Editing Vertical bar (section)

===Mathematics<!--'Double vertical bar' and 'Propositional truncation' redirect here-->===
The vertical bar is used as a [[table of mathematical symbols|mathematical symbol]] in numerous ways.
If used as a pair of brackets, it suggests the notion of the word "size". These are:
* [[absolute value]]: <math>|x|</math>, read "the ''absolute value'' of ''x''"<ref name="Weisstein">{{Cite web|last=Weisstein|first=Eric W.|title=Single Bar|url=https://mathworld.wolfram.com/SingleBar.html|access-date=2020-08-24|website=Wolfram MathWorld|language=en}}</ref>
* [[cardinality]]: <math>|S|</math>, read "the ''cardinality'' of the [[set (mathematics)|set]] ''S''" or "the ''length'' of a [[String (computer science)|string]] ''S''"
* [[determinant]]: <math>|A|</math>, read "the ''determinant'' of the [[Matrix (mathematics)|matrix]] ''A''".<ref name="Weisstein" /> When the matrix entries are written out, the determinant is denoted by surrounding the matrix entries by vertical bars instead of the usual brackets or parentheses of the matrix, as in <math>\begin{vmatrix} a & b \\ c & d\end{vmatrix}</math>.
* [[Order (group theory)#Class equation|order]]: <math>|G|</math>, read "the ''order'' of the [[group (mathematics)|group]] ''G''", or <math>|g|</math>, "the ''order'' of the element <math>g \in G</math>"

Likewise, the vertical bar is also used singly in many different ways:
* [[conditional probability]]: <math>P(X|Y)</math>, read "the [[probability]] of ''X'' ''given'' ''Y''"
* [[distance]]: <math>P|ab</math>, denoting the shortest ''distance'' between point <math>P</math> to line <math>ab</math>, so line <math>P|ab</math> is perpendicular to line <math>ab</math>
* [[divisibility]]: <math>a \mid b</math>, read "''a'' ''divides'' ''b''" or "''a'' is a ''factor'' of ''b''", though Unicode also provides special 'divides' and 'does not divide' symbols ({{unichar|2223}} and {{unichar|2224}})<ref name="Weisstein" />
* [[Function (mathematics)|function]] evaluation: <math>f(x)|_{x=4}</math>, read "''f'' of ''x'', evaluated at ''x'' equals 4" (see [[b:LaTeX/Advanced Mathematics#Subscripts and superscripts|subscripts]] at Wikibooks)
* [[Restriction (mathematics)|restriction]]: <math>f|_{A}</math>, denoting the ''restriction'' of the function <math>f</math>, with a domain that is a superset of <math>A</math>, to just <math>A</math>
* [[set-builder notation]]: <math>\{x|x<2\}</math>, read "the set of ''x'' ''such that'' ''x'' is [[less than]] two". Often, a [[colon (punctuation)|colon]] ':' is used instead of a vertical bar
* the [[Sheffer stroke]] in [[logic]]: <math>a|b</math>, read "''a'' ''nand'' ''b''"
* [[subtraction]]: <math>f(x) \vert _a ^b</math>, read "''f(x)'' ''from'' ''a'' ''to'' ''b''", denoting <math>f(b) - f(a)</math>. Used in the context of a definite integral with variable ''x''.
* A vertical bar can be used to separate variables from fixed parameters in a function, for example <math>f(x|\mu,\sigma)</math>, or in the notation for [[elliptic integrals]].

The '''double vertical bar'''<!--boldface per WP:R#PLA-->, <math>\|</math>, is also employed in mathematics.
* [[Parallel (geometry)|parallelism]]: <math>AB \parallel CD</math>, read "the line <math>AB</math> ''is parallel to'' the line <math>CD</math>"
* [[Norm (mathematics)|norm]]: <math>\|A\|</math>, read "the ''norm'' (length, size, magnitude etc.) of the matrix <math>A</math>". The norm of a one-dimensional [[Vector space|vector]] is the absolute value and single bars are used.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Matrix Norm|url=https://mathworld.wolfram.com/MatrixNorm.html|access-date=2020-08-24|website=Wolfram MathWorld|language=en}}</ref>
* '''Propositional truncation'''<!--boldface per WP:R#PLA-->: (a [[Type theory|type]] former that truncates a type down to a [[proposition (logic)|mere proposition]] in [[homotopy type theory]]): for any <math>a : A</math> (read "term <math>a</math> of type <math> A</math>") we have <math>|a| : \left\| A \right\|</math><ref>{{cite book|author=Univalent Foundations Program|title=Homotopy Type Theory: Univalent Foundations of Mathematics (GitHub version)|year=2013|publisher=Institute for Advanced Study|url=https://hott.github.io/book/nightly/hott-a4-1075-g3c53219.pdf|page=108|access-date=2017-07-01|archive-date=2017-07-07|archive-url=https://web.archive.org/web/20170707022332/https://hott.github.io/book/nightly/hott-a4-1075-g3c53219.pdf|url-status=dead}}</ref> (here <math>|a|</math> reads "''[[Image (mathematics)|image]]'' of <math>a : A</math> in <math>\left\| A \right\|</math>" and <math>|a| : \left\| A \right\|</math> reads "''propositional truncation'' of <math display="inline">A</math>")<ref>{{cite book|author=Univalent Foundations Program|title=Homotopy Type Theory: Univalent Foundations of Mathematics (print version)|year=2013|publisher=Institute for Advanced Study|url=https://books.google.com/books?id=LkDUKMv3yp0C|page=450}}</ref>

In LaTeX [[TeX#math mode|mathematical mode]], the ASCII vertical bar produces a vertical line, and <code>\|</code> creates a double vertical line (<code>a | b \| c</code> is set as <math>a | b \| c</math>). This has different spacing from <code>\mid</code> and <code>\parallel</code>, which are [[relational operator]]s: <code>a \mid b \parallel c</code> is set as <math>a \mid b \parallel c</math>. See below about [[LaTeX]] in text mode.