0 0.0 0.04 0.4 0.4 0.2 0.0 0.0 0.05 0.2 0.4 0.4 0.2 0.0 0.06 0.0 0.2 0.4 0.4 0.2 0.0

: P�! P�P E

������ &('*% ��'*�"!�� ��' �-����� 23� ������� � ����,�23'*� � �Q�)�x�$�pU¡ 2 '*)�+;+ ¿ � 8FO 2 )�T 2 &S+37 2 r�sz$

Rp�+�6�)�7"$98 2 '*)�+cT 2 x�+ �12 +�7=+�T{+3p*O?$*85Y�+c)UT )�p �"» +3&D8?$

X+�T~)�T

)�p �"» +�&S8?$Y��+³)UT 2 $�)ZOS& 2 &?+�7 2 r3s�$Áp�+36U)�7=$*8 2

S'*)�+cT 2 x�+ �"2 +37"+3T{+�p9OS$985YZ$ )Up �=» +3&D8F$

Yp�$ )�p �"» +�&S8?$

Z..[]$�T{x�$98 � r�sz${¤ 2 &S+37 2 r�sz$�'*)�+tT 2 x�+ �"2 $*8wT{+�8?T{$98(+�7=+�T{+3p*O?$*8\YZ+

XpU$98(T{+y8FT{$98\+37"+3T{+3p*OS$98\Y�+

Z!

vw+�7 2 r���+�8�p�+36U)�7=$*8 2 8-x�$ZYZ+3T~8?+3& 0 $9T5x�$98FO 2 8t+�p*O?&S+³8 � x 2 & 2 |}$�&ST 2 &5)�T 2 $�)�O?& 2 &S+37 2 r3s�$ YZ+cY � |}+�&?+�p*O?+�8T 2 p�+ � & 2 8�!�£ $�x�+3& 2 r�sz$Jx�& � p 0 � x 2 7U'*) 2 pUYZ$H+�8FO 2 6�+�7=+ 0 +�T5$*8 0 $9T{x�$*8 � r���+�8�¤ 032 7 0 )�7 2 &�$98�/�& 2 )U8�Y�+ � p 0 7")U8?s�$p 2 &S+37 2 r3s�$�+�p*O?&S+I$*8�+37"+3T{+3p*OS$98wYZ+

X+-YZ+

Z!

® #"® # � #f� ²���� "X��h�el����� m � � ef�£$�x�+3& 2 r3s�$tYZ+ 0 $9T{x�$*8 � r3s�$tYZ+JYZ) 2 8V&S+37 2 r3�*+y8�p�+�6�)�7"$98 2 8]$�6ZO � Y 2�2 x 2 &?O � &�Y 2 8V$�x�+3& 2 r3��+�8XY�+�T 2�¿ :

∨E +

T � p :∧E ¤�T 2 p�+ � & 2 T 2 � 8 0 $9Tt)�TÀYZ+I8F+�$�6ZOS+3& 0 $9T5x�$98 � r3sz$�!�£ 0 $9T5x�$98 � r3sz$�Y 2 8]&?+�7 2 r���+�8

R+

S:R ◦S

E¤H)�T 2 p�$ »�2 &?+�7 2 r3sz$�p�${x�&S$�Y�)ZO?$³[ 2 &FOS+�8 �"2 p�$

X × ZY�+��Up � Y 2 x�+37 2 |}.�&STt)�7 2

R ◦ S =∑

X×Z

y

[µR(x, y) ∧ µS(y, z)]/(x, z): PU! P�� E

Page 60: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

u�� � ����� ��� � � �?� � � � �s��� � � � � ��� �� � ��� � � � � �s��� � �D ��� � � �� ���$�p�YZ+X$I8 �"Tt6�$�7"$

◦8 � /�p � � 0�2(2\0 $�T{x�$*8 � r3s�$(Y 2 8�&S+37 2 r���+y8

R+

S!3�) 2 p�YZ$($\xU&?$ZYZ)ZOS$J[ 2 &FOS+�8 �"2 p�$(¤]Y � 8 0 &S+�OS$$�8 �"Tt6�$�7"$�YZ+H)�p � sz${x�$ZYZ+H8F+�&(8F)�6�8RO � OS) �"Y�$�x�+�7 2 8?$�T 2 + 2 |}.9&?T�)�7 2 � 0�2

R ◦ S =∑

X×Z

y

[µR(x, y) ∧ µS(y, z)]/(x, z): PU! P*u E

Y 2 |}.�&STt)�7 2B290 � T 2 x�$�Y�+3T{$98 » +�& � � 0�2 &�'*)�+ 2 |})�pUr3s�$cY�+ � p 0 7")U8Ssz$;YZ+ 032 Y 2 x 2 &IYZ+{+�7=+�T5+�p*O?$98�p 2 p�$ »�2&S+37 2 r3s�$ »�2 7=+

µR◦S(x, z) =∨

y

[µR(x, y) ∧ µS(y, z)].: P�! P�� E

£À$9x�+�& 2 r3sz$¨YZ+ 0 �z7 0 )U7=$ÁYZ$*8H/�& 2 )�8HY 2 � p 0 7")U8Ssz$ÁYZ$*8-+37"+3T{+�p9OS$98Hp 2 &?+�7 2 r3sz$;¤ 0�2 7 0 )�7 2 Y�$ÁYZ+�|}$�&ST 28 � T � 7 2 & 2 )�T 2 Tt)�7=O � x�7 �"0�2 r3s�$�YZ+HT 2 OS& � @3+y8�$�pUYZ+ 2 $9x�+�& 2 r3s�$5YZ+HT �"p � T5$ 0 $�&S&S+�8?x�$9pUYZ+ 2 Tt)�7=O � x�7 �"0�2 r3s�$+ 2 YZ+JT�� ¿ � T{$ 2 8F$9T 2 !%ºX8FO?+JO � x�$�YZ+J$�x�+3& 2 r3s�$t¤H7 2 &S/ 2 T5+�p*O?+I+3T{x�&S+3/ 2 Y 2 +�T 2 x�7 �1032 r���+�8�YZ+ 0 $�p*O?&S$�7"+'*)�+J)U8 2 TW7".�/ �"0�2 pU+36�)�7"$98 2 !

��� b � "�°l� ® #f� � �� 2 T{$*8 0 $�p�8 � YZ+3& 2 & 2 8wYZ) 2 8(&?+�7 2 r���+�8

R+

ST{$98FO?& 2 Y 2 8Vp 2�> � /9)�& 2 PU! � � 0 )�� 2 8wT 2 O?& � @3+�8wY�+ � p 0 7=)�8?s�$

8Ssz$¨T{$98FO?& 2 Y 2 8Jp 2 8-T 2 O?& � @�+�8-PU! P��³+BP�! P��U!B# 2 & 2 +3p 0 $�p*OS& 2 & 2Á0 $9T5x�$98 � r3sz$¨Y 2 8H&S+37 2 r���+y8T = R ◦ S

�»�2 T{$98�+3T{x�&S+3/ 2 & 2 |}.�&STt)�7 2 T 2�¿ ��T � p�YZ+ 0 $�T{x�$*8 � r3s�$ : P�! P�� E x 2 & 2 +3p 0 $�p*O?& 2 & 2 T 2 O?& � @\Y�+ � p 0 7=)�8?s�$-'9)U+&S+3x�&S+�8?+3p*O 252 &?+�7 2 r3s�$

T!

£\8�T 2 O?& � @�+�8R+

S8?s�$ 2 8�8F+�/�) � p*O?+�8�I

R =

µR(x1, y1) µR(x1, y2) µR(x1, y3)µR(x2, y1) µR(x2, y2) µR(x2, y3)µR(x3, y1) µR(x3, y2) µR(x3, y3)µR(x4, y1) µR(x4, y2) µR(x4, y3)

=

1.0 0.8 0.60.8 1.0 0.80.6 0.8 1.00.4 0.6 0.8

: P�! P*� E

+

S =

µS(y1, z1) µS(y1, z2) µS(y1, z3) µS(y1, z4)µS(y2, z1) µS(y2, z2) µS(y2, z3) µS(y2, z4)µS(y3, z1) µS(y3, z2) µS(y3, z3) µS(y3, z4)

=

1.0 0.8 0.6 0.40.8 1.0 0.8 0.60.6 0.8 1.0 0.8

: P�! P�� E

1.0

0.8

0.6

0.8

1.0

0.8

1.0

0.80.6

0.8

0.60.4

1.0

0.8

0.60.4

0.81.0

0.8

0.6

1.0

0.8

0.80.6

x1

x2

x3

x4

y1

y2

y3

z1

z2

z3

z4

> � /9)�& 2 P�! � I vw+�7 2 r���+�8R+

S!

