# Definition:Element/Notation

## Definition

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 $x \notin S$.

## 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

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 1$: Operations on Sets - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set? - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Basic Notations - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 6$ Significance of the results