Question

I just need help with detailed explanations for b and c

Use the rules of inference and the laws of propositional logic to prove that each argument is valid. Number each line of your argument and label each line of your proof "Hypothesis" or with the name of the rule of inference used at that line. If a rule of inference is used, then include the numbers of the previous lines to which the rule is applied. (a) p q qr Solution (b) p(qAr) -q (c) (pAg)r r q Rule of inference Name Modus ponens -q. Modus tollens Addition .p v q Simplification Conjunction .pAg p q Hypothetical syllogism p v q Disjunctive syllogism p v q Resolution

Answer 1

Hello, please comment if there is any more requirement from the question. I will be happy to help. Please give a thumbs up if the solution helped you.

I am providing an explanation in detail. Please comment if you need step by step solution.
Answer B)

Here we are given two statements-

1 - p implies q and r.

2 - not of q is true

statement 2 implies that q is false.

this implies the AND of q and r is also false.

So, first statement becomes p implies a false statement.

hence if a statment implies something false than the statement itself is false.

There fore, the inference that the not of p is true.

Answer C)

Here again,

Statement 1 - p AND q implies r

Statement 2 - not of R is true

Statement 3 - q is true

here as not of r is true, this implies r is false.

now as first statement , p and q implies r becomes- p and q implies false.

So p and q should also be false.

for an AND to be false, either one or both of the p and q has to be false.

Here from statement 3 we know that q is true, so

p has to be false,

hence the inference that not of p is true.