What is a Set?

A set is a well-defined collection of objects. The objects in a set are called elements or members.

Set theory helps us group objects clearly and compare different groups. In CSEC Mathematics, sets may contain numbers, letters, names, shapes, or any objects that are clearly described.

What You Will Learn

• Understand what a set is
• Identify the elements of a set
• Use membership symbols correctly
• Find the cardinality of a set
• Recognise finite and infinite sets
• Understand the universal set and the empty set
• Write sets using correct notation

Key Words

Term	Meaning
Set	A well-defined collection of objects
Element / Member	An object inside a set
Cardinality	The number of elements in a set
Finite Set	A set with a countable number of elements
Infinite Set	A set that continues without end
Universal Set	The set containing all elements being discussed
Empty Set	A set with no elements

What Makes a Set Well-Defined?

A set must be well-defined. This means it must be clear whether an object belongs to the set or not.

How to Write a Set

Sets are usually written using curly brackets: { }.

This means set A contains the elements 2, 4, 6, and 8.

Diagram: A Simple Set

Membership Symbols

We use special symbols to show whether an element belongs to a set.

Symbol	Meaning	Example
∈	is an element of	4 ∈ A
∉	is not an element of	5 ∉ A

If A = {2, 4, 6, 8}, then:

Cardinality of a Set

The cardinality of a set is the number of elements in the set. It is written as n(A).

For example:

There are 4 elements in set A, so:

Finite and Infinite Sets

A finite set has a limited number of elements. You can count all the elements and stop.

An infinite set continues without end. It has no last element.

Type of Set	Example	Reason
Finite	{1, 2, 3, 4, 5}	The set stops at 5
Finite	{a, e, i, o, u}	There are only 5 vowels listed
Infinite	{1, 2, 3, 4, …}	The counting numbers continue forever
Infinite	{2, 4, 6, 8, …}	The even numbers continue forever

The Universal Set

The universal set is the set of all elements being discussed in a question. It is usually written as U.

For example, if a question is about the numbers from 1 to 10, then the universal set may be:

A smaller set can be chosen from the universal set.

Here, set A contains the even numbers from the universal set.

Diagram: Universal Set

The Empty Set

The empty set is a set with no elements. It can be written in two ways:

For example, the set of months with 40 days is an empty set because no month has 40 days.

Worked Examples

Example 1: A = {2, 4, 6, 8}. State n(A) and whether 5 ∈ A.

1. The elements of A are 2, 4, 6, and 8.
2. There are 4 elements, so n(A) = 4.
3. The number 5 is not listed in set A.
4. Therefore, 5 ∉ A.

Answer: n(A) = 4 and 5 ∉ A

Example 2: List the elements of the set of even numbers between 1 and 11.

1. The even numbers between 1 and 11 are 2, 4, 6, 8, and 10.
2. Write them inside curly brackets.

Answer: {2, 4, 6, 8, 10}

Example 3: Find the cardinality of B = {a, e, i, o, u}.

1. The elements are a, e, i, o, and u.
2. There are 5 elements.

Answer: n(B) = 5

Example 4: Is the set of positive integers divisible by 7 finite or infinite?

1. The set is {7, 14, 21, 28, …}.
2. The pattern continues without end.
3. There is no last multiple of 7.

Answer: Infinite set

Example 5: State whether each set is finite, infinite, or empty.

Set	Answer	Reason
{3, 6, 9, 12}	Finite	It has 4 elements
{5, 10, 15, 20, …}	Infinite	It continues without end
The set of square circles	Empty	No shape can be both a square and a circle

Exam-Style Multiple Choice

Common Mistakes

Practice Questions

1. 1. List the elements of the set of even numbers between 1 and 11.
2. 2. State the cardinality of B = {a, e, i, o, u}.
3. 3. Is the set {10, 20, 30, 40} finite or infinite?
4. 4. Is the set {3, 6, 9, 12, …} finite or infinite?
5. 5. Write the empty set in two different ways.
6. 6. If A = {1, 3, 5, 7, 9}, state whether 4 ∈ A or 4 ∉ A.
7. 7. If P = {Monday, Tuesday, Wednesday}, find n(P).
8. 8. Give one example of a well-defined set.
9. 9. Give one example of a set that is not well-defined.
10. 10. Let U = {1, 2, 3, 4, 5, 6}. Write the set of even numbers in U.