Set Theory • Topic 3 of 3

Venn Diagrams (2 and 3 sets)

Three-set Venn problems are CAT’s favourite Modern Maths setup. Draw three overlapping circles giving seven regions; always fill from the centre outwards. Put n(A∩B∩C) in the middle first, then subtract it to get the "exactly two" slices, then subtract those to get the "only one" slices. The master formula is n(A∪B∪C) = Σ singles − Σ pairs + triple. Two derived results win most questions: exactly-one = Σsingles − 2·Σpairs + 3·triple, and exactly-two = Σpairs − 3·triple. The hardest variant is max/min overlap: given totals and a fixed universe, find the largest or smallest possible "all three". To MAXIMISE the triple overlap, push as many people as possible into the centre; to MINIMISE it, spread people out — use total liking = Σsingles and the rule min(all three) ≥ Σsingles − 2·(Total). Treat percentages as counts out of 100 to keep the arithmetic clean.

✅ Solved examples

1. In a survey of 100, 60 like cricket, 50 like football, 40 like tennis; 30 like cricket and football, 20 football and tennis, 25 cricket and tennis, and 10 like all three. How many like none?
n(C∪F∪T) = (60+50+40) − (30+20+25) + 10 = 150 − 75 + 10 = 85. None = 100 − 85 = 15.
2. Using the same survey, how many like exactly two of the three sports?
Exactly two = Σpairs − 3·triple = (30+20+25) − 3(10) = 75 − 30 = 45.
3. Same survey: how many like exactly one sport?
Exactly one = Σsingles − 2·Σpairs + 3·triple = 150 − 2(75) + 3(10) = 150 − 150 + 30 = 30.
4. In a group of 100, every person likes at least one of tea, coffee, milk. 70 like tea, 60 coffee, 50 milk. What is the minimum number who like all three?
Total likings = 70 + 60 + 50 = 180 across 100 people. Minimum all-three ≥ Σsingles − 2·Total = 180 − 200 = −20 ⇒ floor is 0; but since min(at-least-two) = 180 − 100 = 80 must be double-counted, the minimum who like all three = Σsingles − 2(Total) is negative, so the guaranteed minimum is 0. (At least 80 like two or more, but all three can be 0.)

✏️ Practice — try these, take hints as needed

1. In a class of 50, 25 play chess, 20 carrom, 18 both. None plays neither game beyond these. How many play at least one?
n(A∪B) = 25 + 20 − 18.
= 27.
Two-set inclusion–exclusion.
27
2. Of 150 people: 80 read A, 70 read B, 60 read C; 30 read A&B, 25 B&C, 20 A&C, 10 all three. How many read at least one?
Σsingles − Σpairs + triple.
(80+70+60) − (30+25+20) + 10.
210 − 75 + 10.
145
3. In the previous survey, how many read exactly one magazine?
Exactly one = Σsingles − 2Σpairs + 3·triple.
210 − 2(75) + 3(10).
210 − 150 + 30.
90
4. In a survey of 100, 75 like coffee and 65 like tea. What is the minimum number who like both?
Two-set: min(both) = n(A) + n(B) − Total.
75 + 65 − 100.
Overlap is forced when totals exceed 100.
40
5. In a poll of 100, 80 use app X and 70 use app Y. What is the maximum number who use both?
Both cannot exceed the smaller set.
max(both) = min(80, 70).
Push Y entirely inside X.
70

📝 Topic test — 8 questions

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