Question 1150010
<font face="times" color="black" size="3">
p and q are any truth value statements. In other words, they are a variable that holds T or F
T = true
F = false


Start with a table showing off the various truth value combinations of p and q
<table border = "1" cellpadding = "5">
<tr><td>p</td><td>q</td></tr>
<tr><td>T</td><td>T</td></tr>
<tr><td>T</td><td>F</td></tr>
<tr><td>F</td><td>T</td></tr>
<tr><td>F</td><td>F</td></tr>
</table>


Then add on a ~q column which is the complete opposite of what the q column shows (true flips to false, and vice versa)
<table border = "1" cellpadding = "5">
<tr><td>p</td><td>q</td><td>~q</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>F</td><td>T</td><td>F</td></tr>
<tr><td>F</td><td>F</td><td>T</td></tr>
</table>
We'll use this column later, but for now we'll add on a p v q column next.


The logical disjunction of two truth values is only false when both pieces are false. In other words, p v q = F when both p = F and q = F at the same time. Otherwise, p v q = T
<table border = "1" cellpadding = "5">
<tr><td>p</td><td>q</td><td>~q</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>T</td><td>T</td></tr>
<tr><td>F</td><td>T</td><td>F</td><td>T</td></tr>
<tr><td>F</td><td>F</td><td>T</td><td>F</td></tr>
</table>


Next we'll have a column for ~q --> p. This conditional is only false when ~q = T and p = F. So if we had T --> F, then that whole thing is false. Otherwise, the statement is true.
<table border = "1" cellpadding = "5">
<tr><td>p</td><td>q</td><td>~q</td><td>p v q</td><td>~q -> p</td></tr>
<tr><td>T</td><td>T</td><td>F</td><td>T</td><td>T</td></tr>
<tr><td>T</td><td>F</td><td>T</td><td>T</td><td>T</td></tr>
<tr><td>F</td><td>T</td><td>F</td><td>T</td><td>T</td></tr>
<tr><td>F</td><td>F</td><td>T</td><td>F</td><td>F</td></tr>
</table>


Finally, the last step is to combine the columns p v q and ~q --> p
Let A = p v q and B = ~q --> p. The format we want is A --> B
<table border = "1" cellpadding = "5">
<tr><td>p</td><td>q</td><td>~q</td><td>p v q</td><td>~q -> p</td><td>(p v q) -> (~q -> p)</td></tr>
<tr><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>T</td><td>T</td><td>T</td><td>T</td></tr>
<tr><td>F</td><td>T</td><td>F</td><td>T</td><td>T</td><td>T</td></tr>
<tr><td>F</td><td>F</td><td>T</td><td>F</td><td>F</td><td>T</td></tr>
</table>
which is what the full completed truth table looks like


Note the last column has nothing but T. Each possible outcome leads to (p v q) --> (~q --> p) being a true statement. 


<font color="red" size="4">Answer: Yes it is a tautology</font>

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