# Definition:Element/Class

## Definition

Let $S$ be a class.

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

## Notation

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:

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.

## 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.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory