EQN-05-08

Venn Diagrams & Set Counting

Count overlapping groups without double-counting, using the two-set formula and the "fill the middle first" rule.

~5 min read•Difficulty 2/5

The two-set formula

∣ A or B ∣ = ∣ A ∣ + ∣ B ∣ - ∣ A and B ∣

Subtract the overlap once, because adding |A| and |B| counted it twice.

Fill the middle first

When drawing a Venn diagram, write the intersection (the overlap) first, then subtract it from each circle to get the "only" regions. This avoids double-counting.

30 study French, 25 study Spanish, 10 study both :

French only = 30 - 10 = 20, Spanish only = 25 - 10 = 15,

at least one = 20 + 10 + 15 = 45.

"Neither" and the universal set

total = (at least one) + (neither)

If the group has 50 people and 45 study at least one language, then 5 study neither.

Three sets

The same idea extends, but on the GAT-E two sets (plus "neither") covers almost everything. Always anchor on the overlap.

Diagram

At least one $= ($ only $A) + ($ both $) + ($ only B).

Worked examples

Example 1 (difficulty 2/5)

In a class, 18 play football, 12 play tennis, and 5 play both. How many play at least one sport?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Of 40 students, 22 like tea, 18 like coffee, and 8 like both. How many like neither?

◎ Predict the answer first:

Tips

Use |A or $B ∣ = ∣ A ∣ + ∣ B ∣ - ∣$ both| — the overlap is removed exactly once.
Fill the intersection first, then the "only" regions.
total $= ($ at least one $) + ($ neither) handles the leftover group.

Common mistakes

Adding the two group sizes without subtracting the overlap

Why: The overlap is easy to forget.

Fix: Members in both are counted twice — subtract |both| once.

Forgetting the "neither" group when totals are involved

Why: The diagram only shows the circles.

Fix: Some elements may be outside both sets; account for "neither".

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