Question 568316
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:
<pre>
               A           B        A v B
               T           T          T
               T           F          T
               F           T          T
               F           F          F
</pre>
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:
<pre>
               ~q          p        ~q v p
                T          T           T
                T          F           T
                F          T           T
                F          F           F
</pre>
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:
<pre>
                A            B        A <-> B
                T            T           T
                T            F           F
                F            T           F
                F            F           T
</pre>
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:
<pre>
            (~p v q)         q        (~p v q) <-> q
                T            T                  T
                T            F                  F
                F            T                  F
                F            F                  T
</pre>


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:
<img src = "http://theo.x10hosting.com/2012/feb061.jpg" alt = "$$$$" />


here's a reference on truth tables if you're interested.



<a href = "http://www.algebra.com/algebra/homework/Conjunction/THEO-2011-08-19.lesson" target = "_blank">http://www.algebra.com/algebra/homework/Conjunction/THEO-2011-08-19.lesson</a>