# 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.

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.