Venn Diagrams • Topic 3 of 4

Numerical Venn

A diagram gives counts in each region; you compute totals. Use |A∪B| = |A| + |B| − |A∩B| for two sets. 'Only A' means A minus the overlap.

The two-set picture

only A both only B A B
|A∪B| = (only A) + (both) + (only B) = |A| + |B| − |A∩B|.

The formulas in words

  • Union: |A∪B| = |A| + |B| − |A∩B| — add the two sets, subtract the shared part once.
  • Only A: |A| − |A∩B|. Only B: |B| − |A∩B|.
  • Exactly one: (only A) + (only B) = |A| + |B| − 2|A∩B|.
Never double-count. The single most common error is forgetting to subtract the overlap once in the union. Always remove |A∩B| exactly one time.

✅ Solved examples

1. |A| = 30, |B| = 25, |A∩B| = 10. Find |A∪B|.
30 + 25 − 10 = 45.
2. From the same data, how many are in only A?
30 − 10 = 20.
3. 40 like tea, 30 like coffee, 12 like both. How many like only one?
Only tea 28, only coffee 18; total 46.
4. |A∪B| = 50, |A| = 35, |B| = 25. Find |A∩B|.
35 + 25 − 50 = 10.

✏️ Practice — try these, take hints as needed

1. |A|=20, |B|=15, |A∩B|=5. |A∪B|?
Add, subtract overlap.
—
—
30
2. Only B if |B|=15, overlap 5?
15−5.
—
—
10
3. |A∪B|=60, |A|=40, |B|=30. Overlap?
40+30−60.
—
—
10
4. 25 play cricket, 20 hockey, 8 both. Only one game?
17 + 12.
—
—
29
5. |A|=18, |B|=18, |A∩B|=18. |A∪B|?
Same set.
—
—
18

📝 Topic test — 8 questions

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