Question

-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. For the

arguments stated in English, transform them into propositional logic first.

a)

(10 marks)

(p ∧ q) ⇒r

¬r

q

---------------

∴¬p

b)

If I go rock climbing, then I am sore.

If I am sore, I stay in bed all day.

I did not stay in bed all day.

-----------------------------------------

∴I did not go rock climbing

Answer 1

a)

(p ∧ q) ⇒r

for this we assume that

1. $(p \wedge q) \rightarrow r$

2. $p \wedge q$ be true then we can infer r using

3. using modus pones or implication elimination on 2

4. $\neg r$ premise

5. Contradiction

6. Hence $\neg (p \wedge q)$

7. $\neg (p \wedge q) \equiv \neg p \vee \neg q$

8. $\neg p$ as q is true

b)

Let's say

I go rock climbing is p

I am sore is q

I stay in bed all day is r

then the statements are

1. $p \rightarrow q$

2. $q \rightarrow r$

3. $\neg r$

Then we can see that

4. Assume p

5. q using -> elimination on 1 or modus pones

6. r using -> elimination on 2 or modus pones

7. $\neg r$ copy 3

8. Contradiction

9. $\neg p$ Proof by contradiction

Do give a thumbs up as it matters a lot and in case there are doubts leave a comment.