Logical Definition of Two Disjoint Sets

[MushroomPirat]

Homework Statement

I'm self-studying Daniel Velleman's How to Prove It, and I'm wondering if there is some way to write that "Sets A and B are disjoint" using symbols, other than the $A \cap B = \emptyset$ given in the book.

Homework Equations

The Attempt at a Solution

I'm thinking that if A and B are disjoint, then for every element x, where if x is an element of A, then x is not an element of B. Or:

$\forall x (x\in A \rightarrow x \notin B)$

Thanks !Edit: Right after I posted this, I realized I could just prove the equivalency, and it turned out to be a pretty trivial proof (if I did it right). So, I think this thread can be deleted now.

 

[lippyka]

A \cap B = \emptyset \begin{aligned}\forall x (x\in A \rightarrow x \notin B) &\iff \\\forall x (x\in A \lor x \notin B) &\iff \\\forall x ((x\in A \land x \notin B) \lor (x \notin A \land x \notin B)) &\iff \\\forall x (x\in A \land x \notin B) &\iff \\\forall x \neg(x\in A \land x \in B) &\iff \\\neg \exists x (x\in A \land x \in B) &\iff \\A \cap B = \emptyset\end{aligned}