Tableau exercises Propositional logic

The following sequents provide practice in the art of constructing tableaux. Answers are given, but of course the idea is to come up with solutions of your own before looking them up. As ever, the answers are not unique: in some cases, alternative tableaux are just as good as the ones given.

p ∨ q, ¬q ∨ r ⊢ p ∨ r	Try it	Answer
p → ¬q ⊢ q → ¬p	Try it	Answer
p → ¬q ⊢ ¬p → q	Try it	Answer
(p ∧ q) → ¬r ⊢ (p ∧ r) → ¬q	Try it	Answer
¬p ∨ ¬q ⊢ ¬(p ∨ q)	Try it	Answer
¬p → q ⊢ ¬(p → q)	Try it	Answer
(p → q) → q ⊢ (q → p) → p	Try it	Answer
(p → q) → r ⊢ p → (q → r)	Try it	Answer
p → (q → r) ⊢ (p → q) → r	Try it	Answer
p → ¬q, q → ¬p ⊢ ¬(p ↔ q)	Try it	Answer
(p ∧ q) → ¬ r ⊢ (r ∧ ¬ p) → ¬q	Try it	Answer
⊢ (p → (q → r)) → ((s → q) → (p → (s → r)))	Try it	Answer
(p ∧ q) → s, (p ∧ r) → s ⊢ (p ∧ ¬ s) → (q ↔ r)	Try it	Answer
p ↔ (q ∨ r) ⊢ (p ↔ q) ∨ (p ↔ r)	Try it	Answer
p ↔ (q ∧ r) ⊢ (p ↔ q) ∧ (p ↔ r)	Try it	Answer
p ↔ (q ↔ r) ⊢ (p ↔ q) ↔ (p ↔ r)	Try it	Answer
p ↔ (q ↔ r) ⊢ (p ↔ q) ↔ r	Try it	Answer

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