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Sets

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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).

Example

If we only care about factors of 24, then ℰ = {1, 2, 3, 4, 6, 8, 12, 24}. Nothing outside this list exists as far as the problem is concerned.

n(A) — the size of a set

Rule

n(A) = the number of elements in set A

Just count what's inside the bag.

Example: if A = {1, 4, 9}, then n(A) = 3.

∈ and ∉ — membership

Symbol	Meaning	Example
∈	"is an element of"	3 ∈ {1, 2, 3} — true, 3 is in the set
∉	"is NOT an element of"	4 ∉ {1, 2, 3} — true, 4 isn't in the set

∅ — the empty set

∅ is the set with nothing in it. It's a completely valid, legitimate set — just an empty one. Think of it as an empty bag, not a "missing" bag.

⊆ — subsets

Rule

A ⊆ B means "every element of A is also in B"

A is entirely contained within B — like a smaller bag that fits completely inside a bigger bag.

Example

If A = {1, 2, 3} and B = {0, 1, 2, 3, 4}, then every element of A (1, 2, 3) is also found in B. So A ⊆ B.

The crossed-out version, A ⊄ B, simply means "A is not a subset of B" — at least one element of A is missing from B.

Practice Question

Let ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10}, B = {1, 2, 3, 4}. (a) Find n(A). (b) Is B ⊆ A? Explain.

3. Combining Sets: Intersection, Union, Complement

This is the heart of the chapter. There are three ways to combine or modify sets, and mixing them up is the #1 mistake students make. Let's nail each one with a clear phrase you can say in your head.

A ∩ B — Intersection ("AND")

Rule

A ∩ B = elements that are in BOTH A and B

Say it as "A and B." It's the overlap — only what the two sets share.

Example

A = {1, 2, 3, 4, 5} and B = {-1, 1, 4, 7, 8}.
Which numbers appear in both lists? Just 1 and 4.
So A ∩ B = {1, 4}.

A ∪ B — Union ("OR")

Rule

A ∪ B = elements that are in A, or B, or both

Say it as "A or B." Combine every element from both sets into one big list — but if something appears in both, only write it down once.

Example

A = {5, 6, 7, 8} and B = {3, 7, 11}.
List everything from A, then add anything new from B (7 is already there, so skip it).
So A ∪ B = {3, 5, 6, 7, 8, 11}.

Memory trick

∪ looks like a cup — it "collects" everything from both sets (union, more).
∩ looks like an upside-down cup that only catches what falls into the shared middle (intersection, less).

A′ — Complement ("NOT")

Rule

A′ = every element in ℰ that is NOT in A

Say it as "not A." Look at the universal set and cross out anything that's in A — what's left is A′.

Example

ℰ = {1, 2, 3, 4, 5} and A = {1, 3}.
Take everything in ℰ, remove 1 and 3, and you're left with A′ = {2, 4, 5}.

Practice Question

ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8, 10}. Find: (a) A ∩ B (b) A ∪ B (c) A′

4. Venn Diagrams

A Venn diagram is just a picture of everything we just covered. It turns abstract symbols into regions you can literally point at, which makes it much easier to reason about combinations of sets.

How it's built

A rectangle represents the universal set ℰ (everything).
A circle is drawn for each set inside that rectangle.
Circles overlap only if the sets share elements. If two sets have nothing in common, draw the circles apart (or not overlapping) — the diagram should reflect reality, not just look neat.

Reading the regions

         ℰ (the rectangle = everything)
       ______________________
       |    ___     ___     |
       | /   A \/ B   \   |
       | |     (A∩B)   |   |
       | \___/\___/    |
       |______________________|

Region	What it represents
A ∪ B	Both circles shaded entirely — everything in A or B (or both)
A ∩ B	Only the overlapping "lens" shape in the middle
A′	Everything outside the A circle (but still inside the rectangle)
A only	Inside A's circle, but outside B's circle
Outside both	Inside the rectangle, but outside both circles — this represents A′ ∩ B′

Visual trick

Whenever you see a combination like A′ ∩ B′, don't try to work it out purely in your head — sketch a quick Venn diagram, shade "not A" and "not B" separately, then find where the shading overlaps. It's the region outside both circles.

Fully Worked Example (from a real exam-style problem)

Setup

A Venn diagram shows sets A and B. Inside A only: 6, 12. Inside the overlap (A∩B): 14, 28. Inside B only: 7, 21, 35. Outside both circles (but inside ℰ): 5, 8, 1.

So: A = {2, 6, 12, 14, 28} and B = {7, 14, 21, 28, 35}.

