Is this proof correct? Can you help me with this propositional logic exercise?

Question

Problem:

1. ~P
2. Q v (R . P) /Q

Answer:

3. ~ P v ~ R 1, Add
4. ~ (P . R) 3, De Morgan
5. ~ (R . P) 4, Commutation
6. Q 2, 5 Disjunctive Syllogism

My textbook presents another solution

Answer 1

The proof checker associated with the textbook forallx, linked to below, are ways you can check your proof. The steps may not line up exactly but they can show that a proof is very likely correct.

Here is the result of putting your proof in the proof checker:

Note that I did not use the Commutation rule as you did on line 5. It is not in this proof checker. Rather I added the "¬R" on the left side in line 3. Although this is a difference it is minor and you should still feel confident in the correctness of your proof.

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Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

Answer 2

Your direct proof is fine.

It could also be proved indirectly.

Suppose (for RAA) that Q is false. Then ( by disjunctive syllogism) you have (R&P). Therefore P is true ( by & elim). But premise (1) says that P is false. hence a contradiction. By negation introduction, you can infer from this that : Q is true.

Answer 3

Your proof is okay, but uses derived rules of inference. You may be expected to only use the fundamental rules.

Well, you have a disjunction in one premise, so the rule of disjunction elimination ($\vee\mathrm E$) suggests itself. That takes the form of a proof by cases.

$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline #2\end{array}}\fitch{\lnot P\\Q\vee(P\wedge R)}{\fitch{Q}{}\\\fitch{P\wedge R}{~\vdots\\Q}\\Q}$

Well, clearly $Q$ is trivially derived in the left case. I am sure you can also do so in the right case.