Set-Builder Notation

A Set is a collection of things (often numbers). 
Example: {2, 3, 5} is a set.

Here is a simple example of set-builder notation:

General Form: {formula for elements: restrictions} or

{formula for elements| restrictions}


{x | x < 4} or {x : x < 4}


{ }the set of 
xall xjust a place-holder, it could be anything
: or |Such that 
x < 4restrictions 

So {x | x < 4} or {x : x < 4} says “the set of all x less than 4”

Show the Type of Number

You can show what type of number x is.


{ x ∈ R | x ≠ 7}

Element of
RReal Number

So it says “the set of all real numbers except 7”

Number Types

NNatural Numbers
QRational Numbers
RReal Numbers
IImaginary Numbers
CComplex Numbers

Defining a Domain

Set Builder Notation is really useful for defining a domain of a function. The domain is the set of all the values that go into a function.


1/xall the Real Numbers, except 0(because 1/x is undefined at x = 0){x ∈ R | x ≠ 0}
√xall the Real Numbers from 0 onwards(because there’s no square root of a negative number){x ∈ R | x > 0}
1/(x2 − 1)all the Real Numbers, except -1 and 1(because 1/(x2 − 1) is undefined at x = -1 or x = 1){x ∈ R | x ≠ -1, x ≠1}

Learn More

Injective, Surjective & Bijective


Function Transformations

Inverse Functions

Set Index