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 $S$
 * $S$ contains $x$

Beware: "$S$ contains $x$" can be used to mean that $x$ is a subset of $S$.

Historical Note
The symbol originated as $\varepsilon$, first used by Giuseppe Peano in Arithmetices prinicipia nova methodo exposita (1889). It comes from the first letter of the Greek word meaning is.

The stylized version $\in$ was first used by Bertrand Russell in Principles of Mathematics in 1903.

$x \mathop \varepsilon S$ could still be seen in works as late as.

Paul Halmos wrote in in 1960 that:
 * This version [$\epsilon$] of the Greek letter epsilon is so often used to denote belonging that its use to denote anything else is almost prohibited. Most authors relegate $\epsilon$ to its set-theoretic use forever and use $\varepsilon$ when they need the fifth letter of the Greek alphabet.

However, since then the symbol $\in$ has been developed in such a style as to be easily distinguishable from $\epsilon$, and by the end of the 1960's the contemporary notation was universal.

Also see

 * $\in$-Relation