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