Definition:Element

Definition

An element is a member of a set.

We use the symbol $x \in S$ to mean $x$ is an element of the set $S$.

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

but this is rarely seen.

Some texts (usually older ones) use $x \ \overline \in \ S$ or $x \ \in' \ S$ instead of $x \notin S$.

The term member is sometimes used (probably more for the sake of linguistic variation than anything else).

In the context of geometry, elements of a set are often called points, in particular when they are (geometric) points.

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.

$x \ \varepsilon \ S$ could still be seen in works as late as Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951).

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 universal.

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 1$
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 1$: The Axiom of Extension
• W.E. Deskins: Abstract Algebra (1964): $\S 1.1$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.1$
• Seth Warner: Modern Algebra (1965): $\S 1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.2$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Introduction
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.1$
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 1$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.1$
• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.1$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 2 \ \text{(c)}$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.1$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.1$