--- aliases: [complement] --- #sets-functions ## Definition > [!tldr] Definition > If $A$ is a set, the **complement** of $A$ is the set of all points in the [[Universal set|universal set]] that are not in $A$. The notation for the complement of $A$ is $\overline{A}$. > > In [[Set-builder notation|set-builder notation]], $\overline{A} = \{ x \in U \, : \, x \not \in A\}$ (where $U$ is the [[universal set]]). A visual representation of $\overline{A}$ is this Venn diagram: ![[complement.png|300]] (Image credit: [Kyle Shevlin](https://kyleshevlin.com/set-theory)) **Notes:** - The complement of $A$ is also $U \setminus A$, the [[Set difference|set difference]] between the [[Universal set|universal set]] and $A$. - Some alternative notation for the complement of $A$ includes $A'$ , $A^c$, and $A^*$. ## Examples - If $U = \{1,2,3,\dots, 10\}$ and $A = \{1, 2, 3, 4\}$, then $\overline{A} = \{5, 6, 7, 8, 9, 10\}$. - If the [[universal set]] is $\mathbb{N}$ (the [[Natural numbers|natural numbers]]) and $A = \{n \in \mathbb{N} \, : \, n \ \text{is a multiple of 2}\}$ then $\overline{A}$ is the set of [[natural numbers]] that are not multiples of 2 -- that is, the set of all odd [[natural numbers]]. ## Resources <iframe src="https://player.vimeo.com/video/606600971?h=e256db4c8a" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe> <p><a href="https://vimeo.com/606600971">Screencast 3.6: Set operations</a> from <a href="https://vimeo.com/user132700952">Robert Talbert</a> on <a href="https://vimeo.com">Vimeo</a>.</p> Other resources: - Video: [Sets and set operations](https://www.youtube.com/watch?v=QiOfsWm3peE&list=PL2419488168AE7001&index=64&pp=iAQB) - Tutorial: [Complements of sets](https://www.cuemath.com/algebra/complement-of-a-set/)