Obviously I know that I have to use the inclusion-exclusion principle but I'm not sure how to start the proof. I have been told that the answer I have to get to is $$|A ∩ B| + |A ∩ C| + |B ∩ C| − 2|A ∩ B ∩ C|$$ Is this correct? or should I be aiming for another answer?
**EDIT **
so far I have that we want to find $|(A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) - (A ∩ B ∩ C)|$ Is this a correct first step to take?
Expanding on Cosmo5's comment, and against the backdrop of Inclusion-Exclusion, the foundation of the principle is that when each region is added, subtracted, added, subtracted, ... the net effect is that each region is counted once.
From the constraints of the problem, you can consider the following 4 mutually exclusive regions:
Region 1 : in A, in B, not in C.
Region 2 : in A, not in B, in C.
Region 3 : not in A, in B, in C.
Region 4 : in A, in B, in C.
As stated, your formula mishandles Region 4, because the effect of the first three terms is that Region 4 is included three times.
To offset this, the final term must deduct Region 4 twice rather than once.