---
#sets-functions
## Definition
> [!tldr] Definition
> **Set-builder notation** is a way of writing a [[Set|set]] where the set is defined by the rules that its elements must satisfy in order to be [[Set|elements]] of the set. Set-builder notation has two forms:
>
> - ("*Domain/filter*") The [[universal set]] from which elements are selected is given first, then a [[Predicate|predicate]] that filters out all but the elements of the set.
> - ("*Formula/domain*") A formula is given first, then a [[universal set]] on which the formula is applied.
>
> Examples of each form are below.
Note:
- A closely related Python concept is the [list comprehension](https://www.w3schools.com/python/python_lists_comprehension.asp), where a list is defined in terms of rules and formulas rather than explicitly listing the elements. For example the list `[0,3,6,9]` is generated from the list comprehension `[x for x in range(10) if x % 3 == 0]`. This is equivalent to the set, in set-builder notation, of $\{x \in \{0,1,2,\dots,9\} \, : \, 3 \ \text{divides} \ x\}$.
## Examples
- Domain/filter: The set $\{ x \in \mathbb{N} \, : \, x^2 < 100 \}$ takes the natural numbers as its [[universal set]] (or "domain") and filters out everything except those natural numbers whose square is less than 100. In [[roster notation]], this set is equal to $\{0,1,2,3,4,5,6,7,8,9\}$. This way of writing a set is helpful when you are selecting elements from a larger group using rules that can be expressed through predicates.
- Domain/filter: The set $\{a \in \mathbb{Z} \, : \, |a| \leq 1 \}$ is the set $\{-1, 0, 1\}$.
- Formula/domain: The set $\{x^2 \, : \, x \in \mathbb{N}\}$ takes the formula $x^2$ and "maps" it over the set of all natural numbers. The resulting set in [[roster notation]] is $\{0, 1, 4, 9, 16, 25, 36, 49, \dots \}$. This way of writing a set is helpful when the elements of the set can be easily expressed through mathematical computations or constructions.
- Formula/domain: The set $\{(t, t+1) \, : \ t \in \mathbb{N}\}$ takes each natural number and creates a pair or "tuple" with the natural number in the first coordinate and the next natural number in the second coordinate. It is equal to the set $\{(0,1), (1,2), (2,3), (3,4), \dots \}$.
Below are **incorrect** ways to use set-builder notation:
- (*Putting a predicate first, then a domain*) Set builder notation with a predicate first does not make mathematical sense. For example: $\{x < 2 \, : \, x \in \mathbb{Z}\}$ is incorrect.
- (*Putting a set first, then a formula*) Set builder notation with a formula *second* also does not make mathematical sense. For example, $\{x \in \mathbb{Z} \, : \, x+2\}$ is incorrect.
The following flowchart can be used to determine whether the set builder notation is correct and how to convert it to roster notation. (Use the navigation tools to zoom in/out and pan around the flowchart.)
```mermaid
flowchart LR
%% nodes
A("Look at the expression in the notation")
B("Is it a predicate or a formula?")
C1("Does it come first or second?")
C2("Does it come first or second?")
D("Incorrect syntax")
E("Filter the set using the predicate")
F("Apply formula to each element")
G("Does the expression evaluate to True/False?")
H("It's a predicate")
I("It's a formula")
%% edge connections
A --> B
B -->|Predicate| C1
B -->|Formula| C2
C1 -->|First| D
C1 -->|Second| E
C2 -->|First| F
C2 -->|Second| D
B -->|I don't know| G
G -->|Yes| H
G -->|No| I
H --> B
I --> B
```
## Resources
<div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/602744516?badge=0&autopause=0&player_id=0&app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Screencast 3.2: Roster and set-builder notation"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script>
Other resources:
- Video: [Set builder notation](https://www.youtube.com/watch?v=O84JrsT5r1o)
- Tutorial: [Set builder notation](https://www.cuemath.com/algebra/set-builder-notation/)
## Practice
- [This tutorial has some interactive practice exercises at the end](https://www.mathgoodies.com/lessons/sets/set-builder-notation).
- [General practice worksheet on sets that includes set-builder notation practice](https://www.cabrini.edu/globalassets/pdfs-website/math-resource-center/math-111-practice-test-chapter-2-answers.pdf).