The negation of ( p ∧ ( - q ) ) ∨ ( - p ) is equivalent to

The given statement is $\left(p ∧ \left(~ q\right)\right) ∨ \left(~ p\right)$

Let us take \left(A \cap {B}^{c}\right) \cup \left({A}^{c}\right)$.</p><p>Now let us apply distributive law to the above statement.</p><p>$≡ \left(A \cup {A}^{c}\right) \cap \left({B}^{c} \cup {A}^{c}\right)$</p><p>$≡ \left(U\right) \cap \left({B}^{c} \cup {A}^{c}\right)$</p><p>$≡ \left({B}^{c} \cup {A}^{c}\right)$</p><p>Now the given statement \left(p ∧ \left(~ q\right)\right) ∨ \left(~ p\right)$ will be equivalent to $~ q ∨ ~ p$ or $~ p ∨ ~ q$.

Hence the given statement is equivalent to $~ p ∨ ~ q$,

Now negation of given statement by demorgan's law will be,

$~ \left(~ p ∨ ~ q\right) ≡ p ∧ q$