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Homework Help: Help please! Find the truth value of the statement:

  1. Aug 31, 2011 #1
    1. The problem statement, all variables and given/known data

    Given P is true, Q is true, and R is fale, find the truth value of the statement:

    ~(~p ^ q) <-> (q -> ~r)

    2. The attempt at a solution

    Which one is correct- Answer 1 or answer 2?

    answer 1:
    p v ~q <-> (q -> ~r)

    t V f <-> (t -> t)

    t <-> t

    Answer= t

    answer 2:
    p ^ ~q <-> (q -> ~r)

    t ^ f <-> (t -> t)

    f <-> t

    Answer= f
  2. jcsd
  3. Aug 31, 2011 #2


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    Answer 1 is correct. Answer 2 isn't. Why not? And why didn't you just substitute P=t, Q=t and R=f into the original statement?
  4. Sep 1, 2011 #3


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    Staff: Mentor

    Could someone explain explain how I should read this: <->
    and what does this mean: ->

  5. Sep 1, 2011 #4


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    <-> is "if and only if". P<-> Q is true if the true values of P and Q are the same: both T or both F.
    Truth table:
    [tex]\begin{array}{ccc} & T & F\\ T & T & F \\ F & F & T\end{array}[/tex]
    where accross is Q and down is P.

    -> is "implication" P->Q "P implies Q" or "If P then Q" is true in the case that Q is true or both P and Q are false. Truth table:
    [tex]\begin{array}{ccc} & T & F\\ T & T & F \\ F & T & T\end{array}[/tex]

    As for amandamarieve's problem, ~(~p ^ q) <-> (q -> ~r), with p= q= T, r= f,
    "~p ^q" is "F ^ T" which is F so ~(~p ^q) is T. That is, of course, the same as you get by using the fact that ~(~p ^ q)= ~(~p) v (~q)= p v ~q. On the right, q-> ~r, with q T and r F, is T->T which is T. We have T<->T which is T.

    amandamarieve's "answer 2" is wrong because, as I said, ~(~p^q) is pv ~q, not "p^ ~q. In general, ~(p ^ q)= ~p v ~q and ~(p v q)= ~p ^ ~ q.
  6. Sep 1, 2011 #5


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    Staff: Mentor

    Thank you.
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