Question

Write the negation of the following statement using the rule of negation ( p ⇒ q ) and r.

Answer 1

Given statement is (P implies q ) and r

i.e., (p ⇒ q) ∧ r

or (~ p ∨ q) ∧ r. (\(\because\) p ⇒ q ≡ ~ p ∧ q)

Now, the negation of (p ⇒ q) ∧ r is

~((~p ∨ q) ∧ r)

or ~ (~p ∨ q) ∨~ r (\(\because\) ~ (p ∧ q) ≡ ~ p ∨ ~ q)

or (~ (~ p) ∧ ~ q) ∨ (~ r) (\(\because\) ~(p ∨ q) ≡ ~p ∧ ~ q)

or (p ∧ ~p) ∨ ~r (\(\because\) ~ (~p) ≡ p)

or p ∧ ~q ∨ ~r

or p ∧ ~(q ∧ r)