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Консультация № 187373
26.05.2013, 17:03
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Доказать в исчислении высказываний (буквы обозначают произвольные формулы):
Доказать в исчислении высказываний (буквы обозначают произвольные формулы):
Обсуждение
27.05.2013, 00:39
общий
это ответ
Здравствуйте, Martin!
Изменим обозначения конъюнкции и импликации и докажем, что [$172$](([$172$]Z [$8658$] X) [$8743$] [$172$]([$172$]X [$8658$] [$172$]Y)) [$8801$] ((X [$8744$] [$172$]Y) [$8744$] [$172$]Z).
Имеет место следующее свойство, в справедливости которого можно убедиться при помощи таблицы истинности:
а по закону де Моргана
Поэтому
Аналогично
Из выражений (1) и (2) следует, что
Из выражения (3) получаем, что
а это и требовалось доказать.
С уважением.
Изменим обозначения конъюнкции и импликации и докажем, что [$172$](([$172$]Z [$8658$] X) [$8743$] [$172$]([$172$]X [$8658$] [$172$]Y)) [$8801$] ((X [$8744$] [$172$]Y) [$8744$] [$172$]Z).
Имеет место следующее свойство, в справедливости которого можно убедиться при помощи таблицы истинности:
([$172$]Z [$8658$] X) [$8801$] [$172$]([$172$]Z [$8743$] [$172$]X),
а по закону де Моргана
[$172$]([$172$]Z [$8743$] [$172$]X) [$8801$] [$172$]([$172$](Z [$8744$] X)) [$8801$] (Z [$8744$] X).
Поэтому
([$172$]Z [$8658$] X) [$8801$] (Z [$8744$] X). (1)
Аналогично
[$172$]([$172$]X [$8658$] [$172$]Y) [$8801$] [$172$]([$172$]([$172$]X [$8743$] Y)) [$8801$] ([$172$]X [$8743$] Y). (2)
Из выражений (1) и (2) следует, что
(([$172$]Z [$8658$] X) [$8743$] [$172$]([$172$]X [$8658$] [$172$]Y)) [$8801$] ((Z [$8744$] X) [$8743$] ([$172$]X [$8743$] Y)) [$8801$] (((Z [$8744$] X) [$8743$] [$172$]X) [$8743$] ((Z [$8744$] X) [$8743$] Y)) [$8801$]
[$8801$] (((Z [$8743$] [$172$]X) [$8744$] (X [$8743$] [$172$]X)) [$8743$] ((Z [$8744$] X) [$8743$] Y)) [$8801$] ((Z [$8743$] [$172$]X) [$8743$] ((Z [$8744$] X) [$ [$8801$]8743$] Y)) [$8801$]
[$8801$] (((Z [$8743$] [$172$]X) [$8743$] (Z [$8744$] X)) [$8743$] ((Z [$8743$] [$172$]X) [$8743$] Y)) [$8801$]
[$8801$] (((Z [$8743$] [$172$]X [$8743$] Z) [$8744$] (Z [$8743$] [$172$]X [$8743$] X)) [$8743$] ((Z [$8743$] [$172$]X) [$8743$] Y)) [$8801$]
[$8801$] ((Z [$8743$] [$172$]X) [$8744$] 0) [$8743$] ((Z [$8743$] [$172$]X) [$8743$] Y)) [$8801$] ((Z [$8743$] [$172$]X) [$8743$] ((Z [$8743$] [$172$]X) [$8743$] Y)) [$8801$] ((Z [$8743$] [$172$]X) [$8743$] Y). (3)
Из выражения (3) получаем, что
[$172$](([$172$]Z [$8658$] X) [$8743$] [$172$]([$172$]X [$8658$] [$172$]Y)) [$8801$] [$172$]((Z [$8743$] [$172$]X) [$8743$] Y) [$8801$] [$172$](([$172$]X [$8743$] Y) [$8743$] Z) [$8801$]
[$8801$] ([$172$]([$172$]X [$8743$] Y) [$8744$] [$172$]Z) [$8801$] ((X [$8744$] [$172$]Y) [$8744$] [$172$]Z),
а это и требовалось доказать.
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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