Page 61: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�?���?� � � � �s��� � � � � ������ � ��� u9u£ 0 $9T5x�$98 � r3sz${YZ+y8RO 2 8wYZ) 2 8�&?+�7 2 r���+�8 � 0�2 � /9) 2 7 +�p9ODsz$ 2

T = R ◦ S =

1.0 0.8 0.8 0.60.8 1.0 0.8 0.80.8 0.8 1.0 0.80.6 0.8 0.8 0.8

: P�! P�� E

+H$�Y �12 /�& 2 T 2�0 $�&S&S+�8?x�$9pUYZ+3p*OS+ 2 +�8FO 2 &S+37 2 r�sz$5+�8FOS�5T{$98FO?& 2 YZ$5p 2{> � /9)�& 2

x1

x2

x3

x4

z1

z2

z3

z4

1.0

0.80.8

0.6

0.81.0

0.80.8

0.80.8

1.00.8

0.6 0.8

0.80.8

> � /�)�& 2 P�!" �� I � �"2 /�& 2 T 2 Y 2 &S+37 2 r�sz$T

® #"® # � #1_ ²���� "X��h�el����� j � e � � � ! �����������4 +y8ROS+�O � x�$ YZ+ 0 $�T{x�$98 � r�sz$ 2 $�x�+3& 2 r�sz$¨YZ+�x�&S$ZYZ)ZOS$ 8F)U6U8RO � OS) �]2 |})�pUr3s�$ÁYZ+�T �"p � T{$�! �\+y8ROS+�T{$�Y�$ 2+�'*) 2 r3s�$5YZ+ 0 $�T{x�$98 � r�sz$ � 032

R · S =

X×Z

y

[µR(x, y) · µS(y, z)]/(x, z): PU! �*� E

# 2 & 2 )Up �=» +3&D8F$*8�Y � 8 0 &S+�OS$98I$c8 � p 2 7�YZ+ � p*O?+3/9& 2 r3s�$�¤�8F)�6�8RO � OS) �"Y�$cx�+37"$¨8 � p 2 7%Y�+{8?$�T 2 O?.�& � $U!5� »�2 7=$9&IY 2|})�pUr�sz$�YZ+ � p 0 7")U8Ssz${Y 2 &S+37 2 r3s�$�YZ+ 0 $�T{x�$98 � r�sz$ »�2 7=+

µR◦S(x, z) =∨

y

[µR(x, y) · µS(y, z)]: P�! �U E

® #"® # � #"® ²���� "X��h�el����� j � e � � � �����elm4\2¨0 $�T{x�$98 � r�sz$cT{� ¿ � T{$z��T{¤�Y �12c2 $9x�+�& 2 r3s�$cYZ+�T �"p � T5$¨¤�8?)�6U8FO � O?)��"YZ$¨x�+37 2 T{¤�Y �12¨2 & � OST{¤�O �1032;0 $�T{$+�8FOS� � pUY �1032 YZ${p 2 |}.�&STt)�7 2 P�!" �� 2 6 2 � ¿ $

R < + > S =

X×Z

y

{1

2[µR(x, y) + µS(y, z)]}/(x, z)

: PU! ��K E+ 2 |})�pUr�sz$�YZ+ � p 0 7")U8Ssz$ »�2 7=+

Page 62: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

uz� � ����� ��� � � �?� � � � �s��� � � � � ��� �� � ��� � � � � �s��� � �D ��� � � �� ���

Page 63: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� ��������� � �

� ����w��� ���� � �������� � � �

#=��� � -�� *�����# *,6 / #32!$5M��6G-�6 %(* 4)�!��,6�� *,+?-C6# 2 & 2 YZ+�8 0 &S+ » +3& 2 T{+�Y � Y 2 YZ+H)UTW|}+�pM��T{+3pU$�YZ$�Tt)�p�YZ$�&?+ 2 7 � 0 $9T{$ 2 | 2 � ¿�2 YZ+H&S+3p�Y 2 YZ+H)�T 2 | 2 T �"7 �12 �x�&S+ 0 � 8 2 T{$98�YZ+V�R)�p*O 2 & » �z& � $98 0 $9p��R)�p*O?$*8�p�+36�)U7=$*8F$*83!#%$�&{+ ¿ +�T5xU7=$ ��x 2 & 2 2 &S+3pUY 2 Y 2 | 2 T �"7 �12 O?+�&&� 2 T{$985YZ+¨YZ+��Up � &5$*8 0 $�pz�R)�p*O?$9856 2 � ¿�2 ��T{¤�Y �12 + 2 7 O 2

&S+3pUY 2 83!5¤ � T5x�$�&?O 2 p*O?+{p�$zO 2 &H'*)�+ 2 8HYZ+��Up � r���+�8HY�$98 0 $9p��R)�p*O?$*8Jx�$ZYZ+�T¹8?+�8F$96�&?+�x�$9& � x�+3&ST � O � p�YZ$¨'9)U+)�T 2 | 2 T �"7 �12 8F+F� 250 7 2 8?8 � � 032 Y 2�0 $�T{${x�+�&FOS+3p 0 +3p*O?+ 2 YZ$ � 8 0 $�p��R)�p*OS$98 � 0 $�TW/�& 2 )U8�YZ+ � p 0 7")U8Ssz${Y � |}+�&?+�p*O?+�8$�) T{+y8FT{$ � /�) 2 � 8�!¨ºX8FO?+³x�&?$ 0 +�8S8F$ÁYZ+³YZ+��Up � r�sz$ � p 0 7") � +�8 0 $97=¡�+�& 2 8-|}$�&ST 2 8-Y 2 8H|})�p�r���+�8�YZ+ � p 0 7")U8Ssz$ �� 8ROS$B¤ 2 8F+�T�½zp*O �1032 Y�$ 0 $9p��R)�p*O?$�!H��)�O?&S$�x�$9p*O?$ � T{x�$9&FO 2 p*O?+�¤�YZ+��Up � r3s�$³YZ$c«(p �"» +�&S8?$BYZ+ � � 8 0 )�&D8F$BY 2»�2 & � � » +37�!«(T 2t» +3@I'*)�+ 0�2 Y 2t0 $�pz�R)�p*O?$t+ 2 8?) 2 |}$�&ST 2 8F+F� 2 YZ+��Up � Y 2 O?+3T{$*8 2 YZ+��Up � r�sz$ 0 $�T{x�7"+�O 2 Y 2t»�2 & � � » +�7

p�+�6�)�7"$98 2 !� 2 T{$*8 0 $9pU8 � YZ+�& 2 &()�TÂ$�)ZOS&?$B+ ¿ +3T{x�7"$U!(£ »�2 & � � » +�7±p�+�6�)�7"$98 2 O?+3T{x�+3& 2 OS)�& 2 :

TE x�$ZYZ+�8?+3&IYZ+���p � Y 2

0 $9T{$5$ 0 $�p��R)�p*OS${Y�+JO?+3&ST{$98T = {gelado, frio, normal, morno, quente}

+ 0�2 Y 2 )�T YZ+�8FO?+y8 OS+3&ST5$*8��9'9)U+(¤w)�T 0 $�pz�R)�p*O?$-p�+�6�)�7"$98?$ ��OS+3T 2 8?) 2 |})�pUr�sz$-YZ+ � p 0 7=)U8Ssz$HYZ+��Up � Y 2-0 $�T{$� pUY �1032 Y�$5p 25> � /�)�& 2 �U!" �!

geladofrio

normal

morno

quente

0

1

-30 +100

> � /�)U& 2 �U!" I�� 2 & � � » +�7�pU+36�)�7"$98 2 O?+3T{x�+3& 2 OS)�& 2 !

`cb9d%��ef���U� ¶ #l� HmZ��efj���b�° ´¨b*µ%��°l��hym � � > .�O�.�/98BpAO V �! k � Vb2 " . Y 1 _^`� 8B1�.`, V]. e � p�1�/ � ,oV]. �{X, R, U, M}2 � 1 �

u*�

Page 64: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

uz� � ���`� ��� � � �R��� � � ��� � �C ��� � � �� ���

X¤J$�p�$9T{+J8 � Tt6�.�7 �"0 ${Y 2�»�2 & � � » +�7 ��x�$�&w+ ¿ +�T5xU7=$ �

idade;

R¤�$ 0 $9p��R)�p*O?$(Y�+�&?.�O?)�7"$98 � � 8FO?$w¤ $ 0 $9p��R)�p*O?$(Y�+�p�$�T{+y8±Y 2�»�2 & � � » +37

idade�yx�$9&±+ ¿ +�T5xU7=$NI

{novo, velho, meia−idade}

U¤H$�«\p �=» +3&D8F${YZ+ � � 8 0 )U&S8?$58?$�6U&?+J$�'*) 2 7 25»�2 & � � » +37�+�8FOS�{YZ+��Up � Y 2 �Ux�$9&w+ ¿ +3T{x�7"$ ��� 2 OS¤{ yuz���

M8Ssz$ 2 8V&S+3/9& 2 8V8?+3T�½�p9O �"0�2 8�'*)�+ � p�Y �"0�2 T¾${8 � /�p � � 032 YZ${YZ+ 032 Y 2 &?.�O?)�7"$�+�T

R

��� b � "�°l� ¶ #l��� � []$�pU8 � YZ+3&S+t$³+ ¿ +3T{x�7"$cY 2³0 7 2 8?8 � � 032 r�sz$�YZ$*8\6�$�p�8 2 7=)Up�$98�! � 2 T{$98 2 /9$�& 2 | 2 @�+3&I)�T 20 7 2 8S8 � � 0�2 r�sz$�'*)�+H+�pU'*) 2 Y�&?+�O?$ZYZ$*8�$98 2 7")�p�$*83!4 +y8ROS+ 0�2 8?$�OS+3&S+3T{$98�+3p*OSs�$ 2³»�2 & � � » +37�p�+36�)U7=$*8 2X = alunos

!�� 0 $9p��R)�p*O?$cYZ+5O?+3&ST{$98�$�) &?.�O?)�7"$98Y 25»�2 & � � » +37 2 7")�p�$*8�¤ 0 $�T{x�$98FO?${x�+37"$98w8?+3/�) � p*O?+y8VO?+�&?T{$98

R = {pessimos, ruins, medios, bons, excelentes}� «\p �=» +3&D8F$cYZ+ � � 8 0 )�&D8F$c8F$96�&S+�$;'*) 2 7X¤{YZ+��Up � YZ$;+�8FOS�³+3p*O?&S+5�³+c ��³+ 2 8�&S+3/9& 2 8\8?+3T�½zp*O �"0�2 8�x 2 & 2

2 8S8?$ 0 �"2 &]$*8w8 � /�p � � 0�2 YZ$*8���$9) 2 8V|})�pUr3�*+y8wYZ+ � p 0 7")U8Ssz$5+�8FOSsz$ � 7=)�8ROS& 2 Y 2 8Vp 2�> � /�)�& 2 ��! K�!

