Prove $\sim$ (PV( $\sim$ p^q)) and $\sim$ P ^ $\sim q$ are logically equivalent by developing a series of logical equivalence.

solution :

Lets consider,

$\sim$ (pv ($\sim$ p ^ q))

= $\sim$ p ^ $\sim$ (\sim p ^ q) $\rightarrow$ Demorgan's Law

= $\sim$ p ^ p v $\sim$ q $\rightarrow$ Law of negation.

= ($\sim$ p ^ p) v ($\sim$ p ^ $\sim$ q) $\rightarrow$ Distributive law

= F V ($\sim$ p ^ $\sim$ q) $\rightarrow$ $\sim$ p ^ p = F

= $\sim$ p ^ $\sim$ q $\rightarrow$ Identity law.

Note : Verification by truth table.