Venn Diagrams
🔒 Log in to trackThree-circle counting
🔒 Log in to trackWith three sets A, B, C the master formula is
|A∪B∪C| = |A| + |B| + |C| − |A∩B| − |B∩C| − |A∩C| + |A∩B∩C|.
Two derived region-counts appear constantly:
- Exactly two = (|A∩B| − all3) + (|B∩C| − all3) + (|A∩C| − all3): each pairwise figure includes the triple region, which must be stripped from both pairs it contaminates.
- Exactly one = union − exactly-two − all-three.
Exam data usually gives the seven numbers directly (three singles, three pairs, one triple); plug them into the 8 regions and add or subtract as asked.
The eight regions
Three circles A, B, C give seven regions inside and one outside:
- Only A, only B, only C — exactly one set.
- A-B only, B-C only, A-C only — exactly two sets.
- A-B-C — all three.
- None — outside all circles.
Always fill the diagram from the centre outwards: all three first, then the exactly-two regions, then the only regions, then none.
The master formula
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)
Why the triple is added back: it is counted 3 times in the singles, removed 3 times in the pairs, so it would be lost without the last term.
Region formulas
- A-B only = n(A∩B) − all three (and the same for the other two pairs).
- Only A = n(A) − n(A∩B) − n(A∩C) + all three.
- Exactly two = (sum of pairwise) − 3 × all three.
- At least two = (sum of pairwise) − 2 × all three.
- Exactly one = union − exactly two − all three.
Worked example
In a club of 60 members, 30 play cricket (C), 25 football (F), 20 hockey (H); 10 play C and F, 8 play F and H, 7 play C and H, and 4 play all three.
- Centre: 4.
- Exactly two: C-F only 10 − 4 = 6; F-H only 8 − 4 = 4; C-H only 7 − 4 = 3.
- Only C = 30 − 6 − 3 − 4 = 17; only F = 25 − 6 − 4 − 4 = 11; only H = 20 − 4 − 3 − 4 = 9.
- Union = 17 + 11 + 9 + 6 + 4 + 3 + 4 = 54 (the formula gives 30 + 25 + 20 − 10 − 8 − 7 + 4 = 54 as well).
- None = 60 − 54 = 6. Forgetting the "+ 4" gives a union of 50 and a wrong answer of 10.
Reverse questions
- Find the triple from the union: rearrange the master formula. With singles 25, 30, 20, pairs 10, 8, 7 and union 55: 75 − 25 + x = 55, so x = 5.
- Exactly-counts to sum of circles: if 60 people are in exactly one set, 30 in exactly two and 10 in all three, then n(A) + n(B) + n(C) = 60 + 2 × 30 + 3 × 10 = 150, because a person in two circles is counted twice.
Percent data
When the three sets are given in percent, take the total as 100 and use the same formulas. If 50%, 40% and 30% like three kinds of songs, with pairs 20%, 15%, 10% and all three 5%, the union is 80% and 20% like none. Convert to people only at the end.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Three-set union and none
Three totals, three pairwise overlaps and the triple; asks at least one or none.
Union = ΣA − Σpairs + triple.
None = total − union.
Check that no region is negative.
Why it works: the triple is added in 3 times and removed 3 times, so it is added back once.
60 members: 30, 25, 20; pairs 10, 8, 7; all three 4. None?
Show solutionHide solution
Union = 75 − 25 + 4 = 54; none = 6.
Exactly one / exactly two / at least two
Asks how many belong to exactly one, exactly two or at least two sets.
Exactly two = Σpairs − 3 × triple.
At least two = Σpairs − 2 × triple.
Exactly one = union − exactly two − triple.
Why it works: each pairwise figure contains the triple once.
Pairs 15, 12, 10; triple 5. Exactly two?
Show solutionHide solution
37 − 15 = 22.
Fill one region
Asks for one specific region: only A, or A-and-B-but-not-C.
A-B only = n(A∩B) − triple.
Only A = n(A) − n(A∩B) − n(A∩C) + triple.
Fill from the centre outwards if more regions are needed.
Why it works: subtracting both pairs removes the triple twice, so it is added back once.
n(A) = 45, n(A∩B) = 15, n(A∩C) = 12, triple 6. Only A?
Show solutionHide solution
45 − 15 − 12 + 6 = 24.
Reverse three-set puzzle
The union or the exactly-counts are given and the triple or the circle sum is asked.
Rearrange the master formula for the unknown term.
Or use ΣA = exactly one + 2 × exactly two + 3 × all three.
Verify by filling the eight regions.
Why it works: each person is counted once for every circle they are in.
Everyone in at least one of three sets; 60 in exactly one, 30 in exactly two, 100 people. Sum of the three circle totals?
Show solutionHide solution
All three = 10; ΣA = 60 + 60 + 30 = 150.
Formula sheet
Add singles, subtract pairs, re-add the triple.
Sum of the three pairwise figures minus three times the triple.
Strip the multi-set members from the union.
Shortcuts that save time
Whenever a question says 'exactly two', your reflex is: pairwise sums overcount the core by 3. Compute pairwise sums, subtract 3×(all three), done.
Q. |A∩B| = 8, |B∩C| = 7, |A∩C| = 6, all three = 3.
Sol. Exactly two = (8+7+6) − 3×3 = 12.
Show solutionHide solution
'Exactly' always means: subtract every richer region, as many times as it was counted.
Mistakes to avoid
Where most students lose marks on this subtopic.
Using pairwise figures as 'exactly two' without stripping the triple region.
Sign slips on the +all-three term in the union formula.
Adding 'neither/none' into the union before subtracting from the grand total.
Quick revision
Read this the night before the exam.
Fill from the centre: triple → exactly-two → only → none.
Union = ΣA − Σpairs + triple.
Exactly two = Σpairs − 3 × triple; at least two = Σpairs − 2 × triple.
Only A = A − AB − AC + triple.
ΣA = exactly one + 2 × exactly two + 3 × all three.
Check: no negative region; seven regions sum to the union.
Practice: 13 questions
Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.
Topic test · 13 questions
Suggested time 11 min · wrong answers go to your mistake notebook automatically.