1 2 3 4 5 6 7 8 9 10

1

0

NotasBons alunos

bons

excelentes

regulares

ruins

pessimos

> � /�)�& 2 �U!LK I�� 2 & � � » +37�p�+�6�)�7"$98 2�0 $�&S&?+y8Fx�$�p�YZ+3p*O?+ 2{0 7 2 8S8 � � 0�2 r3s�$�YZ+H)UT 2 7")�p�${+�TÀ|})�pUr�sz$�YZ+H8?) 2 p�$zO 2 !

� ��� ��� � ��� % ! ) '*2�'��*!�� �-��� � � !3���*� !��� 2 T5+y8FT 2 T 2 pU+ � & 2 '*)�+¨$*8 0 $9p��R)�p*O?$*8{p�+36�)U7=$*8F$*8�x�$*8?8?)�+�T�)�T 0 $9p��R)�p*O?$ YZ+¨xU&?$9x�& � +�Y 2 YZ+y8 � T{x�$�&?�O 2 p*O?+�8 � 2 8 »�2 & � � » + � 8Jp�+�6�)�7"$98 2 8JO 2 T�6�¤�T OS¼3T 2 7"/�)�T 2 8Hx�&S$�x�& � +yY 2 Y�+�8 � T5x�$�&?O 2 p*O?+�8�'*)�+�x�&S+ 0 � 8 2 T 8F+�&2 xU&?+y8F+�p9O 2 Y 2 83!`cb9d%��ef���U� ¶ # _²���� "�°lb*������b��

� > .dO�.�/98BpAO V �o k � V]2 " . Y H�2 > ,oV :&. "# ,�.�/c.NH0.�1�.x ∈ X

3�8 " : E�?> H�2 ��)9�?� :&2 �o k � V]2 " 27:f.AV e �� �µ(x) > 0��6 » �"2 T5+�p*O?+�� '*) 2 pUYZ$Á)�T 2 »�2 & � � » +37�p�+�6�)�7"$98 2 pUs�$Á¤ 0 $�T{x�7"+�O 2 �%¡U� +3p*O?& 2 Y 2 8Hx 2 & 2 2 85'9) 2 � 8tpUs�$

¡ 2�» +3&D� � p*OS+3&Sx�&?+3O 2 r3sz$B7 � p�/ K��"8FO �1032 ¡U�cx 2 &?O � &JYZ$ 0 $�p��R)Up9OS$cY�+�OS+3&ST{$98�+5x�$�&?O 2 p*O?$ 2 8 2 �1Y 2 Y�+5'*) 2 7"'*)�+�&8 � 8FO?+�T 2 '*)�+H¤J6 2 8F+ 2 YZ$5p�+�8FO?+y8VO?+�&?T{$98w8?+3&D� � pUY�+��Up � Y 2 !£ > � /9)�& 2 ��! P5T{$98FO?& 2 )UT 2{»�2 & � � » +�7�p�+�6�)�7"$98 2{0 $�T{x�7"+�O 2 +I)UT 2 p�sz$ 0 $9T5xU7=+3O 2 !

`cb9d%��ef���U� ¶ #1® ! mZ����ef�9��� ��m �c��ef��m���b���% O�.�/98BpAO V �! k � V]2 " . � 2�/ > . �?> . �> ,�.�/9:I8(aGM�2C1�. �?� 8B1�.�1 ,.�V K��?> . " O ��U 9" H�jR. > .�1�.C1 ,�.�/9:I8(aUM�2 �o k � V]2 " .0/ "� ,R.A/c.PH�.�1�. #� :_/c.�1�.

x

i=1

µAi(x) ≡ 1

2 � 1 pY 2 ���?>d /c271 H�2 � )9�?� :&2 "J�! k � V]2 " 2 " .Ne �� 2lO�.�V]2A/

x, /9: �� H [

Page 65: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�R�b��� � ��� � � � �C ��� � � �� ��� � � � � ��� � ��� u��

x

(x)µ

x

(x)µ

Variavel Completa Variavel Incompleta

> � /�)�& 2 �U! P I�� 2 & � � » +37�p�+�6�)�7"$98 250 $�T{x�7"+�O 2 +JpUs�$ 0 $�T{x�7"+�O 2 !

ºX8FO 2 ¤H)�T 2�0 $9pUY � r3s�$�T 2 � 8V|}$�&?O?+-'*)�+ � 2 > ,!V : � 1 + 0 $9pUY � r3sz$�8?)�� 0 � +�p9OS+Jx 2 & 2 []$�T{x�7"+�O?)�YZ+J¤-'9)U+25»�2 & � � » +37�p�+36U)�7=$*8 2 |}$�&ST{+H)�T 2 x 2 &?O � r3s�$5Y 2 )�p � Y 2 YZ+�!£ x 2 &?O � r3sz$�Y 2 )�p � Y 2 YZ+t+y8ROD� 2 8S8F$ 0 �"2 Y 2�0 $9T 2 8F$96�&?+�x�$*8 � r3s�$�YZ$98 0 $9p��R)�p*O?$*8(p�+�6�)�7"$98?$98�!\£ > � /�)�& 2

�U! � � 7=)U8FO?& 2�0 $9T{${Y�$ � 8 0 $9p��R)�p*O?$*8(8F+�8F$96�&?+�x���+�Tc!���6U8F+�& »�2 &('9)U+-p�$�x�$�p*O?$�YZ+tT{� ¿ � T 2 8F$96�&S+3x�$98 � r�sz$5$»�2 7"$�&wY 2 8?$�T 2 YZ$98 »�2 7"$�&S+�8�Y 2 8V|})�pUr3�*+y8(YZ+ � p 0 7")U8Ssz$5¤H)�T;!4 s�$¨¡U�Á)UT 2 7=/9$�& � OST{$cxU&?+ 0 � 8F$Áx 2 & 2 YZ+�OS+3&ST � p 2 &t$ÁT�� ¿ � T{$Á$�) T �"p � T{$ /�& 2 ) YZ+³8?$�6U&?+�x�$*8 � r3s�$ �p�$¨+�p9O 2 p*O?$¨+y8RO 2 8?$�6�&S+3x�$98 � r�sz$;YZ+ » +�&?+��U+�O � &t$ÁT 2 � 8 0 $9&?&S+�O 2 T{+3p*O?+{x�$98S8 � » +�7 2 8?+3T�½zp*O �1032 Y 2;»�2 & � � » +�72 8S8?$ 0 �"2 Y 2 !ºX8FO 2 8F$96�&S+3x�$98 � r�sz$tp�sz$5¤�)�T 2 � T5x�$98 � r3sz${Y 2 O?+�$�& �"2 Y 2 7=.9/ �1032 pU+36�)�7"$98 2 ��T 2 8wY�+ » +I&S+��U+3O � & 2 &S+ 2 7 � �

Y 2 YZ+HYZ+ 032 Y 2 )�T¾YZ$*8 0 $9p��R)�p*O?$*8�+3p » $97 » � YZ$98�!£¦+ ¿ x�+3& � ¼3p 0 �"2 T{$*8ROS& 2 '*)�+ 2 8F$96�&S+3x�$98 � r�sz${+3p*O?&S+-$*8 0 $9p��R)�p*O?$*8 »�2 & �12 +3p*O?&S+�K�u��À+5uz���WY 2 8(6 2 8F+y8

YZ$ 0 $9p��R)�p*O?$�! 4\2{> � /9)�& 2 ��! �5$�/�& 2 )cYZ+-8?$�6�&S+3x�$98 � r�sz$ »�2 7=+-u����B!

0

0.75

0.5

0.25

1.0

x y(x+y)/2

> � /9)�& 2 ��! �NI QZ$�6�&S+3x�$98 � r�sz$5YZ+tuz� � +3T 0 $�p��R)Up9OS$98�p�+�6�)�7"$98?$98�!# 2 & 2 )�TÂ/�& 2 )ÁY�+58?$�6�&S+3x�$98 � r3sz$�YZ+�uz� � ��$³x�$9p9OS$B$9pUYZ+t$ 0 $�&S&?+H$³/�& 2 )¨T�� ¿ � T5$cYZ+ � p 0 7=)U8Ssz$Bx 2 & 2)�T 0 $�pz�R)�p*O?${&?+�x�&S+�8?+3p*O 2 ${x�$9p9OS$�YZ+HT �=p � T{$�/�& 2 )cYZ+ � p 0 7")U8?s�$�x 2 & 2 $*8 0 $�pz�R)�p*O?$98 » � @ � p�¡�$*83!3�) 2 7"'*)�+�& »�2 & � � » +37Vp�+�6�)�7"$98 2 0 $�T{x�7"+�O 2 x�$�Y�+Á8F+�&�OS& 2 pU8F|}$�&ST 2 Y 2 +�T�)�T 2 »�2 & � � » +�7]pU+36�)�7"$98 2 '9)U+

|}$�&ST 2 )UT 2 x 2 &FO � r�sz$�Y 2 )Up � Y 2 YZ+Jp�$9&?T 2 7 � @ 2 p�YZ$z�q8F+ 2 8]|})�pUr3�*+y8�YZ+ � p 0 7=)�8?s�$5YZ+ 290 $�&DYZ$ 0 $�T 2 |}.�&STt)U7 2 I

µAi(x) =

µAi(x)

∑pj=1 µAj

(x)

: ��!= Ex 2 & 25032 Y 2

i = 1, . . . , p!

