[variation]

Prove validity of teh following conclusions:

Prove the validity of the following:

1. It rains, Ali is sick. Ali was not sick. ⊢ It didn't rain.

2. I like maths, I study. I study or don't make an exam. ⊢ I don't make an exam, I do not like Maths.

3. I study, I do not fail in maths. I don't play soccer, I study. I failed in maths. ⊢Therefore I played soccer.

My attempts at solutions so far:

1. ((p → q) Λ ¬q) → ¬p This statement is a tautology so this conclusion is true?

2. ((p → q) Λ (¬q V ¬r)) → (¬r → ¬p) This is not a tautology but has only one place that is false so is the argument true or not?

3. ((p → ¬q) Λ (¬r → p) Λ q) → r This is also a tautology so this argument is valid?

[Kolya]

To prove validity you either need to construct a truth table or a derivation of the conclusions from the premises.

[variation]

(Original post by Kolya)
To prove validity you either need to construct a truth table or a derivation of the conclusions from the premises.

I first changed the statements into a boolean algebra, then constructed a truth table and that is why I got as you can see there. 1 and 3 are a tautology, 2 is not. So my questions are first: is my change of statements into a boolean algebra correct? And if it is, does that mean only those statements that are a tautology are right, as is in 1 and 3, and the other as in 2, is not correct?