Question 1204214
<font color=black size=3>
Method 1


Informal approach:


If ~p ^ r is the case then ~p is the case and r is also the case.


Using ~p and p v q, we find that q comes out of that.
p v q means "p or q". We then know that p isn't the case since ~p is, so q must be the case.


If q is true then so is q v r. 
In fact, we can replace r with any other logical statement. There's nothing really special about the r.


----------------------------


Method 2


Truth Table
<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>r</td><td>~p</td><td>~p ^ r</td><td>p v 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>T</td><td>T</td></tr><tr><td>T</td><td>T</td><td>F</td><td>F</td><td>F</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>T</td><td>F</td><td>F</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</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>T</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>F</td><td>T</td><td>T</td></tr><tr><td>F</td><td>F</td><td>T</td><td>T</td><td>T</td><td>F</td><td>T</td></tr><tr><td>F</td><td>F</td><td>F</td><td>T</td><td>F</td><td>F</td><td>F</td></tr></table>
Notice that we do not have a situation where all premises are true but the conclusion is false.
Therefore, this is a valid argument.


Here is a review of various truth table rules
<a href = "https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson">https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson</a>


----------------------------


Method 3


Logic Derivation
<table border = "1" cellpadding = "5"><tr><td>Number</td><td>Statement</td><td>Line(s) Used</td><td>Reason</td></tr><tr><td>1</td><td>~p ^ r</td><td></td><td></td></tr><tr><td>2</td><td>p v q</td><td></td><td></td></tr><tr><td>:.</td><td>q v r</td><td></td><td></td></tr><tr><td>3</td><td>~p</td><td>1</td><td>Simplification</td></tr><tr><td>4</td><td>q</td><td>2,3</td><td>Disjunctive Syllogism</td></tr><tr><td>5</td><td>q v r</td><td>4</td><td>Addition</td></tr></table>

Refer to these rules of inference and replacement
<a href="https://logiccurriculum.com/2019/02/09/rules-for-proofs/">https://logiccurriculum.com/2019/02/09/rules-for-proofs/</a>

</font>