Negation & Simplification of Proposition

Question

I am attempting to negate and simplify this proposition:

$ ∃_x((p(x) ∧ q(x)) $

My attempt to solve this:

$ ¬[∀_x(¬(p(x) ∧ q(x)) $

$ ¬[∀_x(¬(¬p(x) ∨ q(x)) $

$ ¬[∀_x(p(x) ∨ ¬q(x)) $

I'm not sure if I have correctly negated and simplified this proposition, so help would be greatly appreciated.

Answer 1

No, you didn't do that correctly.

First of all, you need to start with just putting a negation in front of the statement you have, i.e. start with:

$$\neg \exists x (p(x) \land q(x))$$

Then use the equivalence that:

$$\neg \exists x \ \phi \Leftrightarrow \forall x \ \neg \phi$$

to get:

$$\forall x \ \neg (p(x) \land q(x))$$

Now, at this point you made another mistake, for you went from

$$\neg (p(x) \land q(x))$$

to:

$$\neg p(x) \lor q(x)$$

but it should be:

$$\neg p(x) \lor \color{red}{\neg} q(x)$$

In other words, you get:

$$\forall x (\neg p(x) \lor \neg q(x))$$

Answer 2

Note that for Distributivity law

$$¬[∃_x((p(x) ∧ q(x))]$$ is equivalent to $$∀_x(¬p(x) ∨ ¬ q(x))$$