Set Theory - An example that disprove the statement

Question

P(A\B) = P(A) if and only if A and B are disjoint sets

I am looking for an example to disprove it.

I found that:

A = { n | n * n=1, n ∈ Z}

B = {1}

Is it correct? If not, can you please give me an example of sets A and B that disprove the statement.

Thank you.

Answer 1

This is obviously not true. Normally, the statement on the left would be made if the sets are independent. (We exclude cases where P(B) = 0)

Except for, perhaps, when P(B) =0, when A and B are disjoint, P(A|B) = 0, so P(A|B) does not equal P(A) if P(A) > 0 in this case.

A simple example. A fair coin.

A = {getting a head on the first flip}

B = {getting a tail on the first flip}

P(A) = 1/2

P(B) = 1/2

P(AB) = 0

P(A|B) = P(AB)/P(B) = 0/(1/2) = 0. This does not equal P(A) = 1/2