Can A and B be independent if A is a subset of B? (except P(B)=1 and P(A)=0) I have a question about the independence of events.

Let's say I have positive integers from 1 to 10. Event A is integers below 2 and event B is integers below 3, then P(A)=0.2, P(B)=0.3 and A is a subset of B. 

I am wondering whether A and B are independent. The probability that one event (P(A)) occurs in no way affects the probability of the other event occurring (P(B)) here. So from the definition of the independent events, A and B are independent. But it looks like there is some dependency between A and B because A is a subset of B. A always occurs where B occurs.  

Any suggestions are appreciated.
Assuming independence and $A\subset B$ yields
$$p(A)p(B)=p(A\cap B)=p(A)$$
This implies $p(B)=1$ or $p(A)=0$.
If we have events $A, B$ s.t. $A \subseteq B$ they are independent iff

$$P(A)P(B) = P(A \cap B)$$

$$\iff P(A)P(B) = P(A)$$

$$\iff P(A) = 0 \ \color{red}{\text{or}} \ P(B) = 1$$

Hence, we need not have

$$P(A) = 0 \ \color{red}{\text{and}} \ P(B) = 1$$