Card 8 of 8
Venn diagrams & sets
Reading two- and three-circle Venns, and the overlap formula.
The two-circle picture: left-only, overlap (both), right-only, and outside (neither).
The rule that solves almost every 11+ Venn question:
Total = A + B − Both (+ Neither, if some are outside)
Because everyone in the overlap got counted twice — once in A and once in B — so you subtract them back once.
Worked example: 30 pupils; 18 play football; 15 play tennis; everyone plays at least one. How many play both?
30 = 18 + 15 − Both → Both = 33 − 30 = 3
Then fill the picture: football only = 18 − 3 = 15 · both = 3 · tennis only = 15 − 3 = 12 · check: 15 + 3 + 12 = 30 ✓
With a "neither" group: 32 children; 20 like apples; 17 like bananas; 4 like neither.
Children liking at least one = 32 − 4 = 28 → 28 = 20 + 17 − Both → Both = 9
Reading the regions — the wording that trips children:
| Phrase | Region |
|---|---|
| "A only" / "just A" | left-only (excludes the overlap) |
| "both A and B" | the overlap alone |
| "A or B" | everything in both circles including the overlap |
| "neither" | outside both circles |
"How many play football?" means all football players (15 + 3 = 18). "How many play only football?" means 15. Underline "only".
Carroll diagrams (the 2×2 grid version) work the same way — every child sits in exactly one of four boxes, and the four boxes add to the total.