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Sets

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Senior 1 · Mathematics

Sets

5 lessons

Lesson 1 of 5

Sets, elements and cardinal number

22 min

A collection with a clear rule

Rwanda is divided into four provinces and the City of Kigali. Write the four provinces between curly brackets and you have made a set:

P = \{\text{Northern},\ \text{Southern},\ \text{Eastern},\ \text{Western}\}

Anyone who reads $P$ can decide at once whether a place belongs to it: the Western Province does; the City of Kigali and Lake Kivu do not. A clear yes or no for every object is what makes a collection a set in mathematics.

Definition

Set: a well-defined collection of distinct objects. Well-defined means that, for any object, you can decide without opinion whether it belongs to the collection.

Notation: a set is named with a capital letter, and its members are written between curly brackets (braces), separated by commas.

Example: $V = \{a, e, i, o, u\}$ is the set of vowels of the English alphabet. "The tall learners in S1 A" is not a set, because people disagree about who is tall.

Definition

Element (member): an object that belongs to a set.

Notation: $x \in A$ is read " $x$ is an element of $A$ "; $x \notin A$ is read " $x$ is not an element of $A$ ".

Example: for $V = \{a, e, i, o, u\}$ , $e \in V$ and $k \notin V$ .

A set can also be drawn as a closed loop with its elements written inside. Anything written outside the loop is not an element.

The set $V$ drawn as a loop: $a \in V$ , but $k \notin V$ .

Three ways to describe a set

Method	How it works	The odd numbers between 2 and 12
In words	a sentence that states the rule	"the odd numbers greater than 2 and less than 12"
Listing (roster) method	write every element between braces	$\{3, 5, 7, 9, 11\}$
Set-builder notation	a variable, a colon, then the rule the variable must obey	$\{x : x \text{ is odd},\ 2 \lt x \lt 12\}$

Read $\{x : \ldots\}$ as "the set of all $x$ such that …". Some books use a vertical bar instead of the colon: $\{x \mid x \text{ is odd}\}$ means the same thing.

When a list is long, or never ends, three dots (an ellipsis) show that the pattern continues. $\{1, 2, 3, \ldots, 100\}$ stops at 100, while $\{2, 4, 6, \ldots\}$ goes on forever.

Two rules apply to every list:

Order does not matter: $\{1, 2, 3\}$ and $\{3, 1, 2\}$ are the same set.
Each element is written once: the letters of the word KIGALI form the set $\{K, I, G, A, L\}$ . The second I is the same element, so it is not written again.

Worked example 1: changing the description

(a) List the elements of $A = \{x : x \text{ is a whole number and } x^2 \lt 30\}$ .
(b) Write $B = \{5, 10, 15, 20, 25, 30\}$ in set-builder notation.

Solution

(a) Test the whole numbers in order: $0^{2} = 0$ , $1^{2} = 1$ , $2^{2} = 4$ , $3^{2} = 9$ , $4^{2} = 16$ and $5^{2} = 25$ are all less than 30, but $6^{2} = 36$ is not, and every larger number has a larger square. So

A = \{0, 1, 2, 3, 4, 5\}.

(b) Every element is a multiple of 5, from 5 up to 30:

B = \{x : x \text{ is a multiple of } 5,\ 5 \le x \le 30\}.

How many elements?

Definition

Cardinal number of a finite set: the number of distinct elements in the set.

Notation: $n (A)$ , read "n of A". Some books write $∣ A ∣$ .

Example: $n(\{a, e, i, o, u\}) = 5$ . If $D$ is the set of the districts of Rwanda, then $n (D) = 30$ .

Special sets

Some sets are important enough to have their own names.

Definition

Empty set (null set): the set with no elements.

Notation: $\varnothing$ or ${}$ . Its cardinal number is $n(\varnothing) = 0$ .

Example: $\{x : x \text{ is a month with 32 days}\} = \varnothing$ .

Definition

Singleton (unit set): a set with exactly one element, so $n (A) = 1$ .

Example: $\{x : x \text{ is an even prime number}\} = \{2\}$ .

Definition

Finite set: a set whose elements can be counted to an end, so $n (A)$ is a whole number.

Infinite set: a set that is not finite. However many elements you count, more remain.

Examples: the districts of Rwanda form a finite set; $\{2, 4, 6, 8, \ldots\}$ is an infinite set.

Warning

Three ways to get the empty set wrong

${0}$ is not empty. It has one element, the number 0, so $n(\{0\}) = 1$ .
$\{\varnothing\}$ is not empty either: it is a box that contains an empty box, so $n(\{\varnothing\}) = 1$ .
Write the empty set as $\varnothing$ or as ${}$ , never as $\{\varnothing\}$ or ${0}$ .

Comparing two sets: equal and equivalent

Definition

Equal sets: two sets that have exactly the same elements.

Notation: $A = B$ . Example: $\{1, 2, 3\} = \{3, 2, 1\}$ .

Definition

Equivalent sets: two finite sets with the same number of elements, $n (A) = n (B)$ .

Notation: $A \leftrightarrow B$ (some books write $A \sim B$ ).

Example: $\{a, b, c\} \leftrightarrow \{1, 2, 3\}$ : the elements can be paired off one to one, $a$ with 1, $b$ with 2 and $c$ with 3.

