Set Description

These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 9 Mathematics Textbook, Unit 1, Chapter 1.2.

~4 min read

Summary

There are four ways to describe a set: verbal, complete listing (roster), partial listing, and set-builder.
The roster method lists every element inside braces { }.
The partial listing method uses three dots (…) when there are too many elements to list.
The set-builder method gives a rule, like {x | x is a natural number less than 11}.

Key Words

The verbal method describes a set in words (a sentence).
The roster method (complete listing) lists all elements inside braces { }.
The partial listing method lists a few elements and uses three dots (…) for the rest.
The set-builder method writes a rule that the elements must follow.
The number sets are ℕ (natural numbers) = {1, 2, 3, …}, whole numbers = {0, 1, 2, 3, …}, and ℤ (integers) = {…, -2, -1, 0, 1, 2, …}.

The Four Methods at a Glance

Method	How it works	Example
Verbal	Describe the set in words	“the even numbers less than 7”
Roster (complete)	List every element in braces	{2, 4, 6}
Partial listing	List a few, then use …	{1, 2, 3, …, 99}
Set-builder	Write a rule with x	{x \| x ∈ ℕ and x < 11}

Verbal Method (in words)

The verbal method describes the set with an ordinary sentence. For example, “the set of whole numbers greater than 1 and less than 20”.

Listing Methods

Complete listing (roster)

You list every element, separated by commas, inside braces { }. For example, the even numbers less than 7 are {2, 4, 6}.

Partial listing

When a set is too long, or never ends, you list a few elements and use three dots (…). For example, the natural numbers less than 100 are {1, 2, 3, …, 99}, and the whole numbers are {0, 1, 2, 3, …}.

Set-Builder Method (a rule)

You write a stand-in like x, then a vertical line | (or a colon :), then the rule that x must follow.
For example, {x | x ∈ ℕ and x < 11} means “all natural numbers less than 11”, which is {1, 2, 3, …, 10}.
The part before the line is the element. The part after the line is the rule.

Common Mistakes to Avoid

Do not forget the braces { } when you write a set by listing.
Use the three dots (…) only when the next elements are clear from the pattern.
In set-builder form, do not mix up the element (before the line) with the rule (after the line).

Easy Ways to Remember

Roster is like a roll-call: you name every element.
Set-builder is like a recipe: write x, then the rule x must follow.

Quiz

Tap an answer to check it.

1. If M = {x : x is a prime number less than 14}, what is the roster form of M?

2. Which set-builder rule best describes B = {2, 4, 8, 16, 32}?

3. The natural numbers that are divisible by 3 and less than or equal to 30, in roster form, are:

Remember: A set can be described in words, by listing its elements in braces { }, or by a rule in set-builder form like {x | x ∈ ℕ and x < 11}.