I am very new to this and I can't figure out the steps to prove that these reasonings are valid. Thank you for your help.

p → (q→r)

(¬p ∨ ¬q) → r

p → (r ∨ q) 
(q ∨ s ) → t
r → s
p ∨ r

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  • How have you tried to show these? It might also help to know what textbook they came from. If you do not have a proof checker, you might try this one: proofs.openlogicproject.org – Frank Hubeny Mar 25 at 22:40
  • This is unanswerable unless you specify the rules you're allowed to use. Also please say what you have tried. – Eliran Mar 25 at 23:55
  • I don't have any specification about the rules, I guess all rules are allowed ? – MMM Mar 26 at 0:01

Hint for 1, using Natural Deduction rules.

1) p --- premise

2) ¬r --- premise

3) p → (q → r) --- premise

4) q --- assumed [a]

5) (q → r) --- from 1) and 3) by (→E)

6) r --- from 4) and 5) by (→E)

7) contradiction with 2) and 6)

8) ¬q --- from 4) and 7), by (⊥E), discharging assumption [a].

Regarding 2, check it with truth table.

Regarding 3, use "nested" (∨E).

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