Cardinality Properties of Sets

Given sets A, B (and sometimes a third set C, all subsets of a universal set), the following cardinality (counting) formulas hold:

1. $n(A\cup B) = n(A)+n(B)-n(A\cap B)$ — the general two-set union formula.
2. When A and B are DISJOINT ($A\cap B=\phi$, so $n(A\cap B)=0$), this simplifies to $n(A\cup B)=n(A)+n(B)$.
3. $n(A\cap B') + n(A\cap B) = n(A)$ — A splits into the part outside B and the part inside B.
4. $n(A'\cap B) + n(A\cap B) = n(B)$ — similarly, B splits into the part outside A and the part inside A.
5. $n(A\cap B') + n(A\cap B) + n(A'\cap B) = n(A\cup B)$ — the union splits into three disjoint pieces.
6. For any three sets A, B, C: $n(A\cup B\cup C) = n(A)+n(B)+n(C) - n(A\cap B) - n(B\cap C) - n(A\cap C) + n(A\cap B\cap C)$ — each individual set is counted once, each pairwise overlap is then subtracted (since it was counted twice), and the triple overlap is finally added back (since it was subtracted three times in the pairwise step but should be counted once).
7. If $n(A)=m$, then $n[P(A)]=2^m$, where P(A) is the power set of A.
8. $n(A\Delta B) = n(A)+n(B)-2n(A\cap B)$ — since the shared part is excluded entirely from the symmetric difference (subtracted twice, once from each set's count). …