I am reading "How to prove it" by Velleman and am currently in the chapter about quantifiers. I don't understand why $$ \exists x \in A (P(x)) $$ is equivalent to $$\exists x~(x\in A \wedge P(x)), $$ whereas $$ \forall x \in A (P(x)) $$ is equivalent to $$ \forall x~ (x\in A \rightarrow P(x)). $$
I don't understand why for the existential quantifier the two statements are "anded" whereas for the universal quantifier the conditional is used.
$\exists x\in A~\big(P(x)\big)$ versus $\exists x~\big(x\in A\land P(x)\big)$
Some apples are peeled. Somethings are apples and peeled.
$\forall x\in A~\big(P(x)\big)$ versus $\forall x~\big(x\in A\to P(x)\big)$
All apples are peeled. All things are peeled when they are apples.