Prove using formal methods

Question

Prove using formal methods

∀x ¬(P(x) ∧ Q(X)) --> ∀x(¬P(x) v ¬Q(x))

So I tried this problem

1. ∀x ¬(P(x) ∧ Q(X)) P

2. ∀x ¬P(x) v ¬Q(X) Distributing the not. Can I do something like this? I thought the answer was supposed to be much longer than this.

Answer 1

I think your proof defeats the point, since your second step just assumes that the statement is true. Try this- assume that for any $x$, $P(x) \land Q(x)$ is false. Then we can't have $P(x)$ be true and $Q(x)$ be true, otherwise, $P(x) \land Q(x)$ would be true, so at least one of the two is false (essentially, we're looking at a truth table for $P$ and $Q$ and eliminating one possibility, where both are true. The other three possibilities have at least one of the two false.) The rest should be apparent.