Sets & Venn Diagrams

At a glance 10 formulas

1. 1Two sets$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$\text{neither} = \text{total} - n(A \cup B)$
2. 2Smallest and largest overlap$\max(0,\ a + b - N) \le n(A \cap B) \le \min(a, b)$$a, b, c = \text{set sizes},\ N = \text{total};\ \text{all three} \ge a + b + c - 2N$
3. 3Three sets$n(A \cup B \cup C) = S_1 - S_2 + S_3$$S_1 = n(A) + n(B) + n(C),\ S_2 = \Sigma\,\text{pairs},\ S_3 = \text{all three}$
4. 4Exactly two$\text{exactly two} = \Sigma\,\text{pairs} - 3 \times \text{all three}$$\text{at least two} = \Sigma\,\text{pairs} - 2 \times \text{all three}$
5. 5Exactly one$\text{exactly one} = S_1 - 2S_2 + 3S_3$$S_1 = n(A) + n(B) + n(C),\ S_2 = \Sigma\,\text{pairs},\ S_3 = \text{all three}$
6. 6Count by membership$n(A) + n(B) + n(C) = E_1 + 2E_2 + 3E_3$$E_k = \text{in exactly } k \text{ sets};\ \text{union} = E_1 + E_2 + E_3$
7. 7Fill from the centreDraw the three circles and fill the all-three region first, then each pair minus it, then each 'only' region, and the 'none' outside last.
8. 8Extremes: crowd or spreadFor the largest 'none' or overlap, crowd people into the overlaps; for the smallest, spread them out. Every region must stay 0 or more.
9. 9Subsets and relations$n(A) = k \Rightarrow 2^{k} \text{ subsets}$$\text{relations from } A \text{ to } B = 2^{\,n(A) \times n(B)}$
10. 10Element by elementCount pairs of subsets of a $k$-element set element by element: for $A \subseteq B$ each element is in neither, in $B$ only, or in both, so $3^{k}$ pairs.