1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Show that ((p implies q) and (q implies r)) implies (p implies r) is a tautology

  1. Sep 10, 2007 #1
    1. The problem statement, all variables and given/known data

    Show that [tex]\left[\left(p\,\longrightarrow\,q\right)\,\wedge\,\left(q\,\longrightarrow\,r\right)\right]\,\longrightarrow\,\left(p\,\longrightarrow\,r\right)[/tex] is a tautology.



    2. Relevant equations

    Logical equivalences.



    3. The attempt at a solution

    [tex]\begin{array}{l}
    \left[ {\left( {p\; \to \;q} \right)\; \wedge \;\left( {q\; \to \;r} \right)} \right]\; \to \;\left( {p\; \to \;r} \right) \\
    \left[ {\left( {\neg p\; \vee \;q} \right)\; \wedge \;\left( {\neg q\; \vee \;r} \right)} \right]\; \to \;\left( {p\; \to \;r} \right) \\
    \left\{ {\left[ {\left( {\neg p\; \vee \;q} \right)\; \wedge \;\neg q} \right]\; \vee \;\left[ {\left( {\neg p\; \vee \;q} \right)\; \wedge \;r} \right]} \right\}\; \to \;\left( {p\; \to \;r} \right) \\
    \left\{ {\left[ {\left( {\neg p\; \wedge \;\neg q} \right)\; \vee \;\left( {q\; \wedge \;\neg q} \right)} \right]\; \vee \;\left[ {\left( {\neg p\; \wedge \;r} \right)\; \vee \;\left( {q\; \wedge \;r} \right)} \right]} \right\} \to \;\left( {p\; \to \;r} \right) \\
    \left\{ {\left[ {\left( {\neg p\; \wedge \;\neg q} \right)\; \vee \;{\rm F}} \right]\; \vee \;\left[ {\left( {\neg p\; \wedge \;r} \right)\; \vee \;\left( {q\; \wedge \;r} \right)} \right]} \right\}\; \to \;\left( {p\; \to \;r} \right) \\
    \left\{ {\left[ {\neg p\; \wedge \;\neg q} \right]\; \vee \;\left[ {\left( {\neg p\; \wedge \;r} \right)\; \vee \;\left( {q\; \wedge \;r} \right)} \right]} \right\}\; \to \;\left( {p\; \to \;r} \right) \\
    \end{array}[/tex]

    What now?
     
  2. jcsd
  3. Sep 11, 2007 #2
    Without any prior assumptions we need to assume (p->q) and (q->r) and from there show that p imples r. This may not be legit if your instructor wants a symbolic elimination of the "fluff". Symbollically: keep on working, you are no the right track - expand and cancel falsehoods or tautologies like you have been doing.
     
  4. Sep 11, 2007 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    As SiddharthM says, you should just expand all you implications (there are two left) as not ors. Or write out a truth table.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Show that ((p implies q) and (q implies r)) implies (p implies r) is a tautology
Loading...