# Definition:Element/Also known as

## Definition

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

- 1951: J.C. Burkill:
*The Lebesgue Integral*... (previous) ... (next): Chapter $\text {I}$: Sets of Points - 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Sets - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 1$: The Axiom of Extension - 1961: John G. Hocking and Gail S. Young:
*Topology*... (previous) ... (next): A Note on Set-Theoretic Concepts - 1964: Walter Rudin:
*Principles of Mathematical Analysis*(2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction: $1.3$. Notation - 1965: A.M. Arthurs:
*Probability Theory*... (previous) ... (next): Chapter $1$: Set Theory: $1.2$: Sets and subsets - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.1$. Sets - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$. Sets - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Introduction: Set-Theoretic Notation - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $1$ Set Theory: $1$. Sets and Functions: $1.1$: Basic definitions - 1970: Avner Friedman:
*Foundations of Modern Analysis*... (previous) ... (next): $\S 1.1$: Rings and Algebras - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): Chapter $1$: Sets, Functions, and Relations: $\S 1.1$: Sets and Membership - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Sets - 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Sets - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 2$: Introductory remarks on sets: $\text{(c)}$ - 1979: G.H. Hardy and E.M. Wright:
*An Introduction to the Theory of Numbers*(5th ed.) ... (previous) ... (next): Remarks on Notation - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point 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: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**member (element)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**member (element)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**member**