Question 1152672
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p is some logical statement which is either true (T) or false (F)
q is a similar story.


Here are all the ways to have two combos of either T or F
TT
TF
FT
FF
Note how the first column has TT then FF
The second column alternates TFTF


So this is how we form the columns for p and q
<table border = "1" cellpadding = "5">
<tr><td>p</td><td>q </td><td>~p</td><td>~p V q</td></tr>
<tr><td>T</td><td>T</td><td>&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>T</td><td>F</td><td>&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>F</td><td>T</td><td>&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>F</td><td>F</td><td>&nbsp;</td><td>&nbsp;</td></tr>
</table>


Then we negate the first column of values. This means we flip from T to F and vice versa. This gets us the column of ~p truth values.
<table border = "1" cellpadding = "5">
<tr><td>p</td><td>q </td><td>~p</td><td>~p V q</td></tr>
<tr><td>T</td><td>T</td><td>F</td><td>&nbsp;</td></tr>
<tr><td>T</td><td>F</td><td>F</td><td>&nbsp;</td></tr>
<tr><td>F</td><td>T</td><td>T</td><td>&nbsp;</td></tr>
<tr><td>F</td><td>F</td><td>F</td><td>&nbsp;</td></tr>
</table>


Then we apply the disjunction operation to the columns of ~p and q
The result of a disjunction is false only when both ~p and q are false; otherwise, the result is true


This is the fully filled out truth table
<table border = "1" cellpadding = "5">
<tr><td>p</td><td>q </td><td>~p</td><td>~p V q</td></tr>
<tr><td>T</td><td>T</td><td>F</td><td>T</td></tr>
<tr><td>T</td><td>F</td><td>F</td><td>F</td></tr>
<tr><td>F</td><td>T</td><td>T</td><td>T</td></tr>
<tr><td>F</td><td>F</td><td>F</td><td>T</td></tr>
</table>
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