Exercise 1.8 (2)

2. Using the rules of negation, write the negations of the following :

i) (p → r) ∧ q
Ans: ~[(p→r)∧q] ≡ ~(p → r) ∨ ~q (Negation of Conjuction)
                              ≡ (p ∧ ~r) ∨ ~q (Negation of Implication)

ii) ∼ (p ∨ q) → r
Ans: ~ [ ~ (p ∨ q) → r] ≡ ~ (p ∨ q) ∧ ~ r (Negation of Implication)
                                        ≡ ( ~p ∧ ~q) ∧ ~ r (Negation of disjunction)

iii) (∼ p ∧ q) ∧ (∼ q ∨ ∼ r)
Ans: ~ [(~p ∧ q) ∧ (~q ∨ ~r)] ≡ ~(~p ∧ q) ∨ ~(~q ∨ ~r)……(Negation of conjuction) 
                                                   ≡ [ ~(~p) ∨ ~q] ∨ [~(~q) ∧ ~(~r)]……(Negation of conjuction and disjunction) 
                                                   ≡ (p ∨ ~q) ∨ (q ∧ r)                       (Negation on Negation)