---
aliases: [universe, universal set]
---
#sets-functions
## Definition
> [!tldr] Definition
> The **universal set** is the [[set]] consisting of all possible [[Set|elements]] that are being referred to in a particular context. All other sets are [[Subset|subsets]] of the universal set.
Notes:
- There is no single universal set -- it depends on the context of the discussion or problem. See examples below.
- We sometimes use the script letter $\mathcal{U}$ to represent the universal set in a specific use case.
- In a Venn diagram, the universal set is often depicted as a rectangle that contains everything else, for example:
![[venn-diagram-of-universal-set-1627631133.png]]
([Source](https://www.cuemath.com/algebra/universal-set/))
## Examples and Non-Examples
- In some cases the universal set will be stated explicitly. For example, if we are trying to find the complement of the set $B = \{1,2,3\}$, the universal set has to be stated explicitly before this can be done, and the result depends on the choice of the set. For example if $\mathcal{U} = \{0,1,2,\dots,10\}$ then $\overline{B} = \{0,4,5,6,7,8,9,10\}$. But if $\mathcal{U} = \mathbb{N}$ then the complement is $\overline{B} = \{0, 4, 5, 6, 7, \dots\}$.
- Other times, the universal set is implied from context. For example, in the statement "For all $x>0$, $\sqrt{x} < x
quot; the universal set can be assumed to be $\mathbb{R}$, the set of all real numbers unless otherwise specified.