Prove validity of a cononclusion

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The discussion focuses on the validity of three logical conclusions based on given premises. For the first argument, the conclusion is valid as it forms a tautology. The second argument is more complex and not a tautology, leading to uncertainty about its truth. The third argument is confirmed as valid since it also results in a tautology. Overall, the conclusions drawn indicate that the first and third arguments are correct, while the second is deemed false.
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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?
 
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Just a question on number 2. If q is the proposition, I study, and r is the proposition, i make an exam. Then why is "I study or don't make an exam", \neg p \vee \neg r?
 
daveyinaz said:
Just a question on number 2. If q is the proposition, I study, and r is the proposition, i make an exam. Then why is "I study or don't make an exam", \neg p \vee \neg r?

That was my mistake. Thanks for the correction. It should be \ p \vee \neg r?

So are my attempts at solutions correct? Since 1 and 3 are a tautology, they are right. The 2 is false, because it is not a tautology?
 
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