Venn Diagrams • Topic 4 of 4

Region Counting

Three-set diagrams have seven regions: three 'only', three 'pair-only', and one 'all three'. Fill from the centre: subtract the all-three count from each pairwise total to get 'pair-only', then subtract everything from each single total to get 'only'.

The seven regions of a three-set diagram

A B C all 3
3 "only" regions + 3 "pair-only" regions + 1 "all three" = 7 regions.

Fill from the centre outward

  1. Start with the all-three count (the centre).
  2. Pair-only = pairwise total − centre (e.g., A∩B only = |A∩B| − |A∩B∩C|).
  3. Only A = |A| − (its two pair-only regions) − centre.
Sanity check. Add all seven regions (plus any "none" outside) — it must equal the grand total. If it doesn't, you've double-counted an overlap.

✅ Solved examples

1. All three = 5. A∩B (total) = 12. A∩B only?
12 − 5 = 7.
2. Sum of regions 10, 8, 6 (only) + 4, 3, 2 (pairs) + 1 (centre). Total?
10+8+6+4+3+2+1 = 34.
3. |A∩B∩C| = 6, |A∩B| = 14. Members in A and B but not C?
14 − 6 = 8.
4. A∩C total 9, centre 3. A∩C only?
9 − 3 = 6.

✏️ Practice — try these, take hints as needed

1. All-three=4, A∩B total=11. A∩B only?
11−4.
—
—
7
2. Regions 5,6,7 (only) +2,3,4 (pairs) +1 (centre). Total?
Add all seven.
—
—
28
3. A∩C total 9, centre 3. A∩C only?
9−3.
—
—
6
4. Only-A 12, A∩B-only 3, A∩C-only 2, centre 1. |A|?
12+3+2+1.
—
—
18
5. B∩C total 10, centre 4. B∩C only?
10−4.
—
—
6

📝 Topic test — 8 questions

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