Question 1204272: solve plz? with rules of inference and replacement...
1. (~D + A) v (~D + R)
2. D v ~A /∴ K > R
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1. ~(G + M)
2. M v ~G /∴ ~G
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I know y'all only answer two questions...but it's worth a shot to ask...If not, thank you anyways!!
1. P = (R + S)
2. P /∴ R = S
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Normally I would mention that the rule of this website is "one question per post", but the problems aren't too lengthy here.
Problem 1
| Number | Statement | Line(s) Used | Reason | | 1 | (~D + A) v (~D + R) | | | | 2 | D v ~A | | | | :. | K > R | | | | 3 | ~D + (A v R) | 1 | Distribution | | 4 | ~D | 3 | Simplification | | 5 | ~A | 2,4 | Disjunctive Syllogism | | 6 | A v R | 3 | Simplification | | 7 | R | 6,5 | Disjunctive Syllogism | | 8 | R v ~K | 7 | Addition | | 9 | ~K v R | 8 | Commutation | | 10 | K > R | 9 | Material Implication |
Refer to these rules of inference and replacement
https://logiccurriculum.com/2019/02/09/rules-for-proofs/
The notation is slightly different. That reference sheet has a center dot instead of a plus sign. Also, that sheet uses a horsehoe instead of a greater than sign.
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Problem 2
| Number | Statement | Line(s) Used | Reason | | 1 | ~(G + M) | | | | 2 | M v ~G | | | | :. | ~G | | | | 3 | ~G v ~M | 1 | De Morgan's Law | | 4 | ~G v M | 2 | Commutation | | 5 | G > ~M | 3 | Material Implication | | 6 | G > M | 4 | Material Implication | | 7 | ~M > ~G | 6 | Transposition | | 8 | G > ~G | 5,7 | Hypothetical Syllogism | | 9 | ~G v ~G | 8 | Material Implication | | 10 | ~G | 9 | Tautology |
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Problem 3
This logical argument is invalid
Proof of this is shown in the truth table below.
| | | | Premise | Premise | Conclusion | | P | R | S | R+S | P = (R+S) | P | R = S | | T | T | T | T | T | T | T | | T | T | F | T | T | T | F | | T | F | T | T | T | T | F | | T | F | F | F | F | T | T | | F | T | T | T | F | F | T | | F | T | F | T | F | F | F | | F | F | T | T | F | F | F | | F | F | F | F | T | F | T |
Here is a review of various truth table rules
https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson
Focus on row 2. I've highlighted this in red
The premises P = (R+S) and P are true, but the conclusion is false.
Any time there is a situation with all true premises but a false conclusion, it means the argument is invalid.
Therefore, you would not be able to find a logical derivation when presented with these premises and conclusion. Attempting to do so will have you endlessly searching for something that doesn't exist. It's possible that your teacher made a typo somewhere.
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