Page 66: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

��� � ���`� ��� � � �R��� � � ��� � �C ��� � � �� ���

Page 67: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� ��������� � �

������� ��� � � �� �(��� �¾��� � �¢��� � � ������ � �

�"��� �i�G� ��@4! �����4 $*8 032 x�� OS)�7=$*8 2 p*O?+�& � $�&S+�8 » � T{$98 2 8JYZ+��Up � r���+�8I6U�98 �1032 8�YZ+57".�/ �1032 YZ+5p�+36�)U7=$*8 2 � 2 x�&S+3pUY�+3T{$98 2 $9x�+�& 2 &0 $9T 0 $�p��R)Up9OS$98HpU+36�)�7"$98?$98�! � � T5$*8 0 $9T{$¨x�$ZYZ+�T5$*8-+y8 0 $�7"¡�+3&-+�p*O?&S+�Y � |}+�&?+�p*O?+�8tT 2 p�+ � & 2 8HYZ+B+ ¿ + 0 )ZO 2 &2 8V$9x�+�& 2 r���+�8V8F$96�&S+ 0 $9p��R)�p*O?$*83! 4 +�8FO?+ 032 x � OS)�7"$ � &?+�T{$98]T5$*8ROS& 2 & 0 $�T{$5)ZO � 7 � @ 2 &�+�8FO?+�8 0 $�p 0 + � O?$98]6U�98 �10 $*8x 2 & 250 $9pU8ROS&?) � &(8 � 8FO?+�T 2 8V+�8?x�+ 0 �12 7 � 8FO 2 83!

�"��� A ��4+?# *�� ��6�� ��6G*,+.��6 @$# = *,6��7#?> -C6 / #32!$5M��6G��6£À6 2 8F+³YZ$Áx�+3pU8 2 T{+3p*OS$¨p�+36�)U7=$*8F$ 8Ssz$ 2 8H&S+3/�& 2 8H'*)�+B+�8FO 2 6�+37"+ 0 +3T &?+�7 2 r���+�8J+�p*O?&S+BY �"» +3&D8 2 8 »�2 & � � » + � 8p�+�6�)�7"$98 2 8V+H)�T 2 $�);T 2 � 8�&?+�/ � �*+y8V$�) 0 $�pz�R)�p*O?$98�pU+36�)�7"$98?$98�!ºX8FO 2 8�&?+�/�& 2 8V8?s�${Y�${8?+3/9) � p9OS+IO � x�$NI£�: �> , /0.�: � /c.{+�8FOS� e �� #� : !Q�+ 2 O V]24H�8B1�.�1 +y8ROD� >-� 8;:&2P.�V�:f.{+�p*OSsz$ � / .A/ >i� 8;:&2 � 2A/9: [

�I8VO?+�&?T{$*8�+3T¢p�+�/�& � OS$�8?s�$ 2 8 »�2 & � � » + � 8V+J$98 0 $�p��R)Up9OS$98�p�+�6�)�7"$98?$98�!º�TWO?+3&ST{$98�/9+3& 2 � 8 0�2 Y 2 &S+3/�& 2 O?+3T 2 |}$�&ST 2

Se x1 e Ai1 AND · · · AND xn e Ai

n entao y eBj : u�!= E4\2 +�'*) 2 r�sz$ÁuZ!"

Aij

� pUY �1032 $ 0 $9p��R)�p*O?$ÁpU+36�)�7"$98?$iY 2¨»�2 & � � » +37�pU+36�)�7"$98 2

Aj+

Bj ¤B$ 0 $�p��R)�p*OS$p�+�6�)�7"$98?${Y 2�»�2 & � � » +�7�Y�+J8 2 �1Y 2 !«(T~8 � 8ROS+3T 2 p�+�6�)�7"$98?$ ¤c+3p*OSs�$ )�T 0 $�p��R)Up9OS$ Y�+³&S+3/9& 2 8-YZ$ O � x�$�8F+3��+�p*OSsz$ '9)U+³T 2 x�+ �12 +3p*O?& 2 Y 2 8

:}» +37"$ 0 � Y 2 YZ+ E +3TÂ8 2 �"Y 2 8 : |}&S+ 2 & E !XºX8FO 2 8w&S+3/�& 2 8wY�+��Up�+�TW&S+3/ � ��+�8�pU$�+�8?x 2 r3${+3p*O?& 2 Y 2×8 2 �"Y 2¨:

X × YE !

«(T 8 � 8FO?+�T 2 p�+36�)U7=$*8F$F : X → Y

2 x�&?$ ¿ � T 2 2 |})�pUr3s�$f : X → Y

0 $96�& � pUYZ$ 8?+3) /9&S� � 0 $ 0 $9T|}& 2 /�T{+�p9OS$98�$�)t&S+3T{+�pUYZ$98 :�� +3T{x�¡�x 2 O 0 ¡�+y8 ��E '*)�+V8?+�8F$96�&?+�x���+�Tc! [ 2 Y 2 &?+�/�& 2 YZ+3O?+3&ST � p 2 )�T &?+�T5+�pUYZ$�!£ 2 x�&?$ ¿ � T 2 r3s�$IY 2 |})�pUr�sz$HT{+37"¡�$�& 2H2 T{+�Y � Y 2 '9)U+w$98 |}& 2 /9T{+3p*O?$*8 2 )�T{+3p*O 2 Tª+3Tªp DUT5+�&?$H+�Y � T � p�)�+3T+3TWO 2 T 2 p�¡�$ 0 $9T{$�+y8ROD� � pUY �1032 YZ${p 2�> � /9)�& 2�� &?+3|�3| I |}& 2 / � !� 0 $9p��R)�p*O?$5YZ+\&S+3/9& 2 8%O?& 2 O 2 YZ+ 0 $9p��R)�p*O?$*8Xp�+�6�)�7"$98?$98 : '*)�+3p*OS+�� 2 7=O?$ �9&D�zx � YZ$ � 0�2 &S$ �*+�O 0yE ��p�$�+3p*O 2 p9OS$

O 2 p*O?$ 2 8-+3p*OS& 2 Y 2 8�YZ$ 8 � 8FO?+�T 2 0 $9T5$ 8F) 2 8t8 2 �1Y 2 8t8?s�$ »�2 & � � » + � 8J+�8 032 7 2 &S+�8�!¨#%$�&�+ ¿ +3T{x�7"$ �%$ 8 � 8ROS+3T 2&S+ 0 +�6�+ 0 $�T{$ � pZ|}$9&?T 2 r�sz$;)�T 2 OS+3T{x�+�& 2 O?)�& 2 YZ+

35o +�pUsz$ÁO?+3T{x�+3& 2 OS)�& 2 '*)�+�p*O?+��%$9) 2 7=O?)�& 2 YZ+Á 9!L�zP0 T +5pUs�$ 2 7=O?)U& 2 T5¤yY �12 !H£ªT5+y8FT 2³0 $ � 8 2B2�0 $�p*O?+ 0 +-p 2 8 2 �1Y 2 !-� 8 � 8FO?+�T 2 YZ+ 0 $9p9OS&?$97=+�p�+36U)�7=$*8F$cYZ+ » +|}$�&Sp�+ 0 +3&])�T 0 $9T 2 pUYZ$-+y8 0�2 7 2 &�+\pUsz$�)�T &?.�O?)�7"$-p�+�6�)�7"$98?$U!�#�$9&X+ ¿ +�T{x�7=$ � 2-» +37"$ 0 � Y 2 Y�+\YZ$tT{$�O?$�&VYZ+ » +8?+3&w yK�����&?xUT +wpUsz$ » +37"$ 0 � Y 2 YZ+�&D�zx � Y 2 !�#%$�& +�8FO 2 & 2 @ysz$�O?+3T{$*8 '9)U+ � pU8?+3& � & )UT T{.ZYZ)�7"$-'9)U+VpU+36�)�7 � 8 2 $*8»�2 7"$�&S+�8VYZ+I+3p*O?& 2 Y 2 +�$9)ZO?&S$�'*)�+�O?& 2 p�8R|}$9&?T 2-2 8]&S+�8?x�$98FO 2 8Xp�+36U)�7=$*8 2 8X+�T¢&S+�8?x�$*8RO 2 8�+y8 0�2 7 2 &S+�8Xp 2 8 2 �1Y 2 !£ > � /9)�& 2�� &?+3|�3| I 8 � 8FpU+36 � T{$98FO?& 2 )UT¾Y �12 /9& 2 T 2 +3T¾6�7=$ 0 $98wYZ+J)�T�8 � 8ROS+3T 2 p�+36U)�7=$*8F$�!� 2 T{$*8 0 $9pU8 � YZ+�& 2 &�$-+ ¿ +�T{x�7=$5YZ+ 0 $�pU8FO?&S) � &X)�TÀ8 � 8FO?+�T 2 '*)�+ 0 $�p*O?&S$�7"+ 2 OS+3T{x�+�& 2 O?)�& 2 YZ+�)�T 2 T�6 � �

+3p*OS+ 2 )�T{+3p*O 2 pUYZ$5$�)cY � T � p�) � p�YZ$ 2�» +37"$ 0 � Y 2 YZ+JYZ$5T{$zO?$9&wYZ$ 2 x 2 &?+�7=¡�$5YZ+ 2 &?� 0 $�pUY �"0 � $9p 2 YZ$�!�Q�+�/�)�p�YZ$�U

Page 68: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�9K � �*�`� ��� � � � � �� � � � � ��� ��� � ��� � � ��� � � �� ��

X

Y Y

X

(a) Aproximacao menos precisa (b) Aproximacao mais precisa

> � /�)�& 2 uZ!" I�¬I&D��� 0 $98VT{$98FO?& 2 pUYZ$ 0 $9T{$ 2 )�T{+3p*O 2 & 2 xU&?+ 0 � 8?s�$5Y 2{2 x�&?$ ¿ � T 2 r3s�$tp�+�6�)�7"$98 2 !

