Problems on Sets

Lessons

Senior 3 · Mathematics

Problems on Sets

4 lessons

Lesson 3 of 4

Solving problems with three sets

25 min

When the centre is the unknown

In S1 you filled a three-set diagram when every overlap was given: you wrote the centre first and worked outwards. Real surveys are rarely so tidy. Often the number in all three sets is exactly what the question asks for, and some regions are given only as "exactly two" or "only". This lesson shows how to set up such problems with one unknown and solve them with an equation.

Definition

In exactly $k$ of the sets: inside $k$ of the sets and outside all the others.

Notation: with three sets, "exactly one" is regions 1, 2, 3; "exactly two" is regions 4, 5, 6; "all three" is region 7.

Example: a farmer who grows maize and beans but not cassava is in exactly two of the sets (region 4).

Definition

At least two of the sets: in two or in all three of the sets.

Notation: $(A \cap B) \cup (A \cap C) \cup (B \cap C)$ , which is regions 4, 5, 6 and 7.

Example: "at least two" = "exactly two" + "all three".

Important

Two counting facts for three sets. Write $E_{1}$ , $E_{2}$ and $E_{3}$ for the numbers of elements in exactly one, exactly two and all three of the sets. Then

n(A \cup B \cup C) = E_1 + E_2 + E_3

and

n (A) + n (B) + n (C) = E_{1} + 2 E_{2} + 3 E_{3} .

Why: the union is made of regions 1 to 7, which are exactly the elements counted by $E_{1}$ , $E_{2}$ and $E_{3}$ . In the sum $n (A) + n (B) + n (C)$ , an element that lies in exactly $k$ of the sets is counted once for each set it is in, that is $k$ times.

Worked example 1: finding the centre

A cooperative in Rwamagana has 80 members (illustrative figures). 40 grow maize ( $M$ ), 38 grow beans ( $B$ ) and 30 grow cassava ( $C$ ). 15 grow maize and beans, 12 grow maize and cassava, 10 grow beans and cassava, and 4 grow none of the three. Find the number who grow (a) all three crops (b) exactly one crop (c) at least two crops.

Solution Let $x$ be the number who grow all three. "Maize and beans" (15) includes those who also grow cassava, so maize and beans only is $15 - x$ . In the same way, maize and cassava only is $12 - x$ and beans and cassava only is $10 - x$ . Then

M \text{ only} = 40 - (15 - x) - (12 - x) - x = 13 + x,

B \text{ only} = 38 - (15 - x) - (10 - x) - x = 13 + x,

C \text{ only} = 30 - (12 - x) - (10 - x) - x = 8 + x.

Every region written in terms of $x$ ; $n (U) = 80$ .

(a) The eight regions add up to 80:

\begin{aligned} (13 + x) + (13 + x) + (8 + x) & \\ {}+ (15 - x) + (12 - x) + (10 - x) + x + 4 &= 80 \\ 75 + x &= 80 \\ x &= 5 \end{aligned}

So 5 members grow all three crops. Check with the rule: $40 + 38 + 30 - 15 - 12 - 10 + 5 = 76 = 80 - 4$ . ✓

The finished diagram: $18 + 18 + 13 + 10 + 7 + 5 + 5 + 4 = 80$ . ✓ The shaded regions are those in at least two sets.

(b) Exactly one crop: $18 + 18 + 13 = 49$ members.

Worked example 2: "exactly" totals

A school has three clubs: Debate ( $D$ ), Science ( $S$ ) and Environment ( $E$ ). In one S3 class of 50 learners, 5 are in all three clubs, 20 are in exactly two clubs and 18 are in exactly one club. (a) How many are in no club? (b) The Debate club has 30 members from this class and the Science club has 25. How many are in the Environment club?

Solution

(a) $n(D \cup S \cup E) = E_1 + E_2 + E_3 = 18 + 20 + 5 = 43$ , so $50 - 43 = 7$ are in no club.

(b) $n (D) + n (S) + n (E) = E_{1} + 2 E_{2} + 3 E_{3} = 18 + 40 + 15 = 73$ . So $n (E) = 73 - 30 - 25 = 18$ .

Notice that part (b) needed no diagram at all: the counting fact did the work.

Worked example 3: the unknowns in the "exactly two" regions

80 S3 learners were asked which of Mathematics ( $M$ ), Physics ( $P$ ) and Chemistry ( $C$ ) they enjoy. 20 enjoy Mathematics only, 15 Physics only and 12 Chemistry only; 6 enjoy all three and 7 enjoy none. The number who enjoy Mathematics and Physics only is equal to the number who enjoy Mathematics and Chemistry only, and is twice the number who enjoy Physics and Chemistry only. Find (a) the number who enjoy Physics and Chemistry only (b) $n (M)$ (c) the number who enjoy Physics but not Chemistry.

