|
Question 483294: if p is true, q is false, and r is true, find the truth value of the statement. (r → ∼p) ∧ (q → ∼r)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your truth table looks like this:
p q r ~p ~r (r->~p) (q->~r) (r->~p)^(q->~r)
- - - -- -- ------- ------- ---------------
T T T F F F F F
T T F F T T T T
T F T F F F T F
T F F F T T T T
F T T T F T F F
F T F T T T T T
F F T T F T T T
F F F T T T T T
if p is true then ~p is false.
if p is false, then ~p is true.
if r is true, then ~r is false.
if r is false, then ~r is true.
if r is false then r->~p is true, whether or not ~p is true.
the only time r->~p is false, is if r is true and ~p is false.
if q is false, then q->~r is true, whether or not ~r is true.
the only time q->~r is false is if q is true and ~r is false.
this is your truth table for the implied statement.
A B A->B
T T T
T F F
F T T
F F T
you can see that A->B is only false when A is true and B is false.
when you apply the truth table for the implied statement, you assign different variables to A and B.
The A and B are just place holders to show you the logic.
for example:
A->B becomes r->~p after you replace A with r and B with ~p.
this is your truth table for the and statement.
A B A^B
T T T
T F F
F T F
F F F
You can see that A^B is only true when A is true and B is true.
in your problem above:
A^B becomes (r->~p)^(q->~r) after you replace A with (r->~p) and you replace B with (q->~r).
in your problem there is a cascading of statements.
here's your truth table again for ease of reference.
only the fourth row is shown.
p q r ~p ~r (r->~p) (q->~r) (r->~p)^(q->~r)
- - - -- -- ------- ------- ---------------
T F F F T T T T (fourth row)
it states that (r->~p)^(q->~r) is true.
this can only happen if (r->~p) is true and (q->~r) is true.
(r->~p) is true if:
r is false.
r is true and ~p is true.
this particular row has r is false.
that's enough to make (r->~p) true.
(q->~r) is true if:
q is false.
q is true and ~r is true.
this particular row has q is false.
that's enough to make (q->~r) true.
once you have (r->~p) column and (q->~r) column, then you use those columns to find the truth of the and statement of (r->~p)^(q->~r).
|
|
|
| |
