Turn Disjunctive Normal Form into only AND and NOT statements

Question

I'm working on implementing logic gates and want to implement an OR gate using only AND and NOT gates.

I came up with this canonical form for the OR function. $$Or(x, y) = (\lnot{x} \land y) \lor (x \land \lnot y) \lor (x \land y)$$ $$= y \lor (x \land \lnot y)$$

Then, working with DeMorgan's laws, I tried (following from above):

$$= \lnot(\lnot y) \lor \lnot(\lnot(x \land \lnot y))$$ $$= \lnot(\lnot y \land \lnot(x \land \lnot y))$$

But I can't figure out how to simplify it any further. Am I missing something or is this it?

Answer 1

I would just start with the double negation and then apply De Morgan's law:

$$x\lor y = \lnot(\lnot( x\lor y)) = \lnot (\lnot x \land \lnot y)$$