Sets

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand and use set language, notation and Venn diagrams to describe sets
  2. Use set notation including: n(A), A′, ξ, A ∪ B, A ∩ B
  3. Understand and use set language, notation and Venn diagrams to represent relationships between sets
  4. Use extended set notation including: ∈, ∉, ∅, A ⊆ B, A ⊈ B, and work with Venn diagrams for two or three sets

What is a Set?

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A set is a collection of distinct objects or numbers. The key word here is "distinct" – this means you cannot have repeated items in a set. Each item appears only once.

For example:

  • The set of vowels in the English alphabet is {a, e, i, o, u}
  • The set of even numbers less than 10 is {2, 4, 6, 8}

We write sets using curly brackets { } and list the items inside, separated by commas.

Set Notation and Symbols

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Sets have their own special language with symbols that help us write about them quickly. Here are the main symbols you need to know:

Basic Symbols

ξ (xi) – The universal set
This is the "universe" for your question – it contains all the items you're considering. Think of it as the big box that holds everything relevant to the problem.

n(A) – The number of elements in set A
This tells you how many items are in the set. For example, if A = {2, 4, 6, 8}, then n(A) = 4.

A′ (read as "A prime" or "A complement") – The complement of set A
This means everything in the universal set that is NOT in A. It's "everything except A."

∅ – The empty set
A set with no elements at all. We write it as { } or use the symbol ∅. For example, the set of negative natural numbers is empty.

Membership Symbols

∈ – "is an element of" or "belongs to"
For example: 3 ∈ {1, 2, 3, 4} means "3 belongs to the set {1, 2, 3, 4}."

∉ – "is not an element of" or "does not belong to"
For example: 5 ∉ {1, 2, 3, 4} means "5 does not belong to the set {1, 2, 3, 4}."

Subset Symbols

A ⊆ B – "A is a subset of B"
This means every element in set A is also in set B. Think of A as being contained completely inside B.

A ⊈ B – "A is not a subset of B"
This means at least one element in A is not in B.

Example:
If A = {1, 3, 5} and B = {1, 2, 3, 4, 5}, then A ⊆ B (A is a subset of B).
If C = {1, 3, 7} and B = {1, 2, 3, 4, 5}, then C ⊈ B (C is not a subset of B, because 7 is in C but not in B).

Set Operations

A ∪ B (read as "A union B") – The union of A and B
This means combine both sets together, including all elements from A and all elements from B. If an element appears in both sets, you only write it once (remember, no repeats in sets!).

Example:
If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}

A ∩ B (read as "A intersection B") – The intersection of A and B
This means only the elements that appear in BOTH set A AND set B at the same time.

Example:
If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}

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