Definition:Element
This page is about element in the context of set theory. For other uses, see Chemical Element.
Definition
Let $S$ be a set.
An element of $S$ is a member of $S$.
Class-Theoretic Definition
In the context of class theory the definition is the same:
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.
Also see
- $\in$-Relation
- Definition:New Element
- Definition:Element of Matrix
- Definition:Generalized Element
- Definition:Global Element
- Definition:Element in Abelian Category
- Results about elements can be found here.
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
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