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.
Detailed notes
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.
Checks that catch mistakes
- No region can be negative. If "only A" comes out negative, a number was misread.
- The seven inside regions must add up to the union.
- "At least two" is always greater than or equal to "exactly two".
Quick revision
- 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.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Three-set union and nonevery common2 practice Q
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.
Example: 60 members: 30, 25, 20; pairs 10, 8, 7; all three 4. None?
Union = 75 − 25 + 4 = 54; none = 6.
Type 2: Exactly one / exactly two / at least twocommon2 practice Q
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.
Example: Pairs 15, 12, 10; triple 5. Exactly two?
37 − 15 = 22.
Type 3: Fill one regioncommon2 practice Q
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.
Example: n(A) = 45, n(A∩B) = 15, n(A∩C) = 12, triple 6. Only A?
45 − 15 − 12 + 6 = 24.
Type 4: Reverse three-set puzzleoccasional2 practice Q
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.
Example: Everyone in at least one of three sets; 60 in exactly one, 30 in exactly two, 100 people. Sum of the three circle totals?
All three = 10; ΣA = 60 + 60 + 30 = 150.
Formulas
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.
Shortcut tricks
⚡ Triple-strip subtraction
Whenever a question says 'exactly two', your reflex is: pairwise sums overcount the core by 3. Compute pairwise sums, subtract 3×(all three), done.
Example: Q. |A∩B| = 8, |B∩C| = 7, |A∩C| = 6, all three = 3.
Sol. Exactly two = (8+7+6) − 3×3 = 12.
'Exactly' always means: subtract every richer region, as many times as it was counted.
Where students lose marks
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.
Practice sets — 13 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 13 questions
Suggested time 11 min · wrong answers go to your mistake notebook automatically.