From ProofWiki
Jump to navigation Jump to search

This page is about Element in the context of Set Theory. For other uses, see Chemical Element.


Let $S$ be a set.

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

Class-Theoretic Definition

In the context of class theory the definition is the same:

Let $S$ be a class.

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 $S$ 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 as a synonym for element (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

  • Results about elements can be found here.

Historical Note

The symbol for is an element of originated as $\varepsilon$, first used by Giuseppe Peano in his Arithmetices prinicipia nova methodo exposita of $1889$. 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 $1903$.

See Earliest Uses of Symbols of Set Theory and Logic in Jeff Miller's website Earliest Uses of Various Mathematical Symbols.

$x \mathop \varepsilon S$ could still be seen in works as late as 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts and 1955: John L. Kelley: General Topology.

Paul Halmos wrote in Naive Set Theory 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 practically universal.