Bancode Regras

Difusas

Bancode

Semânticas

Avaliadorde Regras

Difusas

Nebulisador

EntradasEscalares

SaídasEscalares

Desnebulisador

> � /9)�& 2 u�! K I � �12 /�& 2 T 2 +3T¢6U7=$ 0 $*8wYZ+H)�T�8 � 8ROS+3T 2 p�+36U)�7=$*8F$�!

2 YZ+��Up � r3s�$�YZ+ »�2 & � � » +37 pU+36�)�7"$98 2 Y 2 Y 2 +3T � &S+�|��Y I »�2 &?pU+36 � $�x�& � T5+ � &S$�x 2 8?8?$�¤ 0 & �12 & 2 8 »�2 & � � » + � 8�p�+��6�)�7"$98 2 8�!�# 2 & 2 � 8ROS$5YZ+ » +�T5$*8]Y�+��Up � & 2 8 »�2 & � � » + � 8�+H8?+3)U8Vp�$9T{+�8�8 � Tt6�.�7 �10 $98�! 4 +�8FO?+J+ ¿ +�T{x�7=$�O?+�&?+�T{$98 2»�2 & � � » +37±pU+36�)�7"$98 2 YZ+�+3p*O?& 2 Y 2

temperatura'9)U+t&S+3x�&S+�8?+3p*O 2�2 O?+�T5x�+3& 2 OS)�& 2�2 T�6 � +�p*O?+�Y�$B7"$ 0�2 7 2 8?+3&

&S+�|}& � /9+3& 2 YZ$5+ 2 YZ+ »�2 & � � » +37±Y�+-8 2 �"Y 2velocidade

'9)U+-&?+�x�&S+�8?+3p*O 252{» +�7=$ 0 � Y 2 YZ+-YZ$�T{$zOS$�&\YZ$ 2 x 2 &S+37"¡�$YZ+ 2 &F� 0 $9pUY �10 � $�p 2 YZ$U!�x�&S. ¿ � T{$tx 2 8S8F$�¤IYZ+��Up � &w$98 0 $9p��R)�p*O?$*8]pU+36�)�7"$98?$98 : &S.zOS)�7"$98 E YZ+ 032 Y 2 )�T 2 YZ+y8RO 2 8VYZ) 2 8 »�2 & � � » + � 8 �

'*)�+�8?s�$�8?)�6Z� 0 $�pz�R)�p*O?$98;YZ$*8c+�8?x 2 r3$98BY�+ +�p*O?& 2 Y 2 + 8 2 �1Y 2 !¾�I8c&S.zOS)�7=$*8³'*)�+�YZ+��Up�+3T $98 0 $9p��R)�p*O?$*8p�+�6�)�7"$98?$98�Y 2 8wY�) 2 8 »�2 & � � » + � 8V8?s�$�$*8w8F+�/�) � p*O?+y8MI

temperatura = {gelado, frio, normal, quente, fervendo}

velocidade = {pare, lenta, media, rapida, disparada}

�I8�«(p �"» +�&S8?$98VYZ+ ��$�T �=p � $�Y 2 8 »�2 & � � » + � 8XO?+�T5x�+3& 2 OS)�& 2 + » +�7=$ 0 � Y 2 YZ+�+y8RODsz$5T{$98FO?& 2 YZ$*8Xp 2 8 > � /�)U& 2 8uZ! P5+-uZ! �U!£ 8?+3/�) � &�YZ+��Up � T{$98 2 8?+3T�½�p9O �"0�2 YZ+ 032 Y 2 )�T�Y�$98 0 $�p��R)�p*OS$98�p�+�6�)�7"$98?$98 � 2 O?& 2�» ¤y85Y�+¨+�'*) 2 r3��+�8 �

/�&D��� 0 $98]$�);+�p*)UT5+�& 2 r3s�$B!�£\8 > � /�)U& 2 8�u�! P5+-uZ! �{T{$98FO?& 2 T 2 8VY�+��Up � r���+y8(YZ+�8FO?+y8 0 $9p��R)�p*O?$*83!£(/9$�& 2 YZ+���p � T{$*8 2 8�&S+3/9& 2 8�p�+�6�)�7"$98 2 8�! 4 +�8FO?+ 0 $�p��R)Up9OS$�YZ+I&S+3/�& 2 8 032 Y 2t0 $9p��R)�p*O?$5YZ+(OS+3T{x�+3& 2 O?)U& 2 8

¤ 2 8S8?$ 0 �"2 YZ$ 0 $�T¢)UT 0 $9p��R)�p*O?$�YZ+ » +37"$ 0 � Y 2 YZ+y83!

Page 69: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� � �R� � � � � ��� � ���� � �� ��� �� ��� � � ���C ��� � � �� �� �9P

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40Temperatura

gelado

frio

normal

quente

fervendo

> � /9)�& 2 u�! PJI�[]$9p��R)�p*O?$*8�p�+36U)�7=$*8F$*8�'9)U+-YZ+��UpU+3T 2 OS+3T{x�+�& 2 O?)�& 2 !

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100

Velocidade

parada

lenta

normal

r’apida

disparada

> � /�)�& 2 uZ! �JIX[]$�p��R)Up9OS$98�p�+�6�)�7"$98?$98�'*)�+-YZ+��Up�+3T 25» +37"$ 0 � Y 2 Y�+�!

Page 70: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�z� � �*�`� ��� � � � � �� � � � � ��� ��� � ��� � � ��� � � �� ��

Page 71: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� ��������� � �

������� ��� � � � � ��� ���^ � ��� � ���w��� � � � � �

Q�+³$Á¡�$�T{+�T x�+3p�8 2 +3T 0 & �"2 & 0 $9T{x�)ZO 2 YZ$�&S+�8�'*)�+�$ 8?)�6U8FO � O?) 2 T YZ+ » +�$�7"¡ 2 &tx 2 & 2 8 � T{+y8FT{$U!���8F+�&¡�)�T 2 p�${x�&S$ 0 +y8?8 2 � pZ|}$9&?T 2 r3��+�8�YZ+HT 2 p�+ � & 2 ¡ �"6�& � Y 2 ![]$�T{x�)�O 2 Y�$�&S+�8 2 OS) 2 � 8 �V'*)�+Á)�O � 7 � @ 2 T�7".�/ �1032 6�$�$�7"+ 2 p 2 ��pUsz$�$�x 2 8?8 2 T YZ+¨T���'*) � p 2 8�Y�+ 032 7 0 )U7 2 &

/�7"$�& � � 032 Y 2 8 � � p 0�2 x 2 @3+�8�YZ+³8?+ 2 Y 2 x�O 2 & � YZ+B& 2�0 � $ 0 � p 2 & 0 $9T Y 2 YZ$*8 � T{x�&?+ 0 � 8F$*8-+cYZ+ 2 x�&S+3pUYZ+�& 0 $�T 2+ ¿ x�+3& � ¼�p 0 �12 !� 8F+�&(¡�)�T 2 p�$�)ZO � 7 � @ 2�» �z& �12 8�|}$9&?T 2 8wY�+-x�&S$ 0 +�8S8 2 T{+3p*O?$ �Z&?+ 0 +36�+3T � pZ|}$�&ST 2 r���+y8wYZ+H|}$�&ST 2 OS$98(Y � |}+��

&S+3p*O?+y8�+JY�+ » ��& � $98]O � x�$98�!�I8w8 � 8FO?+�T 2 8wYZ+ 0 $9T{x�)ZO 2 r3s�${8?$�T{+�p9OS+Jx�$ZYZ+�&Ss�$ 2 7=T{+F� 2 & 2 O � pU/ � & 2 7"/�)�T 2 |}$�&ST 2 Y�+Hx�&?$ 0 +�8S8 2 T5+�p*O?$

T 2 � 8 � p*OS+37 � /9+3p*O?+ 032 8F$ � p 0 $9&?x�$�&S+3TÀ+y8RO 2 8 0�2 x 290 � Y 2 YZ+y83!�I8t8 � 8ROS+3T 2 8 � p*O?+�7 � /�+�p*O?+�8�x�&S$ 0 )�& 2 T � p 0 $�&Sx�$�& 2 &-T 2 � 8 � p*O?+�7 � /�¼�p 0 �12 2 $98 0 $�T{x�)ZO 2 YZ$9&?+y8H)ZO � 7 � @ 2 pUY�$

|}$�&ST 2 8�YZ+Hx�&S$ 0 +y8?8 2 T{+3p*OS$5'*)�+J+3T�)�7 2 T 2 7=/9)�T 2 8wYZ+y8RO 2 8 032 & 290 OS+3& �"8FO �1032 83!#%$�&H+ ¿ +�T{x�7=$ �±$98t8 � 8ROS+3T 2 p�+36U)�7=$*8F$*8Jx�&S$ 0 +�8S8 2 T � p�|}$�&ST 2 r���+�8 »�2 / 2 8I$9) T{+y8FT{$¨YZ+���p � Y 2 8 � p 0 $9T5�

x�7"+�O 2 T{+3p*O?+9!¦º�7"+�8B8Ssz$ � pU8?x � & 2 Y 2 8�p 2 |}$�&ST 2 0 $9T5$ $*8B8?+3&S+�8�¡�)�T 2 p�$98{| 2 @3+�T 2 �U&?T 2 r���+�8 »�2 / 2 8�Y�$O � x�$�¦Y �12 +�8FOS�5Tt) � O?$�'*)�+�p*O?+J¡�$z�R+ � 2 )�T{+�p9OS+ 2 |}$�&Dr 2 YZ$ 2 & 0 $9pUY �10 � $�p 2 YZ$U!

