Question 1204701: Use conditional proof (CP) together with the eight rules of implication and ten rules of replacement to prove that they are valid. Be sure to include the justification for each line, and offset lines as appropriate for conditional proof.
1. A ⊃ (B ⊃ (C • ~D))
2. (B v E) ⊃ (D v E) /(A • B) ⊃ (C • E)
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
The idea is to assume the antecedent A & B is the case, and show it leads to the consequent C & E
Below is a conditional proof.
| Number | | Statement | Line(s) Used | Reason | | 1 | | A --> (B --> (C & ~D)) | | | | 2 | | (B v E) -> (D v E) | | | | :. | | (A & B) --> (C & E) | | | | 3 | A & B | | Assumption for Conditional Proof | | 4 | B & A | 3 | Commutation | | 5 | A | 3 | Simplification | | 6 | B | 4 | Simplification | | 7 | B --> (C & ~D) | 1,5 | Modus Ponens | | 8 | C & ~D | 7,6 | Modus Ponens | | 9 | ~D & C | 8 | Commutation | | 10 | C | 8 | Simplification | | 11 | ~D | 9 | Simplification | | 12 | B v E | 6 | Addition | | 13 | D v E | 2,12 | Modus Ponens | | 14 | E | 13,11 | Disjunctive Syllogism | | 15 | C & E | 10,14 | Conjunction | | 16 | | (A & B) --> (C & E) | 3 - 15 | Conditional Proof |
We have shown that assuming (A & B) in line 3 leads to (C & E) in line 15.
Therefore, the premises given to us let us conclude that (A & B) --> (C & E) must be the case.
More info:
Logic Rules of Inference and Replacement
A similar logic problem
https://www.algebra.com/algebra/homework/Proofs/Proofs.faq.question.1204702.html
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