The vertical bar, |, is a glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in logic), pipe, bar, or (literally the word "or"), vbar, and others.^[1]

Usage

Mathematics

The vertical bar is used as a mathematical symbol in numerous ways. If used as a pair of brackets, it suggests the notion of the word "size". These are:

• absolute value: ${\displaystyle |x|}$, read "the absolute value of x"^[2]
• cardinality: ${\displaystyle |S|}$, read "the cardinality of the set S" or "the length of a string S"
• determinant: ${\displaystyle |A|}$, read "the determinant of the matrix A".^[2] 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 ${\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}}$.
• order: ${\displaystyle |G|}$, read "the order of the group G", or ${\displaystyle |g|}$, "the order of the element ${\displaystyle g\in G}$"

Likewise, the vertical bar is also used singly in many different ways:

• conditional probability: ${\displaystyle P(X|Y)}$, read "the probability of X given Y"
• distance: ${\displaystyle P|ab}$, denoting the shortest distance between point ${\displaystyle P}$ to line ${\displaystyle ab}$, so line ${\displaystyle P|ab}$ is perpendicular to line ${\displaystyle ab}$
• divisibility: ${\displaystyle a\mid b}$, read "a divides b" or "a is a factor of b", though Unicode also provides special 'divides' and 'does not divide' symbols (U+2223 and U+2224:∣, ∤)^[2]
• function evaluation: ${\displaystyle f(x)|_{x=4}}$, read "f of x, evaluated at x equals 4" (see subscripts at Wikibooks)
• restriction: ${\displaystyle f|_{A}}$, denoting the restriction of the function ${\displaystyle f}$, with a domain that is a superset of ${\displaystyle A}$, to just ${\displaystyle A}$
• set-builder notation: ${\displaystyle \{x|x<2\}}$, read "the set of x such that x is less than two". Often, a colon ':' is used instead of a vertical bar
• the Sheffer stroke in logic: ${\displaystyle a|b}$, read "a nand b"
• subtraction: ${\displaystyle f(x)\vert _{a}^{b}}$, read "f(x) from a to b", denoting ${\displaystyle f(b)-f(a)}$. 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 ${\displaystyle f(x|\mu ,\sigma )}$, or in the notation for elliptic integrals.

The double vertical bar, ${\displaystyle \|}$, is also employed in mathematics.

• parallelism: ${\displaystyle AB\parallel CD}$, read "the line ${\displaystyle AB}$ is parallel to the line ${\displaystyle CD}$"
• norm: ${\displaystyle \|A\|}$, read "the norm (length, size, magnitude etc.) of the matrix ${\displaystyle A}$". The norm of a one-dimensional vector is the absolute value and single bars are used.^[3]
• Propositional truncation: (a type former that truncates a type down to a mere proposition in homotopy type theory): for any ${\displaystyle a:A}$ (read "term ${\displaystyle a}$ of type ${\displaystyle A}$") we have ${\displaystyle |a|:\left\|A\right\|}$^[4] (here ${\displaystyle |a|}$ reads "image of ${\displaystyle a:A}$ in ${\displaystyle \left\|A\right\|}$" and ${\displaystyle |a|:\left\|A\right\|}$ reads "propositional truncation of ${\displaystyle A}$")^[5]

In LaTeX mathematical mode, the ASCII vertical bar produces a vertical line, and \| creates a double vertical line (a | b \| c is set as ${\displaystyle a|b\|c}$). This has different spacing from \mid and \parallel, which are relational operators: a \mid b \parallel c is set as ${\displaystyle a\mid b\parallel c}$. See below about LaTeX in text mode.

Chemistry

In chemistry, the vertical line is used in cell notation of electrochemical cells.

Example,

Zn | Zn²⁺ || Cu²⁺ | Cu

Single vertical lines show components of the cell which do not mix, usually being in different phases. The double vertical line ( || ) is used to represent salt bridge; which is used to allow free moving ions to move.

Physics

The vertical bar is used in bra–ket notation in quantum physics. Examples:

• ${\displaystyle |\psi \rangle }$: the quantum physical state $$Zdroj:https://en.wikipedia.org?pojem=Pipe_character
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