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”
Table of Contents
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} |