if p implies q is true then the truth value of which of the following cannot be determined

Question

a) ~p\/q        b) ~q=>~p

c) ~p=>~q    d) ~(p/\~q)

can someone provide the solution?

Answer 1

We have p -> q as a true proposition.

That means for p as true and q as false we can't have the truth value as True. Because T -> F is False. 

Now since we have p = True and q = False as a test condition to check all the options we can use proof by examples to evaluate the final answer.

Option a: not p or q : not (True) or False = False or False = False. 

Option b: not q -> not p : not(False) -> not(True) = True-> False = False

Option c: not p -> not q : not(True) -> not(False) = False-> True = True (This is the wrong option)

Option d: not (p and not q) : not(True and not(False)) = not(True and True) = not(True) = False

So out of the given options only option c cannot be determined. 

Comments

cannot be determined means what according to you ? I interpreted it to be proposition formulas whose truth value we can't determine. In your solution you are able to determine the truth value. Is my interpretation wrong ?

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"I interpreted it to be proposition formulas whose truth value we can't determine". I think by this you mean propositional formulas where we derive them to be in simpler form but at the end they came out to be dependent on one or more variables that have been used in the formula. For ex: A + AB + (not of A and B). This will come out as A+B on solving. But can't we resolve A+B well put A = True and B = False and we get A+B = True.

So definitely any propositional formula will end up in either True or False. 

But then what is can not be determined? It is relative to what conditions we are provided with.

In the question,

p->q is true which means we can't take p as true and q as false because T->F is false but p->q must be true. However p can be True and q can be false but not at the same time.

But then why option c can't be determined? (By given condition of p as true and q as false)

But wait why we take p as true and q as false? why not other values.

well we can use that but in that case we need to prove the three cases as 

p = True q = True, p = False q = True, p = False q = False,

if we are able to prove that in every of these three cases we get final result as true then we can determine c by our given condition of p as true and q as false.

But note, all these 3 cases needed to be checked.

therefore why not move away with only one case why not just check for p as true and q as false as the first case? If we found that it is false i.e putting p as true and q as false we get output as false then in that case we will move to the remaining three cases. But this will be the worst possibility.

 

So that's why just we will just check with p as true and q as false first. If lucky we will get the answer by this.

 

and "cannot be determined" here means for P as true and Q as false if I put these values then the given option should behave just like what p->q behaves if any of these options will not behave so then we will say that it can't be determined from given condition of p->q as true.

I hope this helps.

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thanks i got this now, that last para about how to interpret the question was where i went wrong