If p and q and q are two statements , prove that (i) ` ~( p vee q) -= (~p) ^^ (`q) " " (ii) ~( p ^^ q) = (~p) v(~q) `

Explanation

The truth tables of ` ~ ( p vee q) and ( ~ p) ^^ ( ~q)` are are as given below :
Truth table for ` ( ~ p vee q)`

Truth table for ` ( ~ p) ^^ (~q)`

We observe that the last columns of the truth tables of `~ ( p vee q) and ( ~ p) vee q) ` are identical .
Hence, `( ~ p vee q ) -= ( ~p) ^^ ( ~q)`
(ii) The truth table of `( ~ p ^^ q) and ( ~p) vee ( ~q )` are so given below :
Truth table for ` ~ ( p ^^ q)`

Truth table for `~ p vee ~ q`

Clearly, entries in the columns of the truth tables of ` ~ ( p ^^ q) and ( ~ p) vee ( ~ q)` are same .
Hence , ` ~ ( p ^^ q) -= ( ~ p) vee ( ~q) `