Independence of disjoint events

I'm taking a class in Probability Theory, and I was asked this question in class today:

Given disjoint events $A$ and $B$ for which $$ P(A)>0\\ P(B)>0 $$ Can $A$ and $B$ be independent?

My answer was:

$A$ and $B$ are disjoint, so $P(A\cap B)=0$.
$P(A)>0$ and $P(B)>0$, so $P(A)P(B)>0$.
$P(A\cap B)\not =P(A)P(B)$, so $A$ and $B$ are not independent.

However, I was told that I am wrong and we cannot know whether or not $A$ and $B$ are independent from the given information, but I did not receive a satisfactory explanation. Is my argument valid? If not, where do I go wrong?

Solution 1:

Your argument is valid, as was already stated in the comments. I am posting this as CW answer so that this question no longer shows as unanswered.

Solution 2:

As so many people have already told you in the comments, your argument is perfectly correct. Perhaps your instructor was thinking of a slightly different question

Given disjoint events $A$ and $B$ for which $ P(A\cup B)>0$, can $A$ and $B$ be independent?

to which the answer is Yes, and it happens when one of $A$ and $B$ is an event of probability $1$ that is a proper subset of the sample space $\Omega$ and the other is a subset of the complement, and hence has probability $0$.

For example, if $X$ is a continuous random variable, then $A = \{X \neq a\}$ and $B =\{X = a\}$ are disjoint events satisfying the condition $P(A \cup B) > 0$, and of course $$P(A\cap B) = P(\emptyset) = 0 = 1\times 0 = P(A)P(B)$$ showing that that $A$ and $B$ are independent.