vw+yYZ+�8Xp�+3)�& 2 � 8X8Ssz$ � pU8?x � & 2 Y 2 8�p�$H|})�p 0 � $9p 2 T{+�p9OS$tY 2 8 0 ¤�7=)U7 2 8�p�+�& » $98 2 8�YZ$ 0 ¤3&S+36U&?$H¡�)�T 2 pU$U!�º�7 2 8�OS+3T2 � p 0 & � » +37 032 x 2�0 � Y 2 YZ+�YZ+ 2 x�&S+3pUY�+3& 0 $9T 2 + ¿ x�+�& � ¼3p 0 �12 +w8Ssz$I6�$ 2 8%+3T &?+ 0 $�pU¡�+ 0 +3&%+ 0 7 2 8S8 � � 032 &�x 2 YZ&S��+�8�!�I8 2 7"/�$�& � O?T{$98X/�+3pU¤�O �10 $*8X8Ssz$�6 2 8?+ 2 YZ$98Xp 2 7"+ � Y 2 8F+�7=+yr3s�$tp 2 O?)�& 2 7 �*Y�+I8?$�6�&S+ » �"» ¼�p 0 �12 YZ$�T 2 � 8 2 xZO?$�!

4 s�$-¤ 0 $�&S&?+3O?$ � 0 $9T{$tTt) � OS$98�+�p�)�p 0 �12 T +y8RO 2 7"+ � � 2 �U&?T 2 &]'*)�+�8?$�6U&?+ » �"» +�T $*8�T 2 � 8�|}$9&FOS+�8�!%QZ$�6�&S+ » �"» +3T$98I'*)�+�8Ssz$BT 2 � 8 2 Y 2 xZO 2 YZ$*8 2 8?)�x�+3& 2 &�Y�+�O?+�&?T � p 2 Y 2 8IY � � 0 )�71Y 2 Y�+�8 ��+5+�8FO?+y8\p�+�T 8F+�T5xU&?+58?s�$�$98�T 2 � 8|}$�&?O?+y83!%£(7"/�$9& � O?T{$*8 /�+�p�¤�O �"0 $98�xU&?$ 0 )U& 2 T O?+y8RO 2 &XY �=» +3&D8 2 8�8F$97=)�r�+�8 ��xU&?$ 0 )U& 2 pUYZ$H+3p 0 $�p*OS& 2 & 2 '*)�+\T{+37"¡�$9&8?+ 2 Y 2 x�O?+ 2 ${x�&S$�6�7"+3T 2 �Z+ ¿�2 O 2 T{+3p*O?+ 0 $9T{$5$ 0 $�&S&?+Ip 2 p 2 OS)�&S+3@ 2 ![]$�T{$�x�$ZYZ+�8?+3& » � 8FO?$ � 032 Y 2 )�T 2 Y 2 8wOS¤ 0 p �"0�2 8(O?+3T 0�2 & 2�0 OS+3& �"8FO �1032 8�x 2 &?O �10 )U7 2 &?+y8w+t8Ssz$ 2 Y 2 xZO 2 Y 2 8

2 Y�+�O?+�&?T � p 2 Y 2B0 7 2 8S8?+�YZ+5x�&S$�6�7"+3T 2 8�!IºX8RO 2 8\O?¤ 0 p �1032 8(OS+3TÂO 2 Tt6�¤3TÂOS+3T YZ+y8 »�2 p*O 2 /�+�pU8I'9)U+ 2 8\| 2 @�+3T� p 032 x 2 @�+�8�YZ+H+3p 0 $�p*OS& 2 &�8?$�7")Ur3�*+y8Vx 2 & 2 Y�+�O?+�&?T � p 2 Y�$98�x�&S$�6�7"+3T 2 83!�I8�x�+y8?'*) � 8 2 YZ$�&S+�8\O?¼�T » $97 O 2 YZ$c8F) 2c2 OS+3pUr�sz$³x 2 & 2 8 � 8FO?+�T 2 8�¡ �=6U& � YZ$98I$9pUYZ+5+�8FO 2 8\O?¤ 0 p �1032 8 0 $�T�6 � �

p 2 Y 2 8 0 $�T{$t|}$9&?T 2 YZ+-8?)�x�+3& 2 &wY � � 0 )�71Y 2 YZ+�8 � pUY �"» � YZ) 2 � 8(YZ+ 032 Y 2 O?¤ 0 p �1032 !� ��� A -�� -�+`� #3� �,6r�7*,+.-�6B@$# = *,6��7#?> -C6 �i�7# 5,*M)�#�i�7# 64 $³7 �"» &?$ : J%)%) 9 � � : �$� '?H5'�� ���� � � �� ��9�9u E Q � 8FO?+�T 2 8\oqp*OS+37 � /9+3p*O?+y8\8Ssz$ 2 p 2 7 � 8 2 YZ$*8(YZ+ 2�0 $9&SY�$ 0 $�T'*) 2 OS&?$ 032 & 290 O?+�&&�18FO �1032 8\x�& � p 0 � x 2 � 8MI 2 '*) � 8 � r3s�$³YZ+ 0 $9p�¡�+ 0 � T5+�p*O?$98 � 032 x 2�0 � Y 2 YZ+{YZ+5+ ¿ x�7 �"0�2 r3s�$ ��6 2 � ¿ $c+2 7=O?${p � » +�7±YZ+H& 2�0 � $ 0 �"p � $�+J|}& 2 / � 7 � Y 2 YZ+9!����� ��� ���,-!��"! � ��' �-� &('*)�-��# !�% ��)/.0'*�£wY�'*) � & � & 0 $�pU¡�+ 0 � T{+3p*OS$98V¤�)UT 2 x 2 &FOS+ � T{x�$9&FO 2 p*O?+Ip�$5x�&?$ 0 +�8S8F$tY�+IYZ+y8F+�p » $�7 » � T{+�p9OS$�YZ+J'*) 2 71'*)�+3&wQ � 8R�O?+�T 2 oqp*OS+37 � /9+3p*O?+5+�+�p » $�7 » +�+ ¿ O?& 2 � & � � p*O?+�&?xU&?+3O 2 &I+{&?+�x�&S+�8?+3p*O 2 &\$ 0 $�pU¡�+ 0 � T{+3p*OS$cYZ+{)�T¹YZ+�OS+3&ST � p 2 Y 2

�*u

Page 72: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

��� � ���`� ��� � � �X� �� ��� � � ��� ��� � � � �� � � � �� �� � � ��z&S+ 2 !wºX8FO 2�2 '9) � 8 � r�sz$�YZ+ 0 $9p�¡�+ 0 � T{+�p9OS$98\+ ¿ � /9+JO?+�T5x�$B+�¤tx�$zOS+3p 0 �"2 7=T{+3p*OS+-pUsz$ 0 $�p ��� » +�7l!\�I8(+y8Fx�+ 0 � �2 7 � 8RO 2 8 2�0 ¡ 2 T Y � | � 0 � 7�O?& 2 YZ)�@ � &J+�T x 2 7 2�» & 2 8IYZ+�OS+3&ST � p 2 YZ$98�O � x�$98HYZ+ 0 $�p�¡�+ 0 � T{+3p*O?$*8 2 7=¤�T pUs�$;8?+3&S+3T2 ) ¿ � 7 �"2 &?+y8 0 $�$�x�+3& 2 O �"» $*83!#%$�&�+ ¿ +�T5xU7=$ ��¤HY � | � 0 � 7±x 2 & 2 )�T¾+�8?x�+ 0 �12 7 � 8FO 2 + ¿ x�7 �1032 & 0 $�T{$5+�8FO 2�0 � $9p 2 &�)�T 032 T � p�¡Us�$ 2 &?O �10 )�7 2 YZ$�!£(7"¤3T�YZ+�8S8?+�8]| 2 OS$�&S+�8 �Z+�8?x�+ 0 �12 7 � 8FO 2 8]x�$ZYZ+3T¾pUsz$�Y � 8?x�$9&(YZ+JO?$ZYZ${$ 0 $9p�¡�+ 0 � T{+�p9OS$U!# 2 & 2 &?+y8F$97 » +3& +y8ROS+�8]x�&?$96�7"+3T 2 8 )UT 8 � 8ROS+3T 2 +�8?x�+ 0 �12 7 � 8RO 2 YZ+ » +()U8 2 &]&?+yYZ+�8Xp�+3)�& 2 � 8 �9'*)�+�8Ssz$ 0�2 x 2 @3+y8

