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Discrete Math Help

  1. Jan 30, 2007 #1
    1. The problem statement, all variables and given/known data

    Use the logical equivalences [tex] p \rightarrow q \equiv \sim p \vee q [/tex] and [tex] p \leftrightarrow q \equiv (p \rightarrow q) \wedge (q \rightarrow p) [/tex] to rewrite the statement form:

    [tex](p \rightarrow (q \rightarrow r)) \leftrightarrow ((p \wedge q) \rightarrow r) [/tex]
    2. Relevant equations
    None


    3. The attempt at a solution

    My initial assumption is to set this up as:

    Let P == [tex](p \rightarrow (q \rightarrow r))[/tex]
    Let Q == [tex]((p \wedge q) \rightarrow r) [/tex]

    Then plug in from there with the equivalences to get:

    [tex] (p \rightarrow q) \wedge (q \rightarrow p) [/tex]

    Furthermore:

    [tex] (\sim p \vee q) \wedge (q \rightarrow p) [/tex]

    Is this the right approach, or am I starting it out wrong?
     
    Last edited: Jan 30, 2007
  2. jcsd
  3. Jan 31, 2007 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You said "Let P= [tex](p \rightarrow (q \rightarrow r))[/tex]" and "Let Q= [tex]((p \wedge q) \rightarrow r) [/tex]" but then wrote "[tex] (p \rightarrow q) \wedge (q \rightarrow p) [/tex]"

    Don't you mean [tex] (P \rightarrow Q) \wedge (Q \rightarrow P) [/tex]?

    Use [tex] p \rightarrow q \equiv \sim p \vee q [/tex] to rewrite each part of that, then use it again to rewrite P and Q separately and plop them in there.
     
  4. Jan 31, 2007 #3
    Heres what I ended up with:

    [tex] \sim ( \sim p \vee ( \sim q \vee r)) \vee (( \sim p \wedge q) \vee r) \wedge ( \sim( \sim p \wedge q)\vee r) \vee ( \sim p \vee ( \sim q \vee r))[/tex]
     
    Last edited: Jan 31, 2007
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