# Definition:Set/Explicit Set Definition

## Definition

A (finite) set can be defined by explicitly specifying all of its elements between the famous curly brackets, known as set braces: $\set {}$.

When a set is defined like this, note that all and only the elements in it are listed.

This is called explicit (set) definition.

It is possible for a set to contain other sets. For example:

$S = \set {a, \set a}$

### Warning

It is important to distinguish between an element, for example $a$, and a singleton containing it, that is, $\set a$.

That is $a$ and $\set a$ are not the same thing.

While it is true that:

$a \in \set a$

it is not true that:

$a = \set a$

neither is it true that:

$a \in a$

## Also known as

Some sources refer to this as a roster for the set.

Others call it an enumeration or a listing.

## Examples

### Example 1

$A := \set {\dfrac 1 2, 1, \sqrt 2, e, \pi}$

### Example 2

$B := \set {\textrm {Romeo}, \textrm {Juliet} }$