# 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

 Symbol Details Note { } the set of x all x just a place-holder, it could be anything : or | Such that x < 4 restrictions

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}

 Symbol Details ∈ Element of R Real Number

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

## Number Types

 Symbols Details N Natural Numbers Z Integers Q Rational Numbers R Real Numbers I Imaginary Numbers C Complex 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) Domain Notation 1/x all the Real Numbers, except 0(because 1/x is undefined at x = 0) {x ∈ R | x ≠ 0} √x all 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}