Maximum and Minimum of n(A∪B) and n(A∩B)

When only $n(A)$ and $n(B)$ are known (but not the exact overlap between A and B), the union and intersection counts are still bounded within a predictable range:

1. $\min\{n(A\cup B)\} = \max\{n(A), n(B)\}$ — the union can never be smaller than the larger of the two individual sets (since that larger set alone is already a subset of the union).
2. $\max\{n(A\cup B)\} = n(A)+n(B)$ — the union can never exceed the sum of the two counts, which happens exactly when A and B are disjoint (no overlap at all).
3. $\min\{n(A\cap B)\} = 0$ — the intersection can be as small as empty (no overlap).
4. $\max\{n(A\cap B)\} = \min\{n(A), n(B)\}$ — the intersection can never exceed the smaller of the two sets (since the overlap is, at most, that entire smaller set). For example, if $n(A)=10$ and $n(B)=20$: $\min\{n(A\cup B)\}=\max\{10,20\}=20$ and $\max\{n(A\cup B)\}=10+20=30$, so $20\le n(A\cup B)\le30$. Also $\min\{n(A\cap B)\}=0$ and $\max\{n(A\cap B)\}=\min\{10,20\}=10$, so $0\le n(A\cap B)\le10$. …