Is 'Some of x are true' a negation of 'All of x are true'? I don't think this necessarily means there exists a false x, just that at least some x are true.
Is my logic correct here in assuming this is NOT a negation? 
 A: The negation of "all $x$ are true" is "some $x$ is false (in the sense that there exists $x$ false, at least one is false)".
A: The $x$s that quantifiers bind are not propositions, they're things, so it doesn't make sense to say "$x$ is true" or "$x$ is false". You need some predicate or formula $P(x)$ involving $x$, whose interpretation is true or false for particular values of $x$.
The negation of $\forall x P(x)$ is $\exists x \neg P(x)$, as this is equivalent to $\neg\forall x P(x)$. In English, the negation of "all $x$ are $P$" is "some $x$ are not $P$".
"Some $x$ are $P$" is the negation of "no $x$ are $P$". The latter is equivalent to the less idiomatic "all $x$ are not $P$".
All of this is true because of the basic equivalence relating quantifiers and negation: $\forall \equiv \neg\exists\neg$, from which it follows that $\neg\forall \equiv \exists\neg$, $\neg\exists \equiv \forall\neg$, and $\exists \equiv \neg\forall\neg$.
A: $$\left(\bigcap A_i\right)' = \bigcup(A_i') \neq \bigcup A_i$$