YZ+ 2 x�&?+�pUYZ+�& 2 x 2 &FO � &(Y�+JY 2 YZ$*8w'9)U+H+ ¿ � 8FO?+�TW8?$�6�&S+�$�xU&?$96�7=+�T 2 !����� �� � � � �*!�2 ! � ���-�Q � 8ROS+3T 2 8c+y8Fx�+ 0 �"2 7 � 8RO 2 8BO?¼�T 8 � YZ$ 2 x�7 �"0�2 YZ$*8 0 $�T &S+37 2 O �"» $ 8F) 0 +�8S8?$ +3T 2 7=/9)�pU8 032 8F$*83! 4 $ +3p*O 2 p*O?$ �Tt) � OS$98�YZ+y8ROS+�8 0�2 8?$985OS+3T )�T�YZ$9T �"p � $ &?+y8ROS& � O?$�+¨$*8�8 � 8FO?+3T 2 8�8?s�$ � p 0�2 x 2 @3+y8�YZ+Á8F+ 2 Y 2 x�O 2 & 0�2 8?$$ 0 $9&?& 2t2 7=/9)�T 2 T{$ZY � � 032 r3s�$�p�$ 2 T�6 � +3p*O?+I$�pUY�+I+�7=+JYZ+ » + 2 OS) 2 &y!�M\$�7"7 2 pUY :G�%)6���$��9 � �� y���9� E�0 )�p�¡�$9)$wOS+3&ST5$ > & 2 / � 7 � Y 2 YZ+ : ��/98;:_:IV ��! #"�"yE x 2 & 2 +y8ROS+ |}+3pM�9T{+3p�$�! > & 2 / � 7 � Y 2 YZ+%x�$ZYZ+]8?+3& » � 8ROS$ 0 $9T{$ 2 � p 2 6 � 7 � Y 2 Y�+YZ$*8w8 � 8ROS+3T 2 8wY�+�OS& 2 6 2 7"¡ 2 & 0 $�T 0 $�p�¡U+ 0 � T{+3p*O?$ � p�+ ¿�2 O?$ � � p 0 $�T{x�7"+�O?${+ � p 0 $9pU8 � 8FO?+�p9OS+�!º�T 8 � 8FO?+�T 2 8�+�8?x�+ 0 �12 7 � 8FO 2 8 0 $�p�¡�+ 0 � T{+3p*O?$³¤t&S+3x�&S+�8?+3p*O 2 YZ$�x�$�&J8��=T�6�$97=$*8�Y � 8 0 &S+�O?$*8(+�O?& 2 O 2 YZ$³x�$9&

T{+ � $�YZ+I+ ¿ x�&S+�8S8?�*+y8�7".�/ �"0�2 8�6�$�$�7"+ 2 p 2 8�!�º]8ROS+\O � x�$5Y�+(O?& 2 O 2 T{+3p*OS$t¤ � p 2 YZ+�'*) 2 YZ$tx 2 & 2 &?+y8F$97 » +3&�x�&S$�6U7=+3�T 2 8wY 25» � Y 2 &?+ 2 7 �Z$�pUYZ+J$ 0 $�p�¡�+ 0 � T{+3p*O?$5¤Hp�+36U)�7=$*8F$�!,±.9/ �1032 p�+36�)U7=$*8 2 O?& 2 O 2 +y8ROS+]x�&S$�6U7=+�T 2 � p 0 $9&?x�$�& 2 pUY�$ 2 p�+36�)U7=$*8 � Y 2 Y�+]p 2 8?) 2 7=.9/ �1032 ! 4 sz$I¡U�I7 � T � O?+�8

& � / �"YZ$*8(+�p9OS&?+ 2 8IYZ+���p � r3��+�8IYZ$98 0 $9p 0 + � OS$98(+ 2 8�$�x�+3& 2 r3�*+y8(7".�/ �"0�2 8(8Ssz$³&?+yYZ+��Up � Y 2 8\YZ+5T 2 p�+ � & 2 '*)�+�¤x�$98S8 � » +37 & 2�0 � $ 0 � p 2 &�)ZO � 7 � @ 2 p�YZ$ 2 �U&ST 2 r���+y8V'*)�+H8Ssz$ »�2 / 2 8V+ � T 0 $9T{x�7=+3O 2 83!vw+yYZ+�85p�+�)�& 2 � 8{&S+�8?$�7 » +�T­$ xU&?$96�7=+�T 2 YZ$ > & 2 / � 7 � Y 2 YZ+;Y � 8ROS& � 6�) � pUYZ$�$ 0 $9p�¡�+ 0 � T5+�p*O?$ x�$9&{O?$ZY 2 2

&S+�YZ+9! 4 +3p�¡�)�T 2 x 2 &?O?+B¤c&S+�8?x�$9pU8?� » +37�x�$�&�O?$9T 2 &t)UT 2 2 Y�+ 0 � 8Ssz$ �%T 2 8�$ 0 $�p��R)Up9OS$ Y 2 8 0 ¤37")�7 2 85'9)U+0 $9T{x���+3T 2 &S+�Y�+�O?$9T 2Á2 YZ+ 0 � 8?s�$U!¨ºX8FO 2 Y � 8ROS& � 6�) � r3s�$ YZ$ 0 $�p�¡U+ 0 � T{+3p*O?$ x�+�&?T � O?+³'*)�+ 2 &S+�YZ+Bx�$*8?8 2O?$9T 2 &�Y�+ 0 � 8?�*+y8�8?$�6U&?+ � pZ|}$9&?T 2�0 $�+�8 � p 0 $�T{x�7"+�O 2 8t6�+3T 0 $�T{$ 0 $�p*O � p�) 2 & 2 |})�p 0 � $9p 2 &5T{+�8?T{$ '*) 2 p�YZ$x 2 &FOS+�8wYZ+H8?) 2 +�8FO?&S)ZO?)�& 2 YZ+ �=¿�2 TWY�+H$�x�+3& 2 &y!4 $ 032 8F$\Y�+ 2 7"/�$9& � O?T{$*8 /�+�p�¤�O �"0 $98 ��M\$�7"7 2 pUY-YZ+3|}+3pUY�+V'*)�+]+37"+�8�x�$ZYZ+�T 8F)�x�+3& 2 &�$(x�&S$�6�7"+3T 2 T 2 p*O?+3p�YZ$

)�T 2 x�$�x�)U7 2 r3sz$wYZ+]8F$97=) �z0 $�+�8 �3$9pUYZ+ 0�2 Y 2 )UT 2 Y�+�8 0 &S+ » + )UT 2 x 2 &?O?+�YZ$wxU&?$96�7=+�T 2 !�£ +y8 0 $�7"¡ 2 Y 2 8?$�7")Ur�sz$YZ+�x�+�pUYZ+-Y 2 ¡ � 8ROS.�& �"2 YZ${x�&S$�6U7=+�T 2 !����� � ��� !���' � 12 .0' ��� �*��2 � ��� ��# !�'�#���)-!�'Q � 8ROS+3T 2 8]+�8?x�+ 0 �12 7 � 8RO 2 8�8Ssz$ 0�2 x 2 @3+y8XYZ+\|}$�&Sp�+ 0 +3&XT{$ZYZ+�7=$*8X'*)�+(OS& 2 O 2 T O 2 &S+�| 2 8X'*)�+(+�p » $�7 » +�T 2 7 OS$-p � » +37YZ+H& 2�0 � $ 0 �"p � $U!�ºXT 0 $�p*O?& 2 8FO?+J&S+�YZ+y8�p�+3)�& 2 � 8w8Ssz$ 032 x 2 @�+�8wYZ+-+ ¿ + 0 )�O 2 &�O 2 &S+�| 2 8VO �=x �"0�2 8wYZ+-6 2 �=¿ ${p � » +37YZ+H& 2�0 � $ 0 �"p � $tO 2 � 8 0 $9T5$ 0 7 2 8?8 � � 032 r�sz$5YZ+Jx 2 YZ&S��+�8�!����� ��� &(� ��� # ! � ���-� � ������� 2�! #���# ��'£ 032 x 2�0 � Y 2 YZ+JYZ+I|}$�&Sp�+ 0 +3&�+ ¿ x�7 �"0�2 r3�*+y8�8F$96�&?+ 0 $9T5$ 2 &?+y8Fx�$98FO 2 |}$�&Sp�+ 0 � Y 2 |}$ � $�6ZO � Y 2 ¤ � T5x�$�&?O 2 p*O?+J+3TTt) � OS$98�x�&S$�6�7"+3T 2 83!£(7"/�)UpU8X+y8RO 2 YZ$*8 2 T5+�& �1032 p�$98X/ 2 & 2 p9OS+3T 2 $ 0 7 � +3p*OS+�$5Y � &S+ � O?$5YZ+J8 2 6�+�&]x�$�&D'*)�+(OS+ » +I8F) 2 x�&S$�x�$98FO 2 Y�+

