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Question 1199080:
Use ordinary truth tables to answer the following problem. Construct the truth tables as per the instructions in the textbook.
Given the statement: [K • (P v ~ R))] • [K > (R • ~ P)]
The statement is:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
Learn the four rules ~, v, •, >
1. ~ means the opposite of what follows the ~
2. FvF is the only false case of v, all others T.
3. T•T is the only true case of •, all others F.
4. T>F is the only false case of >, all others T.
Write the expression across the paper.
Under the K's put TTTTFFFF
Under the P's put TTFFTTFF
Under the R's put TFTFTFTF
[K • (P v ~ R)] • [K > (R • ~ P)]
T T T T T T
T T F T F T
T F T T T F
T F F T F F
F T T F T T
F T F F F T
F F T F T F
F F F F F F
Under the ~'s, put F if ~ is before a T
and F if ~ is before a T
[K • (P v ~ R)] • [K > (R • ~ P)]
T T F T T T F T
T T T F T F F T
T F F T T T T F
T F T F T F T F
F T F T F T F T
F T T F F F F T
F F F T F T T F
F F T F F F T F
Erase the T's and F's in the columns you just use
to get the last columns you put in.
[K • (P v ~ R)] • [K > (R • ~ P)]
T T F T T F
T T T T F F
T F F T T T
T F T T F T
F T F F T F
F T T F F F
F F F F T T
F F T F F T
Staying within the first innermost parentheses (),
under the v, put T everywhere except where v is
between two F's. This is the only time we put F.
[You will notice that this his rule is the exact
opposite of what we will put under •].
[K • (P v ~ R)] • [K > (R • ~ P)]
T T T F T T F
T T T T T F F
T F F F T T T
T F T T T F T
F T T F F T F
F T T T F F F
F F F F F T T
F F T T F F T
Erase the two columns of T's and F's that we used
to get the last column we made:
[K • (P v ~ R)] • [K > (R • ~ P)]
T T T T F
T T T F F
T F T T T
T T T F T
F T F T F
F T F F F
F F F T T
F T F F T
Staying within the other innermost parentheses, ()
under the •, put F everywhere except where • is
between two T's. This is the only time we put T.
[This rule is the exact opposite of what we put
under v].
[K • (P v ~ R)] • [K > (R • ~ P)]
T T T T F F
T T T F F F
T F T T T T
T T T F F T
F T F T F F
F T F F F F
F F F T T T
F T F F F T
Erase the two columns of T's and F's that we used
to get the last column we made:
[K • (P v ~ R)] • [K > (R • ~ P)]
T T T F
T T T F
T F T T
T T T F
F T F F
F T F F
F F F T
F T F F
Staying within the first innermost brackets, [],
under the •, as before, put F everywhere except
where • is between two T's. This is the only
time we put T.
[K • (P v ~ R)] • [K > (R • ~ P)]
T T T T F
T T T T F
T F F T T
T T T T F
F F T F F
F F T F F
F F F F T
F F T F F
Erase the two columns of T's and F's that we used
to get the last column we made:
[K • (P v ~ R)] • [K > (R • ~ P)]
T T F
T T F
F T T
T T F
F F F
F F F
F F T
F F F
Staying within the other innermost brackets, [],
under the >, put T everywhere except where > has
T on the left and F on the right. This is the
only time we put T.
[K • (P v ~ R)] • [K > (R • ~ P)]
T T F F
T T F F
F T T T
T T F F
F F T F
F F T F
F F T T
F F T F
Erase the two columns of T's and F's that we used
to get the last column we made:
[K • (P v ~ R)] • [K > (R • ~ P)]
T F
T F
F T
T F
F T
F T
F T
F T
Now that we have finished all the parentheses
and brackets, we are dow to just 2 columns of T's
and F's. Under the final • outside all parentheses
and brackets we put F's for everything but T•T.
There are none, so we put F's for everything.
[K • (P v ~ R)] • [K > (R • ~ P)]
T F F
T F F
F F T
T F F
F F T
F F T
F F T
F F T
We erase the two columns of T's and F's that we used
to get the last column we made:
[K • (P v ~ R)] • [K > (R • ~ P)]
F
F
F
F
F
F
F
F
Since all the values are F, the statement is a
logical contradiction, which means it is always
false.
[In other problems, when all the values are T, the
statement is a logical tautology, or always true.]
[In other problems when some are T and some are F,
the statement is a contingency, or sometimes true
and sometimes false.]
Edwin
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