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C1.2 Sets & Venn Diagrams

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01 — Explanation

What does set notation actually mean?

A set is a well-defined collection of objects, called its elements. Cambridge Core tier uses these symbols, always with just two sets:

Symbol	Meaning
$\mathscr{E}$	The universal set — everything under consideration
$n(A)$	The number of elements in set $A$
$A'$	The complement of $A$ — everything in $\mathscr{E}$ not in $A$
$A \cup B$	Union — in $A$ or $B$ (or both)
$A \cap B$	Intersection — in both $A$ and $B$

A set can be written by listing its elements, e.g. $B = \{a, b, c\}$ , or using set-builder notation, e.g. $A = \{x : x \text{ is a natural number}\}$ .

Quick way to remember $\cup$ vs $\cap$ : $\cup$ looks like a cup that scoops up everything (union = OR); $\cap$ points down like the roof of an "n" for aNd (intersection = AND).

02 — Explanation

How do I read a two-set Venn diagram?

Two overlapping circles inside a rectangle (the universal set) split the picture into four regions:

        Universal Set ℰ
   ┌───────────────────────────┐
   │   A only    A∩B   B only  │
   │  ┌───────┬──────┬───────┐ │
   │  │       │██████│       │ │
   │  └───────┴──────┴───────┘ │
   │      (outside both circles)│
   └───────────────────────────┘

The overlapping region is $A \cap B$ . Anything inside a circle but outside the overlap is " $A$ only" or " $B$ only". Anything outside both circles is in neither set.

03 — Explanation

How do I fill in a Venn diagram from a worded problem?

Work from the middle outwards:

Find the overlap, $n(A \cap B)$ , and place it in the centre first.
Subtract the overlap from each individual total to get the "only" regions:

n(A \text{ only}) = n(A) - n(A \cap B), \qquad n(B \text{ only}) = n(B) - n(A \cap B)

Whatever remains of the universal set total goes outside both circles.

Example: In a class of $32$ students, $19$ play football ( $F$ ), $15$ play basketball ( $B$ ), and $6$ play neither.

Students playing at least one sport: $32 - 6 = 26$ .
Adding the individual totals: $19 + 15 = 34$ — this is $8$ more than $26$ , because students in both sports were counted twice.
So $n(F \cap B) = 34 - 26 = 8$ .
Football only: $19 - 8 = 11$ . Basketball only: $15 - 8 = 7$ .
Check everything adds back to $32$ : $11 + 8 + 7 + 6 = 32$ ✓

Exam Tip

Never write the raw totals ( $19$ and $15$ ) straight into the circles — that double-counts the overlap. Always find $n(A \cap B)$ first.

04 — Explanation

How do I use a Venn diagram to find a probability?

Once every region of the diagram is filled with a number of elements, a probability is just:

P(\text{event}) = \frac{\text{number of elements in that region}}{n(\mathscr{E})}

Using the class above, the probability a randomly chosen student plays basketball only is $\dfrac{7}{32}$ .

05 — Worked example

Worked exam-style question

In a survey of $50$ people, $28$ own a cat, $22$ own a dog, and $9$ own neither. One person is chosen at random.

(a) Find the number of people who own both a cat and a dog.

(b) Find the probability that the person owns a cat but not a dog.

(a) People owning at least one pet: $50 - 9 = 41$ . Adding individual totals: $28 + 22 = 50$ , which is $9$ more than $41$ , so:

n(\text{cat} \cap \text{dog}) = 50 - 41 = 9

(b) Cat only $= 28 - 9 = 19$ , so:

P(\text{cat, not dog}) = \frac{19}{50}

06 — Common mistakes

Common mistakes to avoid

Placing raw totals straight into each circle instead of removing the overlap first — this double-counts the intersection.
Confusing $n(A)$ with the set itself. $n(A)$ is always a single count (a number), never a list of elements.
Mixing up $\cup$ and $\cap$ under time pressure — use the memory trick above if you're unsure mid-exam.