Exercise 1.8 (4)

4. With proper justification, state the negation of each of the following.

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

ii) (p ↔ q) ∨ (∼ q → ∼ r)
Ans: 
~[(p ↔ q) ∨ (∼ q → ∼ r)]
≡ ~ (p ↔ q) ∧  ~ (∼ q → ∼ r)]………….[Negation of Disjunction]
≡ [(p ∧ ~q) ∨ (q ∧ ~p)] ∧ ~ [~q → ~ r]…………..[Negation of double Implication] 
≡[(p ∧ ~q) ∨ (q ∧ ~p)] ∧ [~q ∧ ~(~r)]………..[Negation of Implication] 
≡[(p ∧ ~q) ∨ (q ∧ ~p)] ∧ (~q ∧ r)……………….[Negation of negation]

iii) (p → q) ∧ r
Ans: 
~[(p → q) ∧ r]
≡ ~(p → q) ∨ ~r……….[Negation of Conjunction]
≡ (p ∧ ~q) ∨ ~r………..[Negation of implication]