Formula

Exactly one, exactly two and at least two of three sets

Algebra • Sets • Operations

Formula

exactly one = P - 2 Q + 3 R, exactly two = Q - 3 R, at least two = Q - 2 R

Write

P = n (A) + n (B) + n (C)

Q = n (A \cap B) + n (B \cap C) + n (C \cap A)

and

R = n (A \cap B \cap C)

. Then exactly one

= P - 2 Q + 3 R

, exactly two

= Q - 3 R

, and at least two

= Q - 2 R

. Each coefficient is what reduces a member's multiple count to exactly one.

How to use it

1Find $R$ first from the union formula; none of the three counts can be written down without it.
2Check the split against the union: (exactly one) $+$ (exactly two) $+ R$ must equal $n (A \cup B \cup C)$ .
3If the coefficients are hard to recall, build the seven Venn regions one at a time instead — the arithmetic is the same and nothing has to be remembered.
4'At least two' is 'exactly two' plus the triple, which is why its coefficient is $- 2$ and not $- 3$ .

For three sets,

P = 150

Q = 63

and the triple overlap is

5

. How many members lie in exactly one set, and how many in at least two?

Step-by-step

1
$exactly one = 150 - 2 (63) + 3 (5) = 150 - 126 + 15 = 39$
Why: A member of exactly two sets is counted twice in $P$ and once in $Q$ , so $- 2 Q$ clears them.
2
$at least two = 63 - 2 (5) = 63 - 10 = 53$
Why: A member of all three is counted three times in $Q$ , so two of those counts are removed.
3
$39 + 53 = 92 = n (A \cup B \cup C)$
Why: The two counts partition the union, which checks both coefficients at once.

Answer

39

in exactly one,

53

in at least two

Watch out

These apply to the FULL pair counts, not to 'exactly two' pair counts. Mixing the coefficient

- 3

(exactly two) with

- 2

(at least two) is the standard error, and the two differ by precisely the triple overlap.

Builds on

Also known as

exactly one of three sets, exactly two sets, at least two sets, Venn region counts, only one subject