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Math Help - logic

  1. #1
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    logic

    My teacher gave me a home work but I couldn't do it.Pls help me

    Question is here:

    1)(pvq)→r/\s (premise)

    2)r→t (premise)

    3)t (premise)
    _________________
    .'. p (conclusion)

    How can we get the conclusion from using the premises (1,2 and 3)?Show that with using rules of inferences.
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  2. #2
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    Here are the steps. You give the reasons.
    \begin{gathered}<br />
  \neg r \hfill \\<br />
  \neg r \vee \neg s \hfill \\<br />
  \neg \left( {r \wedge s} \right) \hfill \\<br />
  \neg \left( {\neg p \vee \neg s} \right) \hfill \\<br />
  p \wedge q \hfill \\<br />
  \therefore p \hfill \\ <br />
\end{gathered}
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  3. #3
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    Quote Originally Posted by Plato View Post
    Here are the steps. You give the reasons.
    \begin{gathered}<br />
\neg r \hfill \\<br />
\neg r \vee \neg s \hfill \\<br />
\neg \left( {r \wedge s} \right) \hfill \\<br />
\neg \left( {\neg p \vee \neg s} \right) \hfill \\<br />
p \wedge q \hfill \\<br />
\therefore p \hfill \\ <br />
\end{gathered}

    I'm sorry but I couldn't understand that Could you explain how can you find, pls?For instance where is t?
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  4. #4
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    Using 3 & 2, with modus tollens gives not r.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Using 3 & 2, with modus tollens gives not r.
    Ok.Then what about the others?There is :

    (pVq)→r/\s

    r

    I'm so sory but I couldn't understand those steps too
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  6. #6
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    Quote Originally Posted by scofield View Post
    My teacher gave me a home work but I couldn't do it.Pls help me

    Question is here:

    1)(pvq)→r/\s (premise)

    2)r→t (premise)

    3)t (premise)
    _________________
    .'. p (conclusion)

    How can we get the conclusion from using the premises (1,2 and 3)?Show that with using rules of inferences.

    Here is another proof using contradiction:

    1) (~pv~q)------> r^s assumption

    2) r---->t.....................assumption

    3) ~t...........................assumption

    4) ~p..........................assumption for contradiction

    5) ~pv~q....................from (4) by using Co...........

    6) r^s.........................from (1) and (5) by using M..........

    7) r..............................from (6) by using A...........

    8) t.............................from (2) and (7) by using M............

    9) t ^ ~t from (3) and (8) by using Ad................

    But t ^ ~t contradiction hence p
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