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Question 568316: Let p, q, and r represent the following statements.
p: Jamie is on the train
q: Sylvia is at the park
r: Nigel is in the car
Construct a truth table for the following
a. ~q V p
b. (~p V q)↔ q
I hope this is in the right section.. I can't understand truth tables for the life of me, so if you could provide some information like step by step that would be fantastic.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! in your problem, you have the following:
Let p, q, and r represent the following statements.
p: Jamie is on the train
q: Sylvia is at the park
r: Nigel is in the car
Construct a truth table for the following
a. ~q V p
b. (~p V q)↔ q
the variables involved are p and q and r
this means your truth table will have 2^3 = 8 rows.
you'll have 1 column for p and 1 column for q and 1 column for r
you should also construct a column for ~q and ~p since these will be involved in the statements.
if p is true, then ~p is false.
if q is true, then ~q is false.
if p is false, then ~p is true.
if q is false, then ~q is true.
make a column for (~q v p)
~q v p is true if either q is false or p is true.
it is only false the statement q is false is false and if the statement p is true is also false.
if the statement q is false is false, this means that q is true.
if the statement p is true is false, this means that p is false.
this means that ~q v p is false if q is true and p is false.
to understand this, you have to understand the truth table for A v B
that truth table looks like this:
A B A v B
T T T
T F T
F T T
F F F
A v B is true in all cases except when both A and B are false.
now if you let ~q be equal to A and you let p be equal to B, then the truth table becomes:
~q p ~q v p
T T T
T F T
F T T
F F F
same truth table with same logic only the names of the variables have been changed which is totally legitimate.
you can see that ~q v p is only false when ~q is false and p is false.
you also know that ~q is false if q is true, so the equivalent statement becomes:
~q v p is false if q is true and p is false.
you construct your ~q v p column based on the OR rules as expressed above.
the ~q v p column is true if either ~q is true or if p is true or if both are true. if both are false, then the ~q v p column is false. you'll see this in the table.
you now want to construct another column for (~p v q).
this is another OR construction, only this time the statement is true if either ~p is true or q is true of both are true. the statement is false if ~p is false and q is false at the same time.
you now want to construct another column for (~p v q) <-> q
you will be comparing columns of (~p v q) and q in order to determine the validity of the statement (~p v q) <-> q
the logic for the if and only if statement is as follows:
A B A <-> B
T T T
T F F
F T F
F F T
If A and B are both true, then the statement A <-> B is true.
if A and B are both false, then the statement A <-> B is true.
In other words, if they are both the same, the statement A <-> B is true.
if they are both different, then the statement A <-> B is false.
this includes:
A is true and B is false.
A is false and B is true.
if you let A = (~p v q) and if you let B = q, then this logic applies to the statement (~p v q) <-> q
the truth table for that would be as follows:
(~p v q) q (~p v q) <-> q
T T T
T F F
F T F
F F T
when you construct your column for, you would follow this logic in setting up the column by comparing the columns for (~p v q) and q.
your final truth table is shown below:
here's a reference on truth tables if you're interested.
http://www.algebra.com/algebra/homework/Conjunction/THEO-2011-08-19.lesson
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