I agree with the other comments and answers. However I would have attacked the question solely by intuition: "if two events are independent, can they also be disjoint?

It is true that this problem can be attacked with math: assuming that events A and B each have a non-zero probability of occurring, they will be regarded as independent $\iff p(A) = p(A|B).$
Since A,B are disjoint, $p(A|B) = 0.$ Since it is assumed that $p(A) > 0, ~p(A) \neq p(A|B).$ Therefore, the two events can't be independent.

However, this problem can also be attacked by considering

event $C = $ the complement of event $B$

and showing, purely by intuition, that events $A$ and $C$ can not be independent.

Consider disjoint events A,B placed in a Venn diagram that represents the universe U.

Informally, $p(A)$ may be regarded as the proportion of the area assigned to event $A$ versus the area of the entire universe $U$ in the Venn diagram.

Since the event $C$ completely encompasses the event $A$, $p(A|C)$ may be similarly regarded as the proportion of the area assigned to event $A$ versus the area assigned to event $C$, rather than versus the area assigned to $U$.

Since $p(B)$ is assumed to be non-zero, the area assigned to event $C$ must be less than the area assigned to $U$. Therefore, the two proportions referred to in the above two paragraphs must be different.

Continuing this *informal* train of thought, suppose you have **any** two events $A$ and $B$, with $C$ = the complement of $B.$

Suppose further that $p(A) \neq 0, p(B) \neq 0, p(C) \neq 0.$

Further suppose that you have *somehow* concluded that events $A$ and $C$ are not independent. That means that the chance of $A$ occurring has been affected (i.e. altered) by whether it is to be assumed that event $C$ has also occurred.

It seems to me that if the chance of $A$ occurring has been affected by whether event $C$ has also occurred, then it is **implied** that the chance of $A$ occurring has also been affected by whether event $B$ has occurred.

In other words, when it is assumed that that $p(A) \neq 0, p(B) \neq 0,$ and $p(C) \neq 0,$ then regardless of any considerations of disjointness,

events $A$ and $B$ are independent $\iff$ events $A$ and $C$ are independent.