Roster method
Best when the elements can be listed without creating a long or unclear expression.
Learning Objectives
- describe sets using words;
- write sets using roster or listing notation;
- use set-builder notation;
- convert from one description method to another;
- use common number-set symbols appropriately.
Introduction
This lesson develops Describing Sets from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🌍 Why This Matters
Examiners often give a set in one form and ask you to rewrite it in another. Being able to move between words, a list and set-builder notation is therefore an essential skill.
🌍 Why Do We Need Different Ways to Describe Sets?
A short set can be listed easily. A large or infinite set is often clearer when described by a rule. Mathematics uses whichever method communicates the collection most efficiently.
Three Ways to Describe a Set
One Set
Words
Mathematical notation
Description
Roster method
Set-builder method
1. Description in Words
A set may be described using a clear sentence.
A is the set of even natural numbers less than 10.
2. Roster or Listing Method
The elements are written inside braces and separated by commas.
A = {2, 4, 6, 8}
List each distinct element once. Use an ellipsis only when the pattern
is unmistakable.
3. Set-Builder Method
Set-builder notation describes the rule that every element follows.
A = {x ∈ ℕ : x is even and x < 10}
This is read as: “A is the set of natural numbers x such that x is even and less than 10.”
Comparing the Methods
Set-builder method
Best when a rule describes the collection more efficiently.
| Words | Roster | Set-builder |
|---|---|---|
| Odd natural numbers below 10 | {1, 3, 5, 7, 9} | {x ∈ ℕ : x is odd and x < 10} |
| Factors of 12 | {1, 2, 3, 4, 6, 12} | {x ∈ ℕ : x divides 12} |
Common Number Sets
Worked Example 1
Write the factors of 18 using roster notation.
1
Find all natural numbers that divide 18 exactly.
2
They are 1, 2, 3, 6, 9 and 18.
3
Write F = {1, 2, 3, 6, 9, 18}.
Worked Example 2
Write B = {5, 10, 15, 20, 25} using set-builder notation.
1
All elements are multiples of 5.
2
The largest listed value is 25.
3
B = {x ∈ ℕ : x is a multiple of 5 and x ≤ 25}.
🔍 How Do I Recognise the Required Method?
- “List the elements” means use roster notation.
- “Describe using set-builder notation” means write a rule.
- “State the set in words” means use a clear verbal description.
- For an infinite set, a rule is usually better than a long list.
⚠️ Common Mistakes
Mistake 1: Writing a vague rule such as “x is small.”
Mistake 2: Forgetting the number system, for example whether x is natural, integer or real.
Mistake 3: Using an ellipsis when the pattern is unclear.
Mistake 2: Forgetting the number system, for example whether x is natural, integer or real.
Mistake 3: Using an ellipsis when the pattern is unclear.
✅ Quick Check
List the prime numbers less than 12.
Describe {2, 4, 6, 8} in words.
Write the integers from −2 to 3 using set-builder notation.
💡 Exam Tip
In set-builder notation, state both the kind of number and the condition it must satisfy. For example, {x ∈ ℕ : x < 6} is clearer than simply writing {x : x < 6}.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Write the vowels of the English alphabet using roster notation.
- List the positive factors of 20.
- Write {1, 4, 9, 16, 25} in words.
- Write the even natural numbers below 14 using set-builder notation.
- Convert {−3, −2, −1, 0, 1, 2, 3} to set-builder notation.
- Write the multiples of 7 from 7 to 49 in roster and set-builder forms.
- Explain why {1, 2, 3, ...} needs a verbal rule to be completely clear.
- Write the set of integer solutions of x² < 10 in roster notation.
✅ Check Your Work
Show practice answers
Foundation
- {a, e, i, o, u}
- {1, 2, 4, 5, 10, 20}
- The first five positive square numbers.
- {x ∈ ℕ : x is even and x < 14}
Confidence
- {x ∈ ℤ : −3 ≤ x ≤ 3}
-
Roster: {7, 14, 21, 28, 35, 42, 49}
Set-builder: {x ∈ ℕ : x is a multiple of 7 and 7 ≤ x ≤ 49} - The dots show continuation but do not state whether the set contains natural numbers, integers or another pattern.
Challenge
-
{−3, −2, −1, 0, 1, 2, 3}
x² < 10, so |x| < √10 ≈ 3.16. The integer values are therefore −3 through 3.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Sets may be described in words, by listing or by a rule.
- Roster notation lists elements inside braces.
- Set-builder notation states the membership rule.
- Choose the method that communicates the set most clearly.