---
tag: 0SJ3RX9TXA
kind: definition
name: Set
msc: ["03E20"]
defines: 0BSJ7RQ9BW
uses: [0EZ758XE73]
sources:
  - url: https://en.wikipedia.org/wiki/Set_(mathematics)
    locator: lead section
---

## Definition

A set is a collection of objects, called the elements of the set.
For an object $x$ and a set $A$, either $x$ is an [[element|0EZ758XE73]] of $A$, written $x \in A$, or it is not, written $x \notin A$.
Two sets $A$ and $B$ are equal when they have the same elements: for each $x$, $x \in A$ if and only if $x \in B$.
A finite set can be written by a list of its elements in braces, for example $\{1, 2, 3\}$.

## Remarks

The sets $\{1, 2, 3\}$, $\{3, 2, 1\}$ and $\{1, 1, 2, 3\}$ are the same set.
In [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], "set" and "$\in$" are primitive notions, and the axioms say how they behave.
Not every collection can be a set. [[Russell's paradox|0HESMMSYEG]] shows this.

## Explanations

### L1 human

A set is like a bag of objects, where only membership counts.
The bag $\{1, 2, 3\}$ contains $1$, $2$ and $3$, and nothing else.
You cannot put the same object in twice, and the objects have no order.