The intersection of two events, $A$ and $B$, is usually represented as $A \cap B$, but sometimes as $AB$ .
Two events are (pairwise) independent if and only if the probability of their intersection equals the product of their probabilities. It means knowledge of the occurrence of one event does not influence the measure of the (conditional) probability of the occurrence of the other. $$A\perp B ~\iff~ \mathsf P(A\,B)=\mathsf P(A)\,\mathsf P(B)\tag{1}$$
Two events are disjoint, or exclusive, if their intersection is an empty set, which in turn infers it to have zero probability. The intersection of disjoint events is impossible. It means the occurrence of one event prohibits the occurrence of the other. $$A\,B=\emptyset ~\implies~ \mathsf P(A\,B)=0\tag{2}$$
These two situations do not occur together, except in the edge case of at least one of the events itself being impossible. (An impossible event is independent of any other event.)