(a) Find n(A).
Count the elements inside the A circle (including the overlap): 2, 6, 12, 14, 28 → 5 elements.
n(A) = 5

(b) What does {14, 28} represent?
These two numbers appear inside both circles — the overlap.
{14, 28} = A ∩ B

(c) Find A ∪ B.
List every element from A, then add any new ones from B (don't repeat 14 and 28, they're already listed):
A ∪ B = {2, 6, 7, 12, 14, 21, 28, 35}

(d) Is A′ ∩ B′ = ∅ true?
A′ ∩ B′ means "everything that's NOT in A, AND NOT in B" — i.e. outside both circles.
Looking at the diagram, there are three elements sitting outside both circles: 1, 5, 8.
Since A′ ∩ B′ = {1, 5, 8}, which is not empty, the statement is false.

Practice Question

ℰ = {whole numbers from 1 to 15}. A = {multiples of 3} and B = {multiples of 5}. (a) List the elements of A and B. (b) Find A ∩ B. (c) Find A ∪ B. (d) How many elements are in neither A nor B?

What to Memorise

Symbol	Name	Meaning
{ }	Curly brackets	Contains the elements of a set
: or \|	"Such that"	Separates the type of element from the rule it follows
ℰ	Universal set	Everything currently being considered
n(A)	Cardinality	The number of elements in set A
∈	Element of	"Is a member of" this set
∉	Not an element of	"Is not a member of" this set
∅	Empty set	A set with zero elements
⊆	Subset	Every element of A is also in B
A ∩ B	Intersection	Elements in both A and B ("and")
A ∪ B	Union	Elements in A, B, or both ("or")
A′	Complement	Elements in ℰ but not in A ("not")

Concepts Checklist

Tick off each concept once you're confident you could explain it without looking at your notes.

I can describe a set by listing its elements in curly brackets.

I can describe a set using rule notation {type : rule} and explain what each part means.

I understand what the universal set ℰ represents, and recognise its alternative symbols (U, S, ξ).

I know the difference between ∈ and ∉, and can use n(A) correctly.

I understand the empty set ∅ and subset notation ⊆.

I can find A ∩ B (intersection) by identifying shared elements.

I can find A ∪ B (union) by combining elements without repeating.

I can find A′ (complement) by removing A's elements from ℰ.

I can draw a Venn diagram with the correct overlapping regions.

I can shade or identify combined regions like A′ ∩ B′ using a Venn diagram.

I can read a completed Venn diagram and extract n(A), A∩B, A∪B from it.

Exam Tips & Common Mistakes

Mixing up ∩ and ∪. This is the single most common error. Before you write an answer, say the phrase out loud: ∩ = "and" (overlap, smaller result), ∪ = "or" (combine, bigger result). If your union answer is smaller than either original set, something's gone wrong.

Writing repeated elements twice in a union. When you find A ∪ B, elements that appear in both sets only get written once in the answer. A set never lists the same element twice.

Forgetting the complement is relative to ℰ, not to "nothing." A′ isn't just "everything that isn't A" in some abstract sense — it's specifically everything in the universal set that isn't in A. Always check what ℰ is defined as first.

Confusing U (universal set) with ∪ (union). Some textbooks use a capital U for the universal set, which looks dangerously similar to the union symbol ∪. Read the question context carefully to work out which one is meant.

Not sketching a Venn diagram for combined-symbol questions. Questions asking you to evaluate something like A′ ∩ B′ or (A ∪ B)′ are much easier and less error-prone if you shade a quick diagram rather than trying to reason abstractly. Examiners often award method marks just for a correctly shaded diagram, even if your final answer has a small slip.

Sloppy subset reasoning. To disprove A ⊆ B, you only need to find one element of A that's missing from B — you don't need to check every element once you've found a counterexample. To prove A ⊆ B, however, you do need to check every single element of A.

What examiners actually look for: correct use of set notation (not plain English) in your final answer, elements listed without repeats and typically in a sensible order, and clearly labelled Venn diagram regions when a diagram is requested or would help show your working.

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Sets

Sets: Notation & Venn Diagrams

Summary — What's in This Chapter

1. What Is a Set — and How Do We Describe One?

Method 1 — Just list the elements

Method 2 — Describe the elements using a rule

2. The Universal Set and Key Symbols

The universal set, ℰ

n(A) — the size of a set

∈ and ∉ — membership

∅ — the empty set

⊆ — subsets

3. Combining Sets: Intersection, Union, Complement

A ∩ B — Intersection ("AND")

A ∪ B — Union ("OR")

A′ — Complement ("NOT")

4. Venn Diagrams

How it's built

Reading the regions

Fully Worked Example (from a real exam-style problem)

What to Memorise

Concepts Checklist

Exam Tips & Common Mistakes

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