Sets of Numbers

Lessons

Senior 1 · Mathematics

Sets of Numbers

5 lessons

Lesson 1 of 5

Natural numbers and their properties

24 min

The numbers you count with

Count the desks in your classroom, the cows in a herd or the districts of Rwanda, and the answer is one of 1, 2, 3, 4, … These counting numbers form the first and most basic set of numbers. A football match, though, can end with no goals at all, so we also need a set that includes 0.

Definition

Natural numbers: the counting numbers.

\mathbb{N} = \{1, 2, 3, 4, \ldots\}

Notation: $\mathbb{N}$ , a double-struck N. The set has a first element, 1, and no last element, so it is infinite.

Example: $30 \in \mathbb{N}$ (the number of districts of Rwanda), but $0 \notin \mathbb{N}$ , $-2 \notin \mathbb{N}$ and $2.5 \notin \mathbb{N}$ .

Definition

Whole numbers: the natural numbers together with 0.

W = \{0, 1, 2, 3, \ldots\} = \mathbb{N} \cup \{0\}

Example: the number of goals in a match is a whole number. Since every natural number is a whole number, $\mathbb{N} \subset W$ .

Warning

Is 0 a natural number? In this course, as in Rwanda's S1 Mathematics textbook, no: $\mathbb{N} = \{1, 2, 3, \ldots\}$ and $0 \notin \mathbb{N}$ . Some books, especially French ones, start $\mathbb{N}$ at 0 and write $\mathbb{N}^*$ for $\{1, 2, 3, \ldots\}$ . When you use another book, check its convention before you answer.

The natural numbers are the separate blue dots 1, 2, 3, …; the whole numbers add the orange dot at 0.

Subsets of $\mathbb{N}$

Many sets you already know are subsets of $\mathbb{N}$ .

Subset	Set-builder notation	Listing
even numbers	$\{2k : k \in \mathbb{N}\}$	$\{2, 4, 6, 8, \ldots\}$
odd numbers	$\{2k - 1 : k \in \mathbb{N}\}$	$\{1, 3, 5, 7, \ldots\}$
multiples of 5	$\{5k : k \in \mathbb{N}\}$	$\{5, 10, 15, 20, \ldots\}$
square numbers	$\{k^2 : k \in \mathbb{N}\}$	$\{1, 4, 9, 16, 25, \ldots\}$
divisors of 12	$\{d \in \mathbb{N} : d \text{ divides } 12\}$	$\{1, 2, 3, 4, 6, 12\}$
prime numbers	$\{p \in \mathbb{N} : p \text{ has exactly two divisors}\}$	$\{2, 3, 5, 7, 11, 13, \ldots\}$

The whole number 0 is even, because $0 = 2 \times 0$ , but it is not in $\mathbb{N}$ , so it does not appear in these subsets.

Definition

Divisor and multiple: $a$ is a divisor (factor) of $b$ if $b = a \times k$ for some $k \in \mathbb{N}$ ; then $b$ is a multiple of $a$ .

Notation: $D (b)$ is the set of divisors of $b$ ; $M (a)$ is the set of multiples of $a$ .

Example: $36 = 4 \times 9$ , so 4 is a divisor of 36 and 36 is a multiple of 4. $D(18) = \{1, 2, 3, 6, 9, 18\}$ .

Definition

Prime number: a natural number with exactly two divisors, 1 and itself.

Example: 13 is prime, since $D(13) = \{1, 13\}$ . 15 is not prime, since $D(15) = \{1, 3, 5, 15\}$ ; such a number is called composite.

Warning

1 is not a prime number. It has only one divisor, itself, not two. And 2 is the only even prime: every other even number has 2 as a third divisor.

Worked example 1: sets of divisors

(a) List $P = \{x \in \mathbb{N} : x \text{ is prime and } 10 \lt x \lt 30\}$ .
(b) Find $D(18) \cap D(24)$ and use it to state the highest common factor of 18 and 24.

Solution

(a) Test 11 to 29, crossing out even numbers and multiples of 3, 5 and 7: $P = \{11, 13, 17, 19, 23, 29\}$ .

(b) $D(18) = \{1, 2, 3, 6, 9, 18\}$ and $D(24) = \{1, 2, 3, 4, 6, 8, 12, 24\}$ . The common divisors are

D(18) \cap D(24) = \{1, 2, 3, 6\},

and the largest of them, 6, is the highest common factor (HCF) of 18 and 24.

The common divisors of 18 and 24 sit in the overlap.

Operations on $\mathbb{N}$ and their properties

Add or multiply two natural numbers and the answer is always a natural number. Subtract or divide them and it may not be: $3 - 5$ and $3 \div 2$ are not in $\mathbb{N}$ , and neither is $3 - 3 = 0$ . This difference has a name.

Definition

Closed: a set is closed under an operation if applying the operation to any two elements of the set always gives an element of the same set.

Example: $\mathbb{N}$ is closed under addition and multiplication, but not under subtraction ( $3 - 5 = -2 \notin \mathbb{N}$ ) or division ( $3 \div 2 = 1.5 \notin \mathbb{N}$ ).

