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Showing logical equivalence

  1. Jun 22, 2012 #1
    1. The problem statement, all variables and given/known data
    (b) Show that (p → q) ∨ (p→ r) is equivalent to p → (q ∨ r).


    2. Relevant equations


    the ~ means negate

    3. The attempt at a solution

    Im not sure if i did this correctly
    (p → q) ∨ (P → r)
    (~p∨q) ∨ (~p∨r) used the conditional law p→q equivalent to ~p∨q
    ((~p∨q)∨~p)∨((~p∨q)∨r)) distributive law
    (~p∨q)∨(~p∨q)∨r
    (~p∨q)∨r
    ~p∨(r∨q) associative law
    p →(q∨r)
     
  2. jcsd
  3. Jun 23, 2012 #2

    tiny-tim

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    hi bonfire09! :smile:

    all your steps are correct

    however, after …
    … don't you notice that they're all ∨ ,

    so you can rearrange them (using the …?… law), and then use ~p∨~p = ~p :wink:
     
  4. Jun 23, 2012 #3

    HallsofIvy

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    Are you required to do it that way? Setting up an 8 case "truth table" shows that both statements are false in case p= T, q= r= F, and true in all other cases.
     
  5. Jun 23, 2012 #4
    That im not sure upon. In Velleman's book he is not so clear about what he wants us to show. But I think what I did suffices.
     
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