MISSION · Data and probability
Sets and Venn Diagrams
The diagram that stops you counting the middle twice
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A Venn diagram draws groups as overlapping circles, so things belonging to more than one group have a place to sit — and get counted once.
The vocabulary
A set is a collection of things. Its members are elements.
- Union, \(A \cup B\) — everything in A or B, or both. The whole of both circles.
- Intersection, \(A \cap B\) — everything in A and B. The overlap only.
- Complement, \(A'\) — everything not in A, inside the surrounding rectangle.
That rectangle matters. It’s the universal set — everything under discussion — and without it, “not A” has no boundary. In a class survey, the rectangle is the class.
The mathematical “or” always means and/or: the union includes the overlap. Everyday English often means “one or the other but not both”, and that difference causes real confusion when reading a question.
Fill the middle first
Thirty students. Eighteen play football, fourteen play tennis, and six play both.
The instinct is to write 18 and 14 into the circles, and it’s wrong — those two numbers already include the six who do both, so the overlap would be counted twice.
Start with the intersection. Put 6 in the middle. Then football-only is \(18 - 6 = 12\), and tennis-only is \(14 - 6 = 8\). Those three regions total 26, so \(30 - 26 = 4\) students play neither, sitting outside both circles but inside the rectangle.
Now every region holds a count that belongs only to it, and any question is a matter of adding the right regions.
The counting rule
The double-counting problem has a formula:
\[n(A \cup B) = n(A) + n(B) - n(A \cap B)\]Add the two groups, then subtract the overlap once, because you added it twice. Here: \(18 + 14 - 6 = 26\), matching the regions.
This is worth knowing in its own right — it’s the same correction that appears in probability as \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\), and the reason the subtraction is there is identical.
Three circles
With three sets the same principle applies, and the order matters more: fill the centre first, then the three two-way overlaps, then the single regions, then outside. Working inwards instead leaves you solving several unknowns at once.
Common mix-ups
- Writing the totals into the circles instead of the region-only counts.
- Forgetting the outside region. “Neither” is part of the picture.
- Reading union as exclusive or. \(A \cup B\) includes both.
- Ignoring the universal set, which makes the complement meaningless.
Check yourself
In a group of 40, 25 like tea, 20 like coffee, and 9 like both. How many like neither?
Tea only is \(25 - 9 = 16\), coffee only is \(20 - 9 = 11\), and with the 9 in the middle that’s 36 accounted for. So \(40 - 36 = 4\) like neither.
LEVEL 3 · REFLECT
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