дискретная мтематика. Множества . A&#8745;B&#8838;C <=> A&#8838;bar{B}&#8746;C

[eek]

Здравствуйте уважаемое сообщество

не получается решить задание. домашняя работа, 1 курс прикладной математики Каунасского Технологического университета

нужно доказать что

`A nn B subseteq C iff A subseteq bar{B} uu C`



попытка решения под катом






\wedge(x&space;\in&space;C)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;(x&space;\notin&space;(A&space;\cap&space;B))\wedge(x&space;\in&space;C)&space;\right&space;]&space;\vee&space;\\&space;\vee&space;\left&space;[&space;(x&space;\notin&space;(A&space;\cap&space;B))\wedge(x&space;\notin&space;C)&space;\right&space;]&space;\Rightarrow&space;\\&space;\Rightarrow&space;\left&space;[&space;((x&space;\in&space;A)&space;\wedge&space;(x&space;\in&space;B)&space\wedge(x&space;\in&space;C)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;(x&space;\in&space;\overline{(A&space;\cap&space;B)})\wedge(x&space;\in&space;C)&space;\right&space;]&space;\vee&space;\\&space;\vee&space;\left&space;[&space;(x&space;\in&space;\overline{(A&space;\cap&space;B)})\wedge(x&space;\notin&space;C)&space;\right&space;]&space;\Rightarrow" title="\\ A \cap B \subseteq C \Leftrightarrow A \subseteq \overline{B} \cup C \\ A \subseteq B \Rightarrow \\ \Rightarrow \left [ (x \in A)\wedge(x \in B) \right ]\vee \\ \vee \left [ (x \notin A)\wedge(x \in B) \right ] \vee \\ \vee \left [ (x \notin A)\wedge(x \notin B) \right ] \\ (a)A \cap B \subseteq C \overset{1}{ \Rightarrow} \\ \overset{1}{ \Rightarrow} \left [ (x \in (A \cap B) )\wedge(x \in C) \right ]\vee \\ \vee \left [ (x \notin (A \cap B))\wedge(x \in C) \right ] \vee \\ \vee \left [ (x \notin (A \cap B))\wedge(x \notin C) \right ] \Rightarrow \\ \Rightarrow \left [ ((x \in A) \wedge (x \in B) )\wedge(x \in C) \right ]\vee \\ \vee \left [ (x \in \overline{(A \cap B)})\wedge(x \in C) \right ] \vee \\ \vee \left [ (x \in \overline{(A \cap B)})\wedge(x \notin C) \right ] \Rightarrow">

\wedge(x&space;\in&space;C)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;\overline{((x&space;\in&space;A)&space;\wedge&space;(x&space;\in&space;B)})\wedge(x&space;\in&space;C)&space;\right&space;]&space;\vee&space;\\&space;\vee&space;\left&space;[&space;\overline{((x&space;\in&space;A)&space;\wedge&space;(x&space;\in&space;B)})\wedge(x&space;\notin&space;C)&space;\right&space;]&space;\overset{2}{\Rightarrow}&space;\\&space;\overset{2}{\Rightarrow}&space;\left&space;[&space;((x&space;\in&space;A)&space;\wedge&space;(x&space;\in&space;B)&space\wedge(x&space;\in&space;C)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;\overline{((x&space;\in&space;A)}&space;\vee&space;\overline{(x&space;\in&space;B)})\wedge(x&space;\in&space;C)&space;\right&space;]&space;\vee&space;\\&space;\vee&space;\left&space;[&space;\overline{((x&space;\in&space;A)}&space;\vee&space;\overline{(x&space;\in&space;B)})\wedge(x&space;\notin&space;C)&space;\right&space;]&space;\Rightarrow&space;\\&space;\Rightarrow&space;\left&space;[&space;((x&space;\in&space;A)&space;\wedge&space;(x&space;\in&space;B)&space\wedge(x&space;\in&space;C)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)&space;\vee&space;(x&space;\notin&space;B))\wedge(x&space;\in&space;C)&space;\right&space;]&space;\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)&space;\vee&space;(x&space;\notin&space;B))\wedge(x&space;\notin&space;C)&space;\right&space;]&space;\overset{3}{\Rightarrow}&space;\\&space;\overset{3}{\Rightarrow}&space;\\" title="\\ \Rightarrow \left [ ((x \in A) \wedge (x \in B) )\wedge(x \in C) \right ]\vee \\ \vee \left [ \overline{((x \in A) \wedge (x \in B)})\wedge(x \in C) \right ] \vee \\ \vee \left [ \overline{((x \in A) \wedge (x \in B)})\wedge(x \notin C) \right ] \overset{2}{\Rightarrow} \\ \overset{2}{\Rightarrow} \left [ ((x \in A) \wedge (x \in B) )\wedge(x \in C) \right ]\vee \\ \vee \left [ \overline{((x \in A)} \vee \overline{(x \in B)})\wedge(x \in C) \right ] \vee \\ \vee \left [ \overline{((x \in A)} \vee \overline{(x \in B)})\wedge(x \notin C) \right ] \Rightarrow \\ \Rightarrow \left [ ((x \in A) \wedge (x \in B) )\wedge(x \in C) \right ]\vee \\ \vee \left [ ((x \notin A) \vee (x \notin B))\wedge(x \in C) \right ] \vee \\ \vee \left [ ((x \notin A) \vee (x \notin B))\wedge(x \notin C) \right ] \overset{3}{\Rightarrow} \\ \overset{3}{\Rightarrow} \\">

