Negation of the Universal Subset definition

Question

I am trying to understand the use of quantifiers within the definition of a subset. The definition of a subset is:

$$ A \subseteq B \equiv \forall x(x \in A \rightarrow x \in B) $$

I am confused about, when you negate the statement of a subset, to an existential quantifier:

$$ \forall x(x \in A \rightarrow x \in B) \equiv \exists x(x \in A \land x \notin B) $$

How do these two have the same meaning? I am struggling to get to grips with the negated version.

To clarify, I understand the conversion and can do that applying the relevant laws but in idiomatic English it makes no sense. For Universal Quantifier, it means For all $x$, if $x$ is a member of $A$, then $x$ is a member of $B$. However, for a Existential Quantifier: There exists some value of such that $x$ is a member of $A$ but is not a member of $B$? That surely goes against the principle of a subset?

Answer 1

Answer: My understanding was incorrect. When you negate a statement that has a quantifier, you look to say the opposite of the original meaning. Quite surprise how I forgot about this. Please refer to the comments below.

Answer 2

Well, the general representation of an all-quantified statement is $\forall x (P(x)\Rightarrow Q(x))$, i.e. $\forall$ combined with $\Rightarrow$, while the general representation of an existential quantified statement is $\exists x (P(x)\wedge Q(x))$, i.e., $\exists$ and $\wedge$.

The above negation is perfectly fine.

Answer 3

Here is used 2 formula: $$(A \Rightarrow B)= (\neg A \lor B)$$ $$\neg \forall x(R(x))= \exists x (\neg R(x))$$ Joining these formulas with negation $\neg (C \lor D)= (\neg C) \land (\neg D)$ gives answer: $$\neg \forall x (x \in A \Rightarrow x \in B)=\\= \exists x (\neg (x \in A \Rightarrow x \in B)) =\\= \exists x (\neg (x \notin A \lor x \in B )) =\\= \exists x(x \in A \land x \notin B)$$

By words: exist x such that it is in A and(but) it is not in B.