Formula

Inclusion-exclusion for three sets

Algebra • Sets • Operations

Formula

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

For finite sets

A

B

and

C

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

; a member of all three is counted three times by the singles and removed three times by the pairs, so one addition restores it.

How to use it

1When six of the seven quantities are known, solve for the seventh — most often the triple overlap, which is rarely given directly.
2'None of the three' is the complement: $n (U) - n (A \cup B \cup C)$ , so a stated failure rate fixes the union before the formula is applied.
3Percentages behave exactly like counts out of $100$ , so take the total as $100$ and count members.
4A pair count such as $n (A \cap B)$ includes the members who are also in $C$ unless the wording says 'but not $C$ '.

Worked example

In a group of

100

people,

60

read the Hindi paper,

50

the English and

40

the Urdu;

25

read Hindi and English,

20

English and Urdu,

18

Hindi and Urdu, and

8

read none. How many read all three?

Step-by-step

1
$n (H \cup E \cup U) = 100 - 8 = 92$
Why: Those who read none are the complement of the union.
2
$92 = (60 + 50 + 40) - (25 + 20 + 18) + x$
Why: Apply the formula with $x$ for the triple overlap.
3
$92 = 150 - 63 + x = 87 + x \Rightarrow x = 5$

Answer

5

Watch out

The sign pattern alternates

+, -, +

; subtracting the triple overlap instead of adding it is the standard error. Every pair count must be the FULL overlap of those two sets, including members of the third, or the formula does not balance.

Builds on

Inclusion-exclusion for two sets

Also known as

three set formula, n(A union B union C), Venn diagram three circles, inclusion exclusion for three sets, triple overlap

Inclusion-exclusion for three sets

Formula

Worked example

Appears in past papers

2019 (I)

2018 (II)