Question 1204214: ~p ^ r
p v q
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∴ q v r
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
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.
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Method 2
Truth Table
| p | q | r | ~p | ~p ^ r | p v q | q v r | | T | T | T | F | F | T | T | | T | T | F | F | F | T | T | | T | F | T | F | F | T | T | | T | F | F | F | F | T | F | | F | T | T | T | T | T | T | | F | T | F | T | F | T | T | | F | F | T | T | T | F | T | | F | F | F | T | F | F | F |
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
https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson
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Method 3
Logic Derivation
| Number | Statement | Line(s) Used | Reason | | 1 | ~p ^ r | | | | 2 | p v q | | | | :. | q v r | | | | 3 | ~p | 1 | Simplification | | 4 | q | 2,3 | Disjunctive Syllogism | | 5 | q v r | 4 | Addition |
Refer to these rules of inference and replacement
https://logiccurriculum.com/2019/02/09/rules-for-proofs/
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