Results 1 to 4 of 4

Math Help - Proofs

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    37

    Proofs

    Prove by direct proof, using only equivalence and inference rules. Remember to name each rule used.
    1. (p ^ q) → r

    2. p ^ r

    Therefore q.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,663
    Thanks
    1616
    Awards
    1
    \begin{gathered}<br />
  \left( {p \wedge q} \right) \to r \hfill \\<br />
  p \wedge \neg r \hfill \\<br />
  \-------  \hfill \\<br />
  p \hfill \\<br />
  \neg r \hfill \\<br />
  \neg \left( {p \wedge q} \right) \hfill \\<br />
  \neg p \vee \neg q \hfill \\<br />
  \neg q \hfill \\ <br />
\end{gathered}
    You fill in the reasons.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,738
    Thanks
    645
    Hello, captainjapan!

    (p \to q) \;\Leftrightarrow\;\sim p \vee q . Alternate definition of implication (ADI)


    Prove by direct proof, using only equivalence and inference rules.
    Remember to name each rule used.

    \begin{array}{c}(p \wedge q) \to r \\ p \:\wedge \sim\! r \\ \hline<br />
\therefore\;\sim\! q \end{array}

    . . \begin{array}{ccc}<br />
(p \wedge q ) \to r & & \text{Given} \\<br />
\sim(p \wedge q) \vee r & & \text{ADI} \\<br />
(\sim\!p \:\vee \sim\!q) \vee r & & \text{DeMorgan} \\<br />
(\sim\! p \vee r) \:\vee \sim\! q & & \text{comm/assoc} \\<br />
\sim(\sim\!p \vee r) \to \:\sim\!q & & \text{ADI} \\<br />
(p \:\wedge \sim\!r) \to \:\sim\! q & & \text{DeMorgan} \\<br />
p \:\wedge \sim r & & \text{Given} \\<br />
\therefore\;\;\sim\!q & & \text{Modus Ponens} <br />
\end{array}

    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,663
    Thanks
    1616
    Awards
    1
    Quote Originally Posted by Soroban View Post
    (p \to q) \;\Leftrightarrow\;\sim p \vee q . Alternate definition of implication (ADI)
    In almost all standard logic textbooks the property
    (p \to q) \;\Leftrightarrow\;\sim p \vee q is known as Material Implication (Impl.).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. proofs
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 2nd 2010, 03:54 AM
  2. lim sup and lim inf proofs
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: February 24th 2010, 07:02 PM
  3. More Proofs
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: February 13th 2008, 07:05 PM
  4. Proofs
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 3rd 2008, 04:23 AM
  5. Replies: 3
    Last Post: October 6th 2007, 02:01 PM

Search Tags


/mathhelpforum @mathhelpforum