Textbook
Membership relates an object to a set. The statement $x\in A$ says that $x$ is an element of $A$; $x\notin A$ denies it. Membership requires well-defined elements and a precise description of the set.
For a finite set given by enumeration, order and repetition do not change its elements: $\{1,2,2,3\}=\{3,2,1\}$. The empty set is written $\varnothing$. For every object $x$, the statement $x\in\varnothing$ is false.
A set can itself be an object: $\varnothing\in\{\varnothing\}$, whereas $\varnothing\notin\varnothing$. Braces change the object rather than decorate it. Membership must be distinguished from inclusion, which compares two sets.
For a set specified by a condition, membership is checked by substituting the object into that condition. For an enumeration, finding a matching element is sufficient. These descriptions do not change the meaning of $\in$. When an element is itself a set, that entire object is compared rather than its contents being automatically unpacked. For example, $2\in\{2\}$ does not imply $2\in\{\{2\}\}$: the latter has the set $\{2\}$ as its sole element, not the number two.
A different relation is described in set inclusion.
In plain language
Membership resembles checking a guest list: the specified person is either listed or absent, and repeating a name creates no extra guest. A guest and a list bearing that name remain different objects, like an item and a box containing it. An empty box contains nothing, while a box containing one empty box has an item: modest possessions, but a valid inventory. This picture distinguishes nesting levels without making a mathematical set a physical container or giving its elements a layout order.
Example
The illustrative set $A=\{1,2,3\}$ is specified by enumeration. Each object is checked against the stated set; in the last step, the object is itself the empty set.
Set membership
An element is present
The number 1 is listed as an element of A.
Step 1 of 3: An element is present
Read all steps
An element is present
The number 1 is listed as an element of A.
- A
- 1current23
An element is absent
The number 4 is absent from A, regardless of its position on a number line.
- A
- 123
- Checking
- 4excluded
A set as an element
The outer set contains one object: the empty set.
- {β }
- β included
- β
- β
Both roles change in the final step: the candidate object is $\varnothing$, and the containing set is $\{\varnothing\}$. The first stepβs answer about one says nothing about this new pair. The entire left-hand object is compared with the right-hand setβs outer-level elements. A common mistake is to notice the empty set inside and declare the whole outer structure empty. The outer braces are precisely what gives it an element.