Definition:Element

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Definition

Let $S$ be a set.


An element of $S$ is a member of $S$.


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:

$S \ni x$ means the set $S$ has $x$ as an element, that is, $x$ is an element of $S$

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 (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 $S$
  • $S$ 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


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.[1]

$x \mathop \varepsilon S$ could still be seen in works as late as 1951: Nathan Jacobson: Lectures in Abstract Algebra: 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.


References


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