Help please Find the truth value of the statement:

[amandamarieve]

Homework Statement

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

 

[Dick]

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?

 

[NascentOxygen]

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

Thanks.

 

[HallsofIvy]

<-> 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:
$$\begin{array}{ccc} & T & F\\ T & T & F \\ F & F & T\end{array}$$
where across 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:
$$\begin{array}{ccc} & T & F\\ T & T & F \\ F & T & T\end{array}$$

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.

 

[NascentOxygen]

<-> 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.

Thank you.