Three-Set Venn Diagrams

At a glance 9 rules

1. 1Eight regionsThree circles cut a group into 8 regions: 3 'only one', 3 'exactly these two', the centre and 'none'. Everyone is in exactly one, so the 8 add to the total (100% in a survey).
2. 2Read the words'A and B' includes the centre. 'A and B but not C' (or 'only A and B') is the pair region alone. 'Only A' is A with neither of the others.
3. 3Only one, from its circle$\text{only } A = n(A) - (x + y + z)$$x, y = A\text{'s two pair-only parts},\ z = \text{all three}$
4. 4Outside a circle$\text{not } A = N - n(A)$$N = \text{group},\ U = \text{union}:\ \text{only } C = U - n(A \cup B)$
5. 5Count the other side$\text{at most two} = N - \text{all three}$$\text{at least one} = N - \text{none}$
6. 6Centre unknown: call it xCall the centre $x$ and write every region in $x$: a pair figure of 15 leaves $15 - x$ in its pair-only part. Then one total fixes $x$.
7. 7Ratio cluesTurn 'in the ratio 3 : 2 : 1' or 'twice as many' into $3k, 2k, k$, one letter per chain of clues, and let a total fix $k$.
8. 8Fraction of a circle'Half the cricket fans follow only cricket' takes the cricket circle as the base: only cricket then equals the other three parts of that circle put together.
9. 9Yes/no traits: a gridYes/no traits (girl or boy, glasses or not) are sets too: keep the group the question asks about and draw a 2 by 2 grid with totals on its edges. Any three cells fix the fourth.