What is the mathematical notation for an intersection set?

Question

I know that we use the notation A∩B = ∅ to express that the sets A and B are disjoint. But could we express it like this?

∀x( x∈A ∧ x∈B )

Because a disjoint set is actually an empty intersection, and since we can express an intersection like this ∀x(x∈A ∧ x∈B), I thought that it would be the same.

What do you think?

Thank you.

Answer 1

We have that $x \in A \cap B \text { iff } x ∈ A \land x ∈ B$.

Thus, to say that $A \cap B = \emptyset$ is to say that no $x$ belongs to $A \cap B$.

Thus, to say that $A$ and $B$ are disjoint, is to say that there are no common elements:

$∀x (x∉A \lor x∉B)$,

or, which is the same:

$∀x (x \in A \to x ∉ B)$.

Answer 2

Writing the expression $∀x( x∈A ∧ x∈B )$ in English, it says

For any $x$, that $x$ belongs to both $A$ and to $B$

which is to say, both $A$ and $B$ contain all possible elements. I don't think that's what you want to say.

Perhaps you forgot a "not" in there. The expression $∀x\lnot( x∈A ∧ x∈B )$ would mean

For any $x$, that $x$ cannot belong to both $A$ and $B$ at the same time

which is the same as $A\cap B = \varnothing$. However, we can get this exact expression even without trying to come up with something from scratch ourselves. Recall that the empty set $\varnothing$ is the unique set with the property $$ \forall x\lnot(x\in \varnothing) $$ and by definition of $\cap$, we have $$ x\in A\cap B \iff x\in A\land x\in B $$ We get $$ A\cap B = \varnothing \iff \forall x\lnot(x\in A\cap B)\iff\forall x\lnot(x\in A\land x\in B) $$

Answer 3

No, but you can do:

$$\neg \exists x (A(x) \land B(x))$$