Question

Prove that (a ∧ (b → ¬a)) → ¬b is a tautology.

Answer 1

$(a ∧ (b → ¬a)) → ¬b \\\equiv (a ∧ (¬b∨ ¬a)) → ¬b \\\equiv ((a ∧ ¬b)∨(a ∧¬a)) → ¬b \\\equiv ((a ∧ ¬b)∨F) → ¬b \\\equiv (a ∧¬b) → ¬b$

$\\\equiv ¬(a ∧¬b) ∨ ¬b \\\equiv (¬a ∨b) ∨ ¬b \\\equiv ¬a ∨ (b ∨¬b) \\\equiv ¬a ∨ T \\\equiv T$

Thus, it is a tautology.

Hence, proved.