\wedge(x&space;\in&space;C)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)&space;\vee&space;(x&space;\notin&space;B))\wedge(x&space;\in&space;C)&space;\right&space;]&space;\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)&space;\vee&space;(x&space;\notin&space;B))\wedge(x&space;\notin&space;C)&space;\right&space;]&space;\overset{4}{\Rightarrow}&space;\\&space;\\&space;\overset{4}{\Rightarrow}&space;\left&space;[&space;((x&space;\in&space;A)&space;\wedge&space;(x&space;\in&space;B)&space\wedge(x&space;\in&space;C)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)\wedge(x&space;\in&space;C))&space;\vee&space;((x&space;\notin&space;B)&space;\wedge(x&space;\in&space;C))&space;\right&space;]&space;\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)\wedge(x&space;\notin&space;C))&space;\vee&space;((x&space;\notin&space;B)&space;\wedge(x&space;\notin&space;C))&space;\right&space;]&space;\vee" title="\\ \overset{3}{\Rightarrow} \left [ ((x \in A) \wedge (x \in B) )\wedge(x \in C) \right ]\vee \\ \vee \left [ ((x \notin A) \vee (x \notin B))\wedge(x \in C) \right ] \vee \\ \vee \left [ ((x \notin A) \vee (x \notin B))\wedge(x \notin C) \right ] \overset{4}{\Rightarrow} \\ \\ \overset{4}{\Rightarrow} \left [ ((x \in A) \wedge (x \in B) )\wedge(x \in C) \right ]\vee \\ \vee \left [ ((x \notin A)\wedge(x \in C)) \vee ((x \notin B) \wedge(x \in C)) \right ] \vee \\ \vee \left [ ((x \notin A)\wedge(x \notin C)) \vee ((x \notin B) \wedge(x \notin C)) \right ] \vee">

правая часть

)&space;\right&space;]\vee&space;\\&space;\vee\left&space;[&space;(x&space;\notin&space;A)\wedge(x&space;\in&space;(\overline{B}&space;\cup&space;C&space)&space;\right&space;]\vee&space;\\&space;\vee\left&space;[&space;(x&space;\notin&space;A)\wedge(x&space;\notin&space;(\overline{B}&space;\cup&space;C&space)&space;\right&space;]&space;\overset{2}{\Rightarrow&space;}&space;\\&space;\overset{2}{\Rightarrow&space;}&space;\left&space;[&space;((x&space;\in&space;A)\wedge&space;(x&space;\in&space;\overline{B}&space)&space;\vee&space;((x&space;\in&space;A)\wedge&space;(x&space;\in&space;C&space)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)\wedge&space;(x&space;\notin&space;B&space)&space;\vee&space;((x&space;\notin&space;A)\wedge&space;(x&space;\in&space;C&space)&space;\right&space;]\vee&space;\\&space;\vee\left&space;[&space;(x&space;\notin&space;A)\wedge(x&space;\in&space;\overline{(\overline{B}&space;\cup&space;C&space})&space;\right&space;]&space;\overset{4}{\Rightarrow&space;}" title="\\ (b) A \subseteq \overline{B} \cup C \overset{1}{\Rightarrow } \\ \Rightarrow \left [ (x \in A)\wedge(x \in (\overline{B} \cup C )) \right ]\vee \\ \vee\left [ (x \notin A)\wedge(x \in (\overline{B} \cup C )) \right ]\vee \\ \vee\left [ (x \notin A)\wedge(x \notin (\overline{B} \cup C )) \right ] \overset{2}{\Rightarrow } \\ \overset{2}{\Rightarrow } \left [ ((x \in A)\wedge (x \in \overline{B} )) \vee ((x \in A)\wedge (x \in C )) \right ]\vee \\ \vee \left [ ((x \notin A)\wedge (x \notin B )) \vee ((x \notin A)\wedge (x \in C )) \right ]\vee \\ \vee\left [ (x \notin A)\wedge(x \in \overline{(\overline{B} \cup C )}) \right ] \overset{4}{\Rightarrow }">

