--- aliases: [sets, set, element, elements] --- #sets-functions ## Definition > [!tldr] Definition > A **set** is an unordered collection of distinct objects. An item that belongs to a set is said to be an **element** of the set. If $x$ is an element of the set $A$, we denote this by $x \in A$ ("$x$ is an element of $Aquot;). If $x$ is not an element of the set $A$, we denote this by $x \not \in A$. Notes * Sets can be described in a few different ways: * In words, for example "the set of all current MTH 225 students at Grand Valley State University" * As a symbol, usually a capital letter ($A$, $X$, etc.) * For some commonly-used sets we use a special letter -- for example $\mathbb{N}$ denotes the set of all [[Natural numbers]] and $\mathbb{Z}$ denotes the set of all [[Integers]]. * In **[[roster notation]]**, where we list the items of the set explicitly within curly braces, for example $\{0, 2, 4, 6\}$. * In [[Set-builder notation|set-builder notation]], where we describe the set using rules for deciding what its elements are, for example $\{n \in \mathbb{N} \, : \, n \ \text{is even and} \ 0 \leq n < 7\}$. This set is equal to $\{0,2,4,6\}$ * The word "distinct" in the definition indicates that **sets cannot contain duplicate items**. For example $\{2,2,2,2,2\}$ is not a properly-written set. This set is the same as $\{2\}$ and should be written this way, without duplicates. ## Examples and Non-Examples - The set of all letters in the word "valley" could be written as $\{v,a,l,e,y\}$. The ordering of the elements doesn't matter; but again we do not allow duplicate items. - There is no restriction on the types of objects allowed in a set. For example, $\{1, x, \{10,11\}, \emptyset\}$ is a set with four elements: The number $1$, the letter $x$, a two-element set, and the [[empty set]]. - Sets can be either finite or infinite. Finite sets, if they are small, are often just written in [[roster notation]]. - If the elements of a set have a pattern, we usually use dots ($\dots$) to indicate this. For example $\{3, 6, 9, 12, \dots, 96, 99\}$ is the set of all positive [[integers]] less than 100 that are multiples of 3. You could also write this set in [[Set-builder notation|set-builder notation]] as $\{a \in \mathbb{N} \, : \, 0 < a < 100 \ \text{and} \ a \% 3 = 0\}$. - Infinite sets that continue a pattern forever in the positive or negative $\infty$ direction also use dots. For example the set of all integer multiples of 10 is $\{\dots, -30, -20, -10, 0, 10, 20, 30, \dots\}$ ## Resources <div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/602725171?badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Screencast 3.1: Sets"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script> Other resources: - Tutorial: [Sets](https://www.mathsisfun.com/sets/sets-introduction.html) ## 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](https://www.cabrini.edu/globalassets/pdfs-website/math-resource-center/math-111-practice-test-chapter-2-answers.pdf).