Venn Counting — 2-Set and 3-Set Overlaps

🎯 Learning Objective

Count 'at least one', 'only', and 'neither' using the union formula for two and three sets.

💡 Concept

• Two sets: n(A or B) = n(A) + n(B) − n(A and B) — subtract the overlap once so it is not counted twice
• Only A = n(A) − n(A and B); Neither = Total − n(A or B)
• Three sets: n(A or B or C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C)
• Add all singles, subtract all pairs, add back the triple (add–subtract–add)
• Only A (3 sets) = n(A) − n(A∩B) − n(A∩C) + n(A∩B∩C)
• Always draw the circles and fill the CENTRE (all three) first, then work outward

🧮 Key Formulas

n(A∪B) = n(A) + n(B) − n(A∩B)

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n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C)

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Neither = Total − n(A∪B∪C)

✏️ Easy Example

Q. In a class of 40, 24 like tea, 20 like coffee and 8 like both. How many like at least one drink?

1. At least one = 24 + 20 − 8
2. = 44 − 8

Answer: 36

🇮🇳 Real-Life Example

A railway canteen tracks passengers ordering tea and samosa. If it double-counts those who took both, the stock never matches — subtracting the overlap once is exactly what balances the register.

📝 Exam-Level Example

Q. In a group of 50 people, 35 speak Hindi, 25 speak English and 15 speak both. How many speak neither?

1. At least one = 35 + 25 − 15 = 45
2. Neither = Total − at least one = 50 − 45

Answer: 5

📝 Exam-Level Example

Q. In a survey of 60 people, 25 like cricket, 26 like hockey, 26 like football; 11 like cricket & hockey, 8 like hockey & football, 9 like cricket & football, and 3 like all three. How many like at least one game?

1. Add singles: 25 + 26 + 26 = 77
2. Subtract pairs: 77 − (11 + 8 + 9) = 77 − 28 = 49
3. Add back the triple: 49 + 3

Answer: 52

📝 Exam-Level Example

Q. From the same survey, how many like ONLY cricket?

1. Only cricket = 25 − (cricket&hockey) − (cricket&football) + (all three)
2. = 25 − 11 − 9 + 3

Answer: 8

🪄 Memory Trick

For two sets the whole game is 'plus, plus, minus the overlap'. For three, chant add–subtract–add. Filling the innermost region first stops all double-counting errors.

⚠️ Common Mistakes

• ❌ Forgetting to subtract the overlap — the single most common Venn error
• ❌ In 3-set problems, subtracting the triple instead of ADDING it back
• ❌ Reading 'like both' as 'like only both' — 'both' includes the all-three people too

🏆 Exam Tips

• ✅ Underline whether the question asks 'at least one', 'only', or 'exactly two' — each needs a different region
• ✅ In 3-set questions, always start by placing the all-three number in the centre

📌 Summary

• 2 sets: n(A∪B) = A + B − (both)
• 3 sets: singles − pairs + triple (add–subtract–add)
• Only A = A minus its overlaps (add the triple back in 3 sets)
• Neither = Total − (at least one)