Property	Addition	Multiplication
Closure	$a + b \in \mathbb{N}$	$a \times b \in \mathbb{N}$
Commutative	$a + b = b + a$	$a \times b = b \times a$
Associative	$(a + b) + c = a + (b + c)$	$(a \times b) \times c = a \times (b \times c)$
Identity element	$a + 0 = a$ : the identity is 0, which is in $W$ but not in $\mathbb{N}$	$a \times 1 = a$ : the identity is 1
Zero property (in $W$ )	—	$a \times 0 = 0$
Distributive	$a \times (b + c) = a \times b + a \times c$ : multiplication distributes over addition

Why multiplication is commutative: an array of dots with 3 rows of 4 is the same array as 4 rows of 3, turned a quarter turn. Turning it does not change the number of dots.

$3 \times 4 = 4 \times 3$ : the same 12 dots, seen two ways.

Why multiplication distributes over addition: a rectangle 7 units by 13 units can be cut into a 7 by 10 piece and a 7 by 3 piece. The area is the same before and after the cut.

$7 \times (10 + 3) = 7 \times 10 + 7 \times 3$

Worked example 2: using the properties to calculate mentally

(a) $25 \times 37 \times 4$ (b) $17 \times 98 + 17 \times 2$

Solution

(a) Change the order (commutative) and regroup (associative) to make an easy product first:

25 \times 37 \times 4 = (25 \times 4) \times 37 = 100 \times 37 = 3700.

(b) Both terms contain 17, so use the distributive property backwards:

17 \times 98 + 17 \times 2 = 17 \times (98 + 2) = 17 \times 100 = 1700.

Worked example 3: properties that fail

Show that subtraction in $\mathbb{N}$ is not closed, not commutative and not associative.

Solution One counterexample is enough for each.

Not closed: $4 - 9 = - 5$ , and $-5 \notin \mathbb{N}$ .
Not commutative: $10 - 4 = 6$ , but $4 - 10 = - 6$ .
Not associative: $(10 - 4) - 2 = 4$ , but $10 - (4 - 2) = 8$ .

Tip

To prove that a property fails, one counterexample is enough. To show that a property always holds, examples are not enough: you need a reason that works for every number, like the dot array or the rectangle above.

Khan Academy India: Closure property of whole numbers (6 min). The video works with the whole numbers 0, 1, 2, … Watch why addition and multiplication always stay inside the set, while subtraction and division can leave it.

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Practice

Show your working. The last two are harder.

List (a) $\{x \in \mathbb{N} : x \lt 6\}$ (b) $\{x \in W : x \lt 6\}$ (c) the prime numbers less than 20.
List $D (30)$ and $D (45)$ . Find $D(30) \cap D(45)$ and the HCF of 30 and 45.
Name the property used: (a) $8 + 5 = 5 + 8$ (b) $(2 \times 7) \times 5 = 2 \times (7 \times 5)$ (c) $9 \times (4 + 6) = 9 \times 4 + 9 \times 6$ (d) $13 \times 1 = 13$
Calculate mentally, naming the properties you use: (a) $4 \times 19 \times 25$ (b) $36 \times 7 + 36 \times 3$
Is $\mathbb{N}$ closed under division? Give a reason.
Is the set of odd numbers closed under addition? Under multiplication? Explain.
Show with an example that division in $\mathbb{N}$ is not associative.

Show answersHide answers

(a) $\{1, 2, 3, 4, 5\}$ (b) $\{0, 1, 2, 3, 4, 5\}$ (c) $\{2, 3, 5, 7, 11, 13, 17, 19\}$
$D(30) = \{1, 2, 3, 5, 6, 10, 15, 30\}$ , $D(45) = \{1, 3, 5, 9, 15, 45\}$ ; $D(30) \cap D(45) = \{1, 3, 5, 15\}$ , so the HCF is 15.
(a) commutative (addition) (b) associative (multiplication) (c) distributive (d) identity element 1 for multiplication
(a) $(4 \times 25) \times 19 = 100 \times 19 = 1900$ (commutative, associative) (b) $36 \times (7 + 3) = 360$ (distributive)
No: for example $7 \div 2 = 3.5 \notin \mathbb{N}$ .
Addition: no, $3 + 5 = 8$ is even. Multiplication: yes. Any two odd numbers can be written $2 a + 1$ and $2 b + 1$ with $a$ , $b$ whole numbers, and $(2 a + 1) (2 b + 1) = 2 (2 a b + a + b) + 1$ is always odd.
$(24 \div 4) \div 2 = 6 \div 2 = 3$ , but $24 \div (4 \div 2) = 24 \div 2 = 12$ .

Sources and credits

7.2: Commutative and Associative Properties (Part 1), from Prealgebra by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
7.4: Distributive Property, from Prealgebra by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
The sections on the natural and whole numbers, their subsets and closure. Written by Ganzaa.
Video: Closure property of Whole Numbers, Khan Academy India, embedded from YouTube.

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

1The numbers you count with

2Subsets of N \mathbb{N} N

3Operations on N \mathbb{N} N and their properties

The numbers you count with

Subsets of $\mathbb{N}$

Operations on $\mathbb{N}$ and their properties