Definition of Disjointness for binary strings

Question

Basically, most of the definition for disjointness are such that $DISJ(A, B) = 1$ if $A \cap B = \emptyset $ and $DISJ(A, B) = 0$ for other case. My confusion is how is $0$'s influence in here. For example, is all-zeros string considered as $\emptyset$ and therefore the result will be $1$ no matter what the other string is ? Another case is if two disjoint string, e.g., $01$ and $10$ becomes not disjoint after same number of $0$ appended to each string.

Currently my thought is, based on most research papers in communication complexity field, that $DISJ(A, B) = 0$ iff there exists one entry is $1$ for both $A, B$. Would like some clarification and related detailed definition with source.

Answer 1

We identify a binary string with a set in the following way: $x \in \{0,1\}^n$ corresponds to the set $\{ i \in [n] : x_i = 1 \}$. In particular, $0^n$ is identified with the empty set.