Formula

Inclusion-exclusion for two sets

Algebra • Sets • Operations

Formula

n (A \cup B) = n (A) + n (B) - n (A \cap B)

For finite sets

A

and

B

n (A \cup B) = n (A) + n (B) - n (A \cap B)

; the number of members of the whole collection

U

lying in neither is then

n (U) - n (A \cup B)

How to use it

1Percentages behave exactly like counts out of $100$ , so take the total as $100$ and count people.
2'Neither' is the complement of the union: find the union first, then subtract it from the total.
3'Only $A$ ' is $n (A) - n (A \cap B)$ , not $n (A)$ .

Worked example

80

people,

50

drink tea,

40

drink coffee and

25

drink both. How many drink neither?

Step-by-step

1
$n (T \cup C) = 50 + 40 - 25 = 65$
Why: The $25$ who drink both were counted in each of the two groups.
2
Neither $= 80 - 65 = 15$

Answer

15

Watch out

Adding the two groups without subtracting the overlap is the standard error, and the quantity subtracted must be the count of both, not the count of exactly one.

Also known as

inclusion exclusion principle, n(A union B) formula, union of two overlapping groups, passed in both failed in both, Venn diagram two circles

Inclusion-exclusion for two sets

Formula

Worked example

Appears in past papers

2020 (I)

2019 (I)

2018 (II)

2017 (II)