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}

Examples

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

Details

SymbolDetailsNote
{ }the set of 
xall x just 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.

Examples

{ x ∈ R | x ≠ 7}

SymbolDetails
Element of
RReal Number

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

Number Types

SymbolsDetails
NNatural Numbers
ZIntegers
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.

Examples

f(x)DomainNotation
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

Intervals

Function Transformations

Inverse Functions

Set Index

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