Nothing is in the correct place and is in excellent condition. ∀ x ¬(C(x) ∧ E(x)).?

Question

Nothing is in the correct place and is in excellent condition.

Generally, this statement ∀ x ¬(C(x) ∧ E(x)). is equivalent to ¬∃ x (C(x) ∧ E(x)).

I wonder if I can use De Morgan law to do the followings.

∀ x ¬(C(x) ∧ E(x)) as ∀ x (¬C(x) ∨ ¬E(x))

thanks.

Answer 1

Yup, that's correct! All four expressions in your question mean the same thing.

Answer 2

Yes you can. Equivalences hold between formulas in general; they don't have to be sentences.

So, it is true that:

$\neg (C(x) \land E(x)) \Leftrightarrow \neg C(x) \lor \neg E(x)$

It is also true that any equivalent formulas can be substituted for each other in larger formulas, while still retaining equivalence (This is called the Substitution Principle for Equivalent Formulas).

So, since you already established that:

$\neg \exists x (C(x) \land E(x)) \Leftrightarrow \forall x \neg (C(x) \land E(x)$

we can fill in the first equivalence, and obtain:

$\neg \exists x (C(x) \land E(x)) \Leftrightarrow \forall x (\neg C(x) \lor \neg E(x))$