Distributing quantifiers in predicate logic?

Question

So I have two questions.

1) If I were to distribute the existential quantifier, would these two propositions be equivalent?

∃x(Blue(x) ∧ Circle(x)) = ∃x(Blue(x)) ∧ ∃x(Circle(x))

2) would the first proposition indicate that there is an x that is both Blue(x) and a Circle(x) or could the x of Blue(x) be distinct of the x that satisfies Circle(x)?

Thanks in advance!!

Answer 1

would the first proposition indicate that there is an x that is both Blue(x) and a Circle"(x) or could the x of Blue(x) be distinct of the x that satisfies Circle(x)?"

The first proposition says that there is an $x$ such that both $Blue(x)$ and $Circle (x)$ holds simultaneously.

The second proposition says that there is an $x$ such that $Bed(x)$ holds AND there is an $x$ such that $Circle(x)$ holds. These $x$ values can be different.

So these propositions are not logically equivalent.

Answer 2

$\def\Blue{\operatorname{\textsf{Blue}}}\def\Circle{\operatorname{\textsf{Circle}}}\exists x~(\Blue(x)\wedge\Circle(x))$ specifies that something simultaneously satisfies both Blue and Circle predicates.

$\exists x~\Blue(x)\wedge\exists x~\Circle(x)$ specifies that something satisfies the Blue predicate, and something satisfies the Circle predicate, but it does not guarantee that the same thing will do both.

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