8?+3/�)U&?$(&S+ 0 )U8 2 Y 2 ! º�T 2 7"/�)�p�8�x 2 �"8?+�8�/ 2 & 2 p9OS+3T '*)�+�OS$�T 2 YZ$9&?+y8±YZ+V+3T{x�&S¤�8FO � T{$98�Y�+ » +�T¦8F+�& � pZ|}$�&ST 2 YZ$*88?$�6�&S+�$98�YZ+�O 2 7"¡�+�8�Y 2 8�YZ+ 0 � 8F��+�8�&?+�7 2 O �"»�2 8 2 $98�8?+3)�8�x�+�Y � YZ$*83!%Q � 8FO?+�T 2 8 � p*O?+�7 � /�+�p9OS+�8 2 x�7 �"0�2 YZ$*8 O T{+yY �"0 � p 2YZ+ » +3T¢|}$�&Sp�+ 0 +3&w/ 2 & 2 p*O �12 8 2 $98�8?+3)U8�)�8F)U��& � $98�8?$�6�&S+J$98VT{¤�OS$�Y�$98(YZ+HYZ+ 0 � 8?s�${+3T{x�&S+3/ 2 YZ$98�!4 +y8ROS+ 032 8F$ ��8 � 8ROS+3T 2 8 � T{x�7"+3T{+3p*O 2 YZ$*8 0 $9TW&S+�YZ+y8�p�+3)�& 2 � 8�pUs�$�x�$ZYZ+�& �12 T�8F+H+�T5xU&?+�/ 2 Y�$98 ���F��'9)U+H¤Y � | � 0 � 7±+ ¿ O?& 2 � &V&S+3/9& 2 8 2 x 2 &?O � &(YZ+-8F) 2 +y8ROS&?)�O?)�& 2 !�£\7=/9)�pU8wx�+�8S'*) � 8 2 Y�$�&S+�8� / 2 71�����+y8RODsz$5O?+�p*O 2 p�YZ$ 0 & �"2 &

T{¤�OS$�Y�$98w'*)�+Hx�+3&ST � O 2 T¢+ ¿ OS& 2 � &w&S+3/9& 2 8 2 x 2 &FO � &(YZ$*8�x�+y8F$*8wY 2 8�+y8ROS&?)�O?)�& 2 8 � p9OS+3&Sp 2 8�!��)ZOS& 2 & 2 @�s�$J¤\'*)�+(x�$�Y�+\8F+�& � T5x�$�&?O 2 p*O?+\'*)�+($*8 x�&S$z�R+3O � 8RO 2 8 +�p9OS+3pUY 2 T 8?+($H& 2�0 � $ 0 �"p � $t+3T{x�&S+3/ 2 YZ$

+�8FOS� 0 $�&S&?+3O?$U!º�TW8 � 8ROS+3T 2 8�p�+�6�)�7"$98?$98VYZ+ 0 � 8?��+�8w8Ssz$tO?$�T 2 Y 2 8 2 x 2 &?O � &wYZ+J&?+�/�& 2 8V8 � T{x�7"+�8wYZ$5O � x�$5o > �q�wM(º 4 +J$*8

)U8?)U�z& � $98Vx�$ZYZ+�T¢| 290 � 7=T{+3p*OS+ » +3& � � 032 &�'*) 2 � 8wY�+37 2 8V|}$�& 2 T 2 O �"» � Y 2 8�+ 0 $9T¾'*)�+Jx�+y8F$�+�7 2 8 0 $9p9OS& � 6�) �=& 2 Tx 2 & 252 8F$97=)¨r3s�$ ��p 2 7�!£ O 2 6�+37 2 �U!= �T{$98FO?& 2 Y 2 +�T :�J%)%) 96� � : ��� '?H�'��6 � � �8� �� ����*u E � 7")U8ROS& 252 8]x�&S$�x�& � +�Y 2 YZ+�8 0 $�T{x�)��

O 290 � $�p 2 � 8�YZ+ 0�2 Y 2 )�T�YZ$98w8 � 8ROS+3T 2 8 2 xU&?+y8F+�p9O 2 YZ$*83!

Page 73: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

�X�b��� � � � � � � � �%� ��� � ��� ��� �� ��� � � ��� � � � �� �� � � � ���

�3,����#.��4. �!2 ( ������2 2��� +. � �8(��!2�42 *#(!*/, ���76 .�9����!,�� � (�2���.K9��� ,�� ������42� (�� . ,����/6 .!0�� 6 2���( *�,�� (���24.������2 . *�,6� (���24.������2 .

+�26�����,�� 2 (��42�68(�+�2689 ,���� �4.! .!� , */,�9 ,"� �8(+2��� � J ,�� # 6 2��.!

� 2 6�+37 2 ��!" I > )�p�r���+�8w�V$�$�7"+ 2 p 2 8([]$�p » +3p 0 � $9p 2 � 83!

Page 74: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

��� � ���`� ��� � � �X� �� ��� � � ��� ��� � � � �� � � � �� �� � � �

Page 75: jeiks.netjeiks.net/wp-content/uploads/2013/10/Apostila_Logica_Fuzzy.pdf · P ! []$9T5x $98 r3sz${YZ+-vw+ 7 2 r + 8 4 +36 ) 7"$98 8 !H!I!H!J!H!H!J!H!J!H!J!H!H!I!H!J!H!J!H!J!H!H!Nu

� � � �¾��� � �W��� � � � � �w�� ��� � � � � �

£\v(oFQZ��.9��ºX,�º]Q�! b*��m � "fhyefk9m !z#%$�&?O?$-£\7=+�/�&S+����V& 2 8 � 7 I�º]Y � OS$�& 2 ¬I7=$96�$ �U y�����U! : � � 6�7 � $zOS+ 0�2 Y�$98XQ�¤ 0 )�7"$98 E !¬-�H� 4 £ ��oR,�£I�Jº �UQ !]���HM\ºX����£\, �UQ !�oqp*O?+37"7 � /9+3p*Ow¡*A�6�& � YB8?A�8FO?+�T�8MI oR8S8F)U+�8 � 0 7 2 8?8 � � 032 O � $�pU8 2 p�Y�|})ZOS)�&S+Y � &?+ 0 O � $9pU83! oqpGIt¬-�H� 4 £ ��oR,�£��Jº ��Q�!b� �HM(ºX����£\, ��Q�! : ºXY ! E ! 3����b�°f°le $�b*���# �� µ���ef� ���±h3�yb � h ![]¡ �10 ¡�+�8FO?+�& ��� +�8FO�Q�)U8S8?+ ¿ �Zoqp�/�7 2�2 O?+�&?& 2 I ��$9¡�p�� � 7"+3A��ªQ�$�p�8�� y���9u�!MI�I,�,�£ 4 � �$��!�º]8 0�2 x � p�/�6�& � OFO?7"+3pU+�8S83!�oqp<Iw§¨o?[VM\£�,�QL�Jo ��vH!b��[V£\v(�(� 4 º],�, � ��!]��§¨oR�\[VM\ºX,�, ���J!: º]Y ! E ! mUk���ef��b � b*m�����ef� $ ! � Q ! 7�! I�§¨$9&?/ 2 p � 2 )Z|}T 2 p�p � ����9��! » !UKZ!�t�JQL�t� �Z�I! ´¨b�����m�°V´¨b*�t�U���±h�m���� ��� ��� � ���±h���b�� h ! 4 +�� ��+�&S8?+3A ��«�QZ£CI�#�&S+3p*O �10 +HM 2 7"7 �± y���*KZ!�t�JQL�t� �Z�I! ��� ��� � � ��el���±el��$ � � ��bÁ´¨b� ��k�elb���k9b¨� � ��� � ��� � ��$�elk !U«IQZ£�I�M 2 &Sx�+3& 2 p�Y¨[]$97=7 � pU8�� ��9�z��!�t�JQL�t� �Z�I! ��� ��� � � � $�el��b�b*��ef� $ ! 4 + � �9+3&D8F+�A���«IQZ£�I�#X&?+�p9O �"0 +HM 2 7"7 �± ��9�*��!§Á£�§ �(£ 4 o �Uº(!�M-!U£\x�x�7 �"0�2 O � $�pc$z|�|})�@�@3A�7"$�/ �"0 O?$ 2 x�x�&S$ ¿ � T 2 OS+�&S+ 2 8F$9p � pU/t)U8 � pU/57 � p�/�) � 8FO �10 8FA�p*O?¡�+y8 � 8�! � ��� � ��m���hym�k���ef����h���� ²���� "�����b*��h ��[X�D �� ��p±!± yK �Zx±!± 9 ��9K��� 9 ��� ����+ 0 +�Tt6�+3&H y�*���Z!v�«�Q�QZºX, �U�I! � 2 /�)�+�p�+�8S8�! �³��h��3��m�°lefm����±��������m�°w��� ! ��el°f��h���"���� � » !± � ��*KzP�!©�º 4 ����!b�U,�£ 4 ¬J£\v(o ��vH! ��� � ��� � ��$�elk �H ����yb*°l°fe%$�b���k�b�� ²������3����°VmU��� �� ���U��� m���ef��� ! 4 +�� �9+3&D8F+�A��«�Q�£�I�#�&S+3p*O �"0 +-M 2 7"7 � ����9��! �£ �\ºXM ��,X!�£-! > )�@3@�A³8?+�OS8�! �� ���U��� mZ�yef��� m���� ²������3����° � » !�� �Zx�!UP�P9����P*uzP �� y���9u�! �£ �\ºXM ��,X!�£t!�£\8?x�+ 0 OD8($�|%pU+�O �V$�&E� 2 pUY¨8?AZ8ROS+3TWO?¡�+�$�&SA�!�oqp<I ! � Q�! 7�! IVv � p�+�¡ 2 &?O 2 p�Y�� � pU8FO?$9p � ����� 9! 0�2 x±!���$.� 2 &DY 2 ��¡�+3$9&?A�$�| > )�@3@�A³Q�AZ8FO?+3T�8�! �oR§Á§ÁºXv(§Á£ 4\4 �]M-! ��! ��� ����� ��b9� � ��b������ �Ám����Àe}��h � " "%°felk�mZ�yef����h ! �V$98FO?$�p ��ºX«\£CI�J7") �V+3&?� 4 � �R¡�$��¨#�)�6U7 � 8F¡ � p�/ �� y���9u�!

�9