Important

Equal sets are always equivalent, but equivalent sets need not be equal.

Why: if $A = B$ , the two sets have the same elements, so they certainly have the same number of them. But $\{a, b, c\}$ and $\{1, 2, 3\}$ have the same number of elements without sharing a single one.

Worked example 2: equal or only equivalent?

Let $R$ be the set of the letters of the word RWANDA, let $D = \{D, W, N, R, A\}$ , and let $K$ be the set of the letters of the word KIGALI. Which pairs of sets are equal, and which are equivalent?

Solution

List each set, writing every letter once:

$R = \{R, W, A, N, D\}$ , so $n (R) = 5$ ;
$D = \{D, W, N, R, A\}$ , so $n (D) = 5$ ;
$K = \{K, I, G, A, L\}$ , so $n (K) = 5$ .

Now compare them.

$R$ and $D$ have exactly the same letters, so $R = D$ , and therefore also $R \leftrightarrow D$ .
$R$ and $K$ both have 5 elements, but $W \in R$ while $W \notin K$ . So $R \leftrightarrow K$ but $R \neq K$ .
In the same way, $D \leftrightarrow K$ and $D \neq K$ .

Worked example 3: naming special sets

Say whether each set is empty, a singleton, finite or infinite, and give its cardinal number where it has one.

(a) $E = \{x : x \text{ is a whole number between 7 and 8}\}$ (b) $F = \{x : x + 4 = 9\}$
(c) $G = \{x : x \text{ is a district of the City of Kigali}\}$ (d) $H = \{5, 10, 15, 20, \ldots\}$

Solution

No whole number lies strictly between 7 and 8, so $E = \varnothing$ and $n (E) = 0$ .
Only $x = 5$ makes $x + 4 = 9$ true, so $F = {5}$ , a singleton: $n (F) = 1$ .
$G = \{\text{Gasabo},\ \text{Kicukiro},\ \text{Nyarugenge}\}$ , a finite set with $n (G) = 3$ .
The multiples of 5 never end, so $H$ is infinite; no whole number counts its elements.

Tip

The symbol $\in$ is a stylised Greek letter epsilon, ε. Read it as "is an Element of". When you list a set from a word or a long list, cross off each letter or number as you write it, so that no element is written twice.

Khan Academy India: What are sets? (6 min). Watch for the idea of a well-defined collection, and for the number sets named near the end, which you will meet again in Unit 2.

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Practice

Work these without looking at the answers. They get harder as you go.

Write in listing form: (a) the set of the vowels in the word MATHEMATICS; (b) $\{x : x \text{ is a whole number},\ 3 \le x \lt 8\}$ .
True or false? (a) $7 \in \{1, 3, 5, 7\}$ (b) $4 \notin \{x : x \text{ is even}\}$ (c) Musanze $\in$ {the districts of the Northern Province}.
Write $\{1, 4, 9, 16, 25\}$ in set-builder notation.
Find $n (A)$ when (a) $A$ is the set of the letters of the word NYABARONGO; (b) $A = \{x : x \text{ is a multiple of } 3,\ 1 \le x \le 30\}$ .
Say whether each set is empty, a singleton, finite or infinite: (a) $\{x : x \text{ is a whole number and } x + 5 = 2\}$ (b) ${0}$ (c) the whole numbers greater than 1000 (d) the months whose names start with J.
Let $A$ be the set of the factors of 6, $B = \{1, 2, 3, 6\}$ , $C$ the set of the letters of the word NOON and $D = \{0, 1\}$ . Which two sets are equal? Which two are equivalent but not equal?
List $A = \{x : x \text{ is a whole number and } 2x + 1 \lt 12\}$ and find $n (A)$ .

Show answersHide answers

(a) $\{A, E, I\}$ (b) $\{3, 4, 5, 6, 7\}$
(a) True (b) False: 4 is even, so $4 \in$ the set (c) True: Musanze is in the Northern Province.
$\{x : x \text{ is a square number},\ 1 \le x \le 25\}$ , or $\{k^2 : k \text{ is a whole number},\ 1 \le k \le 5\}$ .
(a) $\{N, Y, A, B, R, O, G\}$ , so $n (A) = 7$ (b) $\{3, 6, 9, \ldots, 30\}$ , so $n (A) = 10$ .
(a) Empty: $x = - 3$ is not a whole number (b) singleton (c) infinite (d) finite: {January, June, July}, with $n = 3$ .
$A = \{1, 2, 3, 6\} = B$ . $C = \{N, O\}$ and $D = \{0, 1\}$ both have 2 elements, so $C \leftrightarrow D$ , but $C \neq D$ .
$2x + 1 \lt 12$ gives $2x \lt 11$ , so $x \lt 5.5$ . Then $A = \{0, 1, 2, 3, 4, 5\}$ and $n (A) = 6$ .

Sources and credits

1.1: Basic Set Concepts, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
Video: What are sets?, Khan Academy India, embedded from YouTube.

The Rwandan worked examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1A collection with a clear rule

2Three ways to describe a set

3How many elements?

4Special sets

5Comparing two sets: equal and equivalent

A collection with a clear rule

Three ways to describe a set

How many elements?

Special sets

Comparing two sets: equal and equivalent