Set Theory • Topic 1 of 3

Union & Intersection

The union A∪B is everything that lies in A or B (or both); the intersection A∩B is only the common part. The whole subject rests on one counting idea: if you simply add n(A) and n(B), the overlap A∩B gets counted twice, so you must subtract it once. That is the inclusion–exclusion principle, n(A∪B) = n(A) + n(B) − n(A∩B). In CAT this is rarely asked directly — instead you are given three of these four quantities and must find the fourth, or you are told "how many take only Maths" which is n(A) − n(A∩B). The fast habit is to picture the two regions "only A", "only B", and "both", whose sizes always add to n(A∪B). Watch the wording: "either" usually means union, while "both" means intersection. When the universe and a "neither" count are given, remember Total = n(A∪B) + n(neither).

✅ Solved examples

1. In a class, 30 students play cricket, 25 play football, and 10 play both. How many play at least one of the two games?
n(C∪F) = 30 + 25 − 10 = 45 students play at least one game.
2. In a group of 60 people, 35 read The Hindu, 28 read TOI, and 8 read neither. How many read both?
Readers of at least one = 60 − 8 = 52. So 35 + 28 − n(both) = 52 ⇒ n(both) = 63 − 52 = 11.
3. Of 200 customers, 120 bought tea and 90 bought coffee; 40 bought both. How many bought only tea?
Only tea = n(Tea) − n(both) = 120 − 40 = 80 customers.
4. A and B are sets with n(A) = 18, n(B) = 22 and n(A∪B) = 30. Find n(A∩B) and the number in exactly one set.
n(A∩B) = 18 + 22 − 30 = 10. Exactly one = n(A∪B) − n(A∩B) = 30 − 10 = 20.

✏️ Practice — try these, take hints as needed

1. 40 people like apples, 30 like bananas, 12 like both. How many like at least one fruit?
Use n(A∪B) = n(A)+n(B)−n(A∩B).
40 + 30 − 12.
Subtract the overlap once.
58
2. In a survey of 100, 65 use WhatsApp, 45 use Telegram, none use neither. How many use both?
At least one = 100.
65 + 45 − both = 100.
110 − 100.
10
3. n(A) = 50, n(B) = 36, n(A∩B) = 16. Find n(A only).
Only A = n(A) − n(A∩B).
50 − 16.
Remove the common part.
34
4. Of 80 students, 50 take Maths, 40 take Physics, 10 take neither. How many take both?
At least one = 80 − 10 = 70.
50 + 40 − both = 70.
90 − 70.
20
5. n(A∪B) = 48, n(A) = 30, n(B) = 25. How many lie in exactly one of the sets?
First find n(A∩B) = 30 + 25 − 48.
n(A∩B) = 7.
Exactly one = n(A∪B) − n(A∩B).
41

📝 Topic test — 8 questions

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