## 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”

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/(x^{2} − 1) | all the Real Numbers, except -1 and 1(because 1/(x^{2} − 1) is undefined at x = -1 or x = 1) | {x ∈ R | x ≠ -1, x ≠1} |