Solution Let $y$ be the number who enjoy Physics and Chemistry only; then Mathematics and Physics only, and Mathematics and Chemistry only, are each $2 y$ .

$n (U) = 80$

(a) $20 + 15 + 12 + 2 y + 2 y + y + 6 + 7 = 80$ , so $60 + 5 y = 80$ and $y = 4$ .

(b) $n (M) = 20 + 2 y + 2 y + 6 = 20 + 8 + 8 + 6 = 42$ .

Warning

"And" is not "and only". In a three-set problem, "15 grow maize and beans" counts region 4 and region 7, because the farmers who grow all three crops also grow maize and beans. Only "maize and beans only" (or "maize and beans but not cassava") means region 4 by itself. Subtract the centre before you write a number in regions 4, 5 or 6.

Tip

One letter, eight regions, one equation. Put the unknown in the region the question does not give (often the centre). Fill the other regions in terms of it, working from the centre outwards, then add all eight regions and set the total equal to $n (U)$ . Finish by checking every region is a whole number that is zero or more.

ExamRevision: Problem solving Venn Diagrams – 3 sets (22 min). Watch how the diagram is filled from the given information, including the examples that use algebra for an unknown region. Pause before each solution and try it yourself first.

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Practice

Draw a three-set diagram for every question. The last three are harder.

$n (U) = 60$ , $n (A) = 30$ , $n (B) = 25$ , $n (C) = 20$ , $n(A \cap B) = 10$ , $n(A \cap C) = 8$ , $n(B \cap C) = 7$ and $n\big((A \cup B \cup C)'\big) = 9$ . Find $n(A \cap B \cap C)$ .
For the sets in question 1, find the number of elements in exactly one set.
In a group of 70 people, 8 belong to all three of the groups $A$ , $B$ , $C$ , 25 to exactly two and 30 to exactly one. How many belong to none? Find $n (A) + n (B) + n (C)$ .
Of 100 learners, 45 play football, 40 volleyball and 35 basketball. 12 play football and volleyball, 10 football and basketball, 8 volleyball and basketball, and 7 play none of the three. How many play all three? How many play exactly one?
Each of 40 learners takes at least one of three options $A$ , $B$ , $C$ . 10 take $A$ only, 8 take $B$ only and 6 take $C$ only; 4 take all three; and each "exactly two" region has $x$ learners. Find $x$ and $n (A)$ .
Explain why the number of elements in at least two of three sets is $n(A \cap B) + n(A \cap C) + n(B \cap C) - 2\,n(A \cap B \cap C)$ .
Each of 50 learners studies at least one of English, French and Kiswahili. 30 study English, 25 French and 20 Kiswahili, and 5 study all three. How many study exactly two of the languages?

Show answersHide answers

$n(A \cup B \cup C) = 60 - 9 = 51$ . $30 + 25 + 20 - 10 - 8 - 7 + x = 51$ , so $50 + x = 51$ and $x = 1$ .
Pair-only regions: 9, 7, 6. $A$ only $= 30 - 9 - 7 - 1 = 13$ , $B$ only $= 25 - 9 - 6 - 1 = 9$ , $C$ only $= 20 - 7 - 6 - 1 = 6$ . Exactly one: $13 + 9 + 6 = 28$ . Check: $13 + 9 + 6 + 9 + 7 + 6 + 1 + 9 = 60$ . ✓
None: $70 - (30 + 25 + 8) = 7$ . Sum: $30 + 2 \times 25 + 3 \times 8 = 104$ .
$45 + 40 + 35 - 12 - 10 - 8 + x = 100 - 7$ , so $90 + x = 93$ and $x = 3$ . Only: football $45 - 9 - 7 - 3 = 26$ , volleyball $40 - 9 - 5 - 3 = 23$ , basketball $35 - 7 - 5 - 3 = 20$ . Exactly one: 69.
$10 + 8 + 6 + 3 x + 4 = 40$ , so $x = 4$ . $n (A) = 10 + 4 + 4 + 4 = 22$ .
The three pairwise intersections together count each of regions 4, 5, 6 once, but region 7 three times. "At least two" needs region 7 once, so subtract it twice.
$E_{1} + E_{2} + 5 = 50$ and $E_{1} + 2 E_{2} + 15 = 75$ . Subtracting: $E_{2} + 10 = 25$ , so $E_{2} = 15$ (and $E_{1} = 30$ ).

Sources and credits

1.5: Set Operations with Three Sets, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 30 September 2026). Adapted by Ganzaa.
The counting facts for exactly one, two and three sets and the worked examples. Written by Ganzaa.
Video: Problem solving Venn Diagrams- 3 sets HL, ExamRevision, embedded from YouTube.

The Rwandan worked examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1When the centre is the unknown

When the centre is the unknown