Deductive Reasoning in Propositional Logic 73 2.4.4 Show that for any formulae A, B : (a) If A ND B then ¬B ND ¬A (b) If ¬B ND ¬A then A ND B (c) If ¬B ND ¬A then A ND B 2.4.5 Derive each of the following for any formulae A, B, C (Hint: for some of these exercises use derivations of previous exercises.) (a) A → C , B → C ND (A ∨ B ) → C (b) (A → B ) ∨ C ND A → (B ∨ C ) (c) (A ∨ B ) → C ND (A → C ) ∧ (B → C ) (d) A → (B ∨ C ) ND (A → B ) ∨ C (e) A → ¬A ND ¬A (f) ¬A → A ND A (g) ¬A → B , ¬A → ¬B ND A (h) ¬A → B ND A ∨ B (i) (¬A ∧ B ) → ¬C , C ND A ∨ ¬B (j) ¬(A ∧ B ) ND ¬A ∨ ¬B (k) ¬(A ∨ B ) ND ¬A ∧ ¬B (l) ¬A ∨ ¬B ND ¬(A ∧ B ) (m) ¬A ∧ ¬B ND ¬(A ∨ B ) (n) ¬(A → B ) ND A ∧ ¬B (o) A ∧ ¬B ND ¬(A → B ) 2.4.6 Derive the following logical consequences in Natural Deduction (a) p → ¬q, ¬r ∨ p |= (¬p → r ) → ¬q (b) ¬p → ¬q, ¬(p ∧ ¬r), ¬r |= ¬q (c) ¬p ∨ ¬r, r → ¬q, ¬q → p |= ¬r (d) (¬p ∧ q ) → ¬r, r |= p ∨ ¬q 2.4.7 Formalize the following propositional arguments and prove their logical correctness by using Natural Deduction (a) Alice is at home or at work Alice is not at home Therefore, Alice is at work (b) Nina will go to a party or will not go to the office Nina will not go to a party or will not go to the office Therefore, Nina will not go to the office (c) Victor is lazy or Victor is clever If Victor is not successful then Victor is lazy Victor is not lazy Therefore, Victor is clever and successful