Definition:Element

Definition
Let $S$ be a set.

An element of $S$ is a member of $S$.

The symbol universally used in modern mainstream mathematics to mean $x$ is an element of $S$ is:
 * $x \in S$

Similarly, $x \notin S$ means $x$ is not an element of $S$.

The symbol can be reversed:
 * $S \ni x$ means the set $S$ has $x$ as an element, that is, $x$ is an element of $S$

but this is rarely seen.

Some texts (usually older ones) use $x \mathop {\overline \in} S$ or $x \mathop {\in'} S$ instead of $x \notin S$.

Also known as
The term member is sometimes used (probably more for the sake of linguistic variation than anything else).

In the contexts of geometry and topology, elements of a set are often called points, in particular when they are (geometric) points.

$x \in S$ can also be read as:


 * $x$ is in $S$
 * $x$ belongs to $S$
 * $S$ includes $x$
 * $x$ is included in $S$
 * $S$ contains $x$

However, beware of this latter usage: $S$ contains $x$ can also be interpreted as $x$ is a subset of $S$. Such is the scope for misinterpretation that it is mandatory that further explanation is added to make it clear whether you mean subset or element.

Also see

 * $\in$-Relation
 * Definition:New Element
 * Definition:Element of Matrix
 * Definition:Generalized Element
 * Definition:Global Element
 * Definition:Element in Abelian Category