---
tag: 0BPYRMHJ48
kind: definition
name: Universal set
msc: ["03E20"]
defines: 0JNXSS9TJ9
uses: [0SJ3RX9TXA, 0FE8Q5S42Y]
sources:
  - url: https://en.wikipedia.org/wiki/Universal_set
    locator: lead section
---

## Definition

In a given context, fix a [[set|0SJ3RX9TXA]] $U$ such that all the sets under discussion are [[subsets|0FE8Q5S42Y]] of $U$.
The set $U$ is then the universal set, or the universe of discourse, of the context.

## Remarks

In elementary number theory, $U$ can be the set $\mathbb{Z}$ of integers.
The [[complement|0B1FSS1VZW]] of a set is taken relative to the universal set.
In [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], no set contains all sets. This is a consequence of [[Russell's paradox|0HESMMSYEG]].

## Explanations

### L1 human

The universal set is the "whole world" of a problem.
In a problem about real numbers, it is $\mathbb{R}$.
In a [[Venn diagram|0KCT6DPXJK]], it is the rectangle that contains all the circles.