Set Notation & Venn Diagrams

Revise Set Notation & Venn Diagrams for Mathematics 0580 (O Level) — revision notes and instant AI marking.

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1. What is a Set?

A set is simply a well-defined collection of elements. Think of it as a special kind of "group" where the rules are crystal clear about what belongs and what doesn't.

Set: A well-defined collection of distinct elements.

Elements can be numbers, letters, coordinates, or any objects.

📌 How to Write a Set

There are two main ways to describe a set:

Method 1:

List the elements in curly brackets: {1, 2, 3, 6}
This is the set of factors of 6

Method 2:

Use a rule with a colon: {x : x < 30}
This means "x such that x is less than 30"

📌 Understanding Set Notation with Rules

When you see notation like {x : rule}, it means:

Example 1: {x is a positive integer : x² < 30}

This means: positive integers whose square is less than 30
x = 1: 1² = 1 ✓
x = 2: 2² = 4 ✓
x = 3: 3² = 9 ✓
x = 4: 4² = 16 ✓
x = 5: 5² = 25 ✓
x = 6: 6² = 36 ✗ (too big)
Answer: {1, 2, 3, 4, 5}

Example 2: {(x, y) : y = mx + c}

This is the set of ALL coordinates that lie on the line y = mx + c
This is an infinite set!

🧠 Key Point:

The colon (:) or vertical bar (|) means "such that"
Before the colon = what type of element (x, positive integer, etc.)
After the colon = the rule (condition) the element must satisfy
If NO type specified, x can be ANY number (fractions, decimals, irrationals)

Q1: Write out the set {x is a positive integer : x < 5}

Q2: If no type is specified, what does {x : x < 5} include?

2. Essential Set Notation Symbols

You need to know these symbols and what they mean. They're the language of set theory!

Symbol	Meaning	Example
ℰ	Universal set (everything we care about)	ℰ = {1, 2, 3, 4, 5, 6, 7, 8}
n(A)	Number of elements in set A (cardinality)
∈	Is an element of
∉
∅

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Also in the full note

3. Subsets (⊆)
4. Venn Diagrams
5. Complement (A')
6. Worked Example – Reading a Venn Diagram
📋 Chapter Summary
🧠 What to Memorise
✅ Concepts Checklist
🎯 Exam Tips & Common Mistakes

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Set Notation & Venn Diagrams