Question 1190026
<font color=black size=3>
T = true
F = false


We have three variables p, q and r
We need exactly {{{2^3 = 8}}} rows to go through all possible combos of truth values.


The first column for p will have four T's in a row followed by four F's in a row.
The second column for q will have two copies of TTFF
The third column for r will have four copies of TF
This will guarantee to capture every possible outcome


We have this table so far<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>r</td></tr><tr><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>T</td><td>F</td></tr><tr><td>T</td><td>F</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>T</td><td>T</td></tr><tr><td>F</td><td>T</td><td>F</td></tr><tr><td>F</td><td>F</td><td>T</td></tr><tr><td>F</td><td>F</td><td>F</td></tr></table>


Next is to add the column titled ~q which will just flip whatever is in column q<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>r</td><td>~q</td></tr><tr><td>T</td><td>T</td><td>T</td><td>F</td></tr><tr><td>T</td><td>T</td><td>F</td><td>F</td></tr><tr><td>T</td><td>F</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td></tr><tr><td>F</td><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>F</td><td>T</td><td>T</td></tr><tr><td>F</td><td>F</td><td>F</td><td>T</td></tr></table>


Do a similar operation for column ~r<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>r</td><td>~q</td><td>~r</td></tr><tr><td>T</td><td>T</td><td>T</td><td>F</td><td>F</td></tr><tr><td>T</td><td>T</td><td>F</td><td>F</td><td>T</td></tr><tr><td>T</td><td>F</td><td>T</td><td>T</td><td>F</td></tr><tr><td>T</td><td>F</td><td>F</td><td>T</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>T</td><td>F</td><td>F</td><td>T</td></tr><tr><td>F</td><td>F</td><td>T</td><td>T</td><td>F</td></tr><tr><td>F</td><td>F</td><td>F</td><td>T</td><td>T</td></tr></table>


Let's add the p ^ ~q column to the table we're building up<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>r</td><td>~q</td><td>~r</td><td>p ^ ~q</td></tr><tr><td>T</td><td>T</td><td>T</td><td>F</td><td>F</td><td>F</td></tr><tr><td>T</td><td>T</td><td>F</td><td>F</td><td>T</td><td>F</td></tr><tr><td>T</td><td>F</td><td>T</td><td>T</td><td>F</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td><td>T</td><td>T</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td><td>F</td></tr><tr><td>F</td><td>T</td><td>F</td><td>F</td><td>T</td><td>F</td></tr><tr><td>F</td><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>F</td><td>F</td><td>T</td><td>T</td><td>F</td></tr></table>
Recall that p ^ q is only true when p,q are true at the same time; otherwise, it's false.


Next, we can form the q v ~r column<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>r</td><td>~q</td><td>~r</td><td>p ^ ~q</td><td>q v ~r</td></tr><tr><td>T</td><td>T</td><td>T</td><td>F</td><td>F</td><td>F</td><td>T</td></tr><tr><td>T</td><td>T</td><td>F</td><td>F</td><td>T</td><td>F</td><td>T</td></tr><tr><td>T</td><td>F</td><td>T</td><td>T</td><td>F</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td><td>T</td><td>T</td><td>T</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td><td>F</td><td>T</td></tr><tr><td>F</td><td>T</td><td>F</td><td>F</td><td>T</td><td>F</td><td>T</td></tr><tr><td>F</td><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td><td>F</td></tr><tr><td>F</td><td>F</td><td>F</td><td>T</td><td>T</td><td>F</td><td>T</td></tr></table>
The disjunction p v q is only false when p,q are both false; otherwise, it's true.


And finally, here is the table that you'll have as the final answer.<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>r</td><td>~q</td><td>~r</td><td>p ^ ~q</td><td>q v ~r</td><td>(q v ~r) -> (p ^ ~q)</td></tr><tr><td>T</td><td>T</td><td>T</td><td>F</td><td>F</td><td>F</td><td>T</td><td>F</td></tr><tr><td>T</td><td>T</td><td>F</td><td>F</td><td>T</td><td>F</td><td>T</td><td>F</td></tr><tr><td>T</td><td>F</td><td>T</td><td>T</td><td>F</td><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td><td>T</td><td>T</td><td>T</td><td>T</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td><td>F</td><td>T</td><td>F</td></tr><tr><td>F</td><td>T</td><td>F</td><td>F</td><td>T</td><td>F</td><td>T</td><td>F</td></tr><tr><td>F</td><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td><td>F</td><td>T</td></tr><tr><td>F</td><td>F</td><td>F</td><td>T</td><td>T</td><td>F</td><td>T</td><td>F</td></tr></table>
It's definitely a lot to take in, especially because of the 8 rows to keep track of everything. 
For the last column, the statement p -> q is only false when p = T and q = F; otherwise, the conditional statement is true.

</font>