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. Slightly trickier but this was my attempt. ((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?