Venn Diagrams: n(R) Is Not the Same as R Only

The question

A group of 60 Form 4 students was asked about two clubs.

$\xi = \{\text{students in the group}\}$
$R = \{\text{students in the robotics club}\}$
$D = \{\text{students in the debate club}\}$

It is given that $n(R) = 3x$ , $n(D) = 2x + 4$ and $n(R \cap D) = x$ . There are 8 students who belong to neither club.

(a) Find the value of $x$ . [3 marks]

(b) Find the number of students who belong to exactly one of the two clubs. [2 marks]

(Cambridge papers write the universal set as $\mathscr{E}$ rather than $\xi$ . It means the same thing.)

The common wrong answer

The student draws two overlapping circles, writes $x$ in the overlap and $8$ outside, which is right. Then they copy the set totals straight into the outer parts of the circles: $3x$ in “ $R$ only” and $2x + 4$ in “ $D$ only”. Adding the four regions:

\begin{aligned} 3x + x + (2x + 4) + 8 &= 60 \\ 6x + 12 &= 60 \\ x &= 8 \end{aligned}

For part (b) they add the two outer regions: $3(8) + \big(2(8) + 4\big) = 24 + 20 = 44$ .

The error is in the first step. $n(R) = 3x$ counts everyone in the robotics club, including the $x$ students who are also in debate. Writing $3x$ in the “ $R$ only” part counts those $x$ students twice.

The four regions do add to 60, so the mistake hides well. Check the circle instead: with $x = 8$ , the whole of circle $R$ holds $24 + 8 = 32$ students, but $n(R) = 3x = 24$ . The diagram contradicts the question.

Cambridge examiner reports describe exactly this error: candidates writing a set’s total in the outer part of its circle instead of subtracting the overlap, and writing the totals into the regions without understanding what each region represents.

The correct working

(a) Start from the overlap and the outside, which you are given directly. Then work out the “only” parts by subtracting the overlap from each set total:

n(R \text{ only}) = 3x - x = 2x

n(D \text{ only}) = (2x + 4) - x = x + 4

Venn diagram of two overlapping circles R and D inside a rectangle labelled ξ. The part of R only contains 2x, the overlap contains x, the part of D only contains x + 4, and the region outside both circles contains 8. — Each region holds a different group of students: R only, both, D only and neither.

The four regions together make up all 60 students:

\begin{aligned} 2x + x + (x + 4) + 8 &= 60 \\ 4x + 12 &= 60 \\ 4x &= 48 \end{aligned}

x = 12

So $x = 12$ .

Check the circles: $R$ holds $2(12) + 12 = 36 = 3(12)$ ✓ and $D$ holds $16 + 12 = 28 = 2(12) + 4$ ✓.

(b) “Exactly one club” means $R$ only or $D$ only, not the overlap:

2x + (x + 4) = 24 + 16 = 40

So 40 students belong to exactly one club.

Why it works

Each region of a Venn diagram is a separate group of people, and every person is counted in exactly one region. The overlap is the group in both clubs, so it is part of circle $R$ and part of circle $D$ . A total like $n(R)$ is the whole circle, the outer part plus the overlap, so the outer part is what’s left after you take the overlap away.

That’s also why the equation works. Once every region holds a separate group, adding the four regions counts each of the 60 students once. It’s the same idea as the formula $n(R \cup D) = n(R) + n(D) - n(R \cap D)$ : here $36 + 28 - 12 = 52$ students are in at least one club, and $52 + 8 = 60$ ✓.

How to spot it next time

Fill a Venn diagram from the inside out: the overlap first, then the outside, then the “only” parts by subtracting.
Read $n(R)$ as “the whole circle $R$ ”, never as “ $R$ only”.
When you have an answer, add up each whole circle and check it matches the total you were given, not just the grand total.