disjoint sets – Definition of Disjointness for binary strings

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.