• Sets can be described in the following different ways.

  • In this method, the well-defined description of the elements of the set is written in an ordinary English language statement form (in words).

a. The set of integer numbers greater than -3 and less than 10. 

b. The set of female students in this chemistry class.

a) Complete listing method (Roster Method) .

  • In this method, all elements of the set are completely listed. The elements are separated by commas and are enclosed within set braces, { }

a. The set of all prime numbers  less than 8 is described in the complete listing method as {2, 3, 5, 7}. 

b. The set of all vowel letters in the English alphabet is described in the complete listing method as {𝑎,𝑒, 𝑖, 𝑜, 𝑢}.

b) Partial listing method 

  • We use this method if listing all elements of a set is difficult or impossible, but the elements can be indicated clearly by listing a few of them that fully describe the set.

Use partial listing method to describe the following sets. 

a. The set of natural numbers less than 1000. 

{1,2,3,4,..............,1000}

b. The set of integer numbers

{...-3,-2,-1 ,0 ,1,2,3,......}

  • The set-builder method is described by a property that its member must satisfy the common property. 

  • This is the method of writing the condition to be satisfied by a set or property of a set.

  • In set brace, write the representative of the elements of a set, for example 𝑥, and then write the condition that 𝑥 should satisfy after the vertical line (|) or colon ( : )

                   

For example, A = {3, 4, 5, 6, 7, 8, 9} can be described as


The set of natural numbers, whole numbers, and integers are denoted by ℕ, 𝕎, and ℤ, respectively. They are defined as 

            ℕ = {1, 2, 3, . . .} ,

            𝕎 = {0, 1, 2, 3, . . .}, 

             ℤ = {. . . . −3, −2, −1, 0 , 1, 2, 3, . . . }.

Describe the following sets using set builder method. 

i) Set 𝐴 = {-3, -2, -1, …, 6} 

A = {x | x ∈ ℤ and -4 < x < 7}

                Or 

A = {x | x ∈ ℤ and -3 ≤ x ≤ 6}

ii) Let set 𝐵 = {4, 8, 12, 16, ….}

B = {4x | x ∈ ℕ}

         Or 

A = {x | x = 4n for n ∈ ℕ}

1. Describe each of the following sets using a verbal method. 

a. 𝐴 = { 5, 6, 7, 8, 9} 

  • A set of natural numbers between 4 and 10

  • A set of natural number between and including 5 and 9 

b. 𝑀 = {2, 3, 5, 7, 11, 13}

  • A set of prime numbers less than or equal to 13  

c. 𝐺 = {8, 9, 10, … .} 

  • A set of natural numbers greater than 7

  • A set of natural numbers greater than or equal to 8

d. 𝐸 = {1, 3, 5, … , 99} 

  • A set of odd numbers less than 100

2. Describe each of the following sets using complete and partial listing method (if possible): 

a. The set of positive even natural numbers below or equal to 10. 

{2,4,6,8,10}

b. The set of positive even natural numbers below or equal to 30. 

{2,4,6,.....,30}

c. The set of non-negative integers. 

{0,1,2,3,.....} = 𝕎

d. The set of even natural numbers.

{2,4,6,8,.....}

e. The set of natural numbers less than 100 and divisible by 5.

 

{5,10,15,.....,95}

f. The set of integers divisible by 3.

{.....,-6,-3,0,3,6,9,......}

3. List the elements of the following sets:

a. 𝐴 = {3𝑥 | 𝑥 ∈ 𝕎} 

A = {0,3,6,9,12,.....}

b. 𝐵 = {𝑥 | 𝑥 ∈ ℕ and 5 < 𝑥 < 10}

B = {6,7,8,9}

4. Write the following sets using set builder method. 

a. 𝐴 = {1, 3, 5 …. } 

A = {2x - 1 |  x ∈ ℕ }

    = {x ; n ∈ ℕ and x = 2n - 1 }

b. 𝐵 = {2, 4, 6, 8} 

B = {2x ; x ∈ {1,2,3,4} }

    = {x ; x = 2n for n ∈ {1,2,3,4} }

c. 𝐶 = {1, 4, 9, 16, 25}

C = {x² |  x ∈ {1,2,3,4,5} }

d. 𝐷 = {4, 6, 8, 10, … , 52 } 

D = {2x |  x ∈ {2,3,4,......,26}}

    = {x ; x = 2n ; n ∈ {2,3,4,......,26}}

e. 𝐸 = {−10, . . . , −3, −2, −1, 0, 1, 2, … , 5} 

E = {x ;  x ∈ Z and -11 < x < 6

    = {x; x ∈ Z and -10 ≤ x  ≤ 5

f. 𝐹 = {1, 4, 9, … .}


F = {x²|  x ∈ ℕ }



Last modified: Thursday, 25 September 2025, 10:58 AM