With proper justification, state the negation of the following. (p ↔ q) ∨ (~q → ~r)

EXERCISE 1.8Q 4.2   PAGE 21
Exercise 1.8 | Q 4.2 | Page 21

With proper justification, state the negation of the following.

(p ↔ q) ∨ (~q → ~r)


SOLUTION

~[(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]