Breaking down statement in discrete mathematics

Question

Anyone mind helping me break down this statement in discrete mathematics?

Having trouble understanding if I'm doing this correctly or not.

The original statement was:

~(~p -> q) -> (p v r)

And I went on:

Answer 1

You've done fine. A tip, though, following your third step: third statement:

$$\lnot (p\lor q) \rightarrow (p\lor r)\tag{3}$$

$$ \equiv \lnot\lnot (p\lor q) \lor (p \lor r)\tag 4$$

$$\equiv (p\lor q)\lor (p\lor r)\tag 5$$ $$\equiv p\lor q \lor p \lor r \tag{(6) associativity of $\lor$}$$

$$\equiv \underbrace{p \lor p}_{\large \equiv p} \lor q \lor r\tag{(7) commutativity of $\lor$}$$

$$\equiv p\lor q\lor r\tag 8 $$

Answer 2

Well that's correct. Now all you need as a final step is: $$ p \lor q \lor r $$ Since you have $$\lor$$ everywhere, you can eliminate brackets.