Cambridge IGCSE · International Maths (Extended)
Sets: Notation & Venn Diagrams
Big idea: A set is just a labelled bag of things, and set notation is
simply a shorthand language for describing which items are in which bags, whether
the bags overlap, and what's left over.
Summary — What's in This Chapter
- What a set is, and the different ways to describe a set (listing elements, or using a rule with a colon).
- The universal set ℰ — "everything we care about right now."
- Core symbols:
∈, ∉, ∅, ⊆, n(A).
- Combining sets: intersection (∩, "and/overlap"), union (∪, "or/everything"), and complement (A′, "everything NOT in A").
- How to read and draw a Venn diagram, and which region each symbol corresponds to.
- How to combine symbols (e.g. A′ ∩ B′) and reason about them using a diagram.
1. What Is a Set — and How Do We Describe One?
Think of a set as a labelled bag. You can put anything into it —
numbers, letters, coordinates, even other sets — as long as you're clear about
what belongs and what doesn't. The things inside are called elements.
There are two main ways to describe what's in the bag:
Method 1 — Just list the elements
Put the elements inside curly brackets { }, separated by commas.
Example
{1, 2, 3, 6} is the set of factors of 6. That's it — you've listed every element, job done.
Method 2 — Describe the elements using a rule
This is useful when a set is too big (or infinite!) to list out, or when you want
to describe a pattern rather than individual values. The format is:
Format
{ type of element : rule the element must satisfy }
Read the colon as "such that." Everything before it tells you what kind of thing x is (an integer? a coordinate?); everything after it tells you the condition x must meet.
You'll sometimes see a vertical bar | used instead of the colon — it means exactly the same thing. Both of these mean the same set:
{x : x² < 30} = {x | x² < 30}
Worked Example
{x is a positive integer : x² < 30}
Read this step by step:
- "x is a positive integer" — x must be a whole number greater than 0: 1, 2, 3, 4, 5, 6...
- "x² < 30" — but only keep the ones whose square is less than 30.
- Check each: 1²=1 ✓, 2²=4 ✓, 3²=9 ✓, 4²=16 ✓, 5²=25 ✓, 6²=36 ✗ (too big — stop here).
So this set equals {1, 2, 3, 4, 5}.
Watch out
If
no type is specified before the colon — e.g.
{x : x² < 30} — then x can be
any kind of number: a fraction, a decimal, an irrational number like √2. Don't assume it means "positive integer" unless it actually says so!
Set notation can even describe geometry. {(x, y) : y = mx + c}
means "the set of every coordinate pair (x, y) that lies on the line y = mx + c" —
in other words, this describes an entire straight line using set language.
Practice Question
Write out the elements of the set {x is a positive integer : x² < 50}.
2. The Universal Set and Key Symbols
The universal set, ℰ
ℰ means "everything we're currently interested in" — the full playing field.
Every other set in the question is a smaller collection drawn from inside ℰ. You might also see
it written as U (careful — that's different from the union symbol ∪!),
S, or the Greek letter ξ (xi).
n(A) — the size of a set
Rule
4. Venn Diagrams