I dug up an old problem, we were discussing a while ago. Unfortunately, I still don't understand it.

If A and B are events in a probability space such that 0 < P(A) = P(B) = P(A intersect B) < 1, which of the following cannot be true?

A) A and B are independent

B) A is a proper subset of B

C) A != B

D) A intersect B = A union B

E) P(A)P(B) < P(A intersect B)

Despite the correct answer A does make sense, the answer B does not. Would really appreciate if someone gave the counterexample for B.

## My confusion in probability

If you know about measure theory then there does exist a subset A of a set B such that : m(A)=m(B), where m(A) is the measure of A

For example : let B=[0,1/2] and A=(0,1/2) then we have A is a proper subset of A and m(A)=m(B)=m(A intersection B)=1/2

the probability space is just a specific case of measure spaces