)&space;\vee&space;((x&space;\in&space;A)\wedge&space;(x&space;\in&space;C&space)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)\wedge&space;(x&space;\notin&space;B&space)&space;\vee&space;((x&space;\notin&space;A)\wedge&space;(x&space;\in&space;C&space)&space;\right&space;]\vee&space;\\&space;\vee\left&space;[&space;(x&space;\notin&space;A)\wedge&space;(\overline{(x&space;\in&space;\overline{B})&space;\vee&space;(x&space;\in&space;C&space})&space;\right&space;]&space;\overset{5}{\Rightarrow&space;}&space;\\&space;\overset{5}{\Rightarrow&space;}&space;\left&space;[&space;((x&space;\in&space;A)\wedge&space;(x&space;\in&space;\overline{B}&space)&space;\vee&space;((x&space;\in&space;A)\wedge&space;(x&space;\in&space;C&space)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)\wedge&space;(x&space;\notin&space;B&space)&space;\vee&space;((x&space;\notin&space;A)\wedge&space;(x&space;\in&space;C&space)&space;\right&space;]\vee&space;\\&space;\vee\left&space;[&space;(x&space;\notin&space;A)\wedge&space;(\overline{(x&space;\in&space;\overline{B})&space;}&space;\wedge&space;\overline{&space;(x&space;\in&space;C&space})&space;\right&space;]&space;\overset{4}{\Rightarrow&space;}&space;\\" title="\\ \overset{4}{\Rightarrow } \left [ ((x \in A)\wedge (x \in \overline{B} )) \vee ((x \in A)\wedge (x \in C )) \right ]\vee \\ \vee \left [ ((x \notin A)\wedge (x \notin B )) \vee ((x \notin A)\wedge (x \in C )) \right ]\vee \\ \vee\left [ (x \notin A)\wedge (\overline{(x \in \overline{B}) \vee (x \in C )}) \right ] \overset{5}{\Rightarrow } \\ \overset{5}{\Rightarrow } \left [ ((x \in A)\wedge (x \in \overline{B} )) \vee ((x \in A)\wedge (x \in C )) \right ]\vee \\ \vee \left [ ((x \notin A)\wedge (x \notin B )) \vee ((x \notin A)\wedge (x \in C )) \right ]\vee \\ \vee\left [ (x \notin A)\wedge (\overline{(x \in \overline{B}) } \wedge \overline{ (x \in C )}) \right ] \overset{4}{\Rightarrow } \\">

)&space;\vee&space;((x&space;\in&space;A)\wedge&space;(x&space;\in&space;C&space)&space;\right&space;]\vee&space;\\&space;\vee&space;\left&space;[&space;((x&space;\notin&space;A)\wedge&space;(x&space;\notin&space;B&space)&space;\vee&space;((x&space;\notin&space;A)\wedge&space;(x&space;\in&space;C&space)&space;\right&space;]\vee&space;\\&space;\vee\left&space;[&space;(x&space;\notin&space;A)\wedge&space;((x&space;\in&space;B)&space;\wedge&space;(x&space;\notin&space;C&space)&space;\right&space;]&space;\overset{4}&space;{\Rightarrow&space;}&space;\\" title="\\ \overset{5}{\Rightarrow } \left [ ((x \in A)\wedge (x \in \overline{B} )) \vee ((x \in A)\wedge (x \in C )) \right ]\vee \\ \vee \left [ ((x \notin A)\wedge (x \notin B )) \vee ((x \notin A)\wedge (x \in C )) \right ]\vee \\ \vee\left [ (x \notin A)\wedge ((x \in B) \wedge (x \notin C )) \right ] \overset{4} {\Rightarrow } \\">




выражение эквивалентности значит "тогда и только тогда, когда..."

расписал обе части, но одинаковых элементов в полученных выражениях нет.

что нужно сделать дальше? помогите пожалуйста.
я сам всё сделаю, подскажите направление, пожалуйста