For any two statements p and q, the negation of the expressionp∨(~p ∧ q)  is:

The negation of a statement p (read: not p) is the statement ~ p which asserts that p is not true. Sometimes it is helpful to think of ~ p as asserting “it is not the case that p is true”. Thus, ~ p is false when p is true, and true when p is false.

Notice that “~” is not a logical connective. It does not join two statements together. Instead, it applies to a single (possibly compound) statement. Negation has precedence over logical connectives. Thus ~ p ∨ q means (~ p) ∨ q. The negation of ~ p is the statement with the opposite truth value as ~ p, thus ~ (~p) is just another name for p.

The negation of p ∧ q asserts “it is not the case that p and q are both true”. Thus, ~ (p ∧ q) is true exactly when one or both of p and q is false, that is, when ~ p ∨ ~ q is true. Similarly, ~ (p ∨ q) can be seen to the same as ~p ∧ ~q.

So, ~ (p ∨(~p ∧ q)) = ~ (~ p ∧ q) ∧ ~ p

= (~q ∨ p) ∧ ~p

= ~ p ∧ (p ∨ ~q)

= (~q ∧ ~p) ∨ (p ∧ ~p)

= (~p ∧ ~q)