How To Add Dotted Vertical Line In Equation Latex

Typographic symbol ( | )

"｜" redirects here. For the use of a similar-looking graphic symbol in vertical Japanese writing, see Chōonpu.

" ‖ " redirects hither. For the use of a similar-looking character in African languages, see lateral clicks.

|
Vertical bar
In Unicode	U+007C | VERTICAL LINE (|, |, |)
Related
See too	U+00A6 ¦ BROKEN BAR (¦) U+2016 ‖ DOUBLE VERTICAL LINE (‖, ‖) U+2223 ∣ DIVIDES

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

Usage [edit]

Mathematics [edit]

The vertical bar is used as a mathematical symbol in numerous ways:

• absolute value: ${\displaystyle |x|}$ , read "the absolute value of x"^[2]
• cardinality: ${\displaystyle |Southward|}$ , read "the cardinality of the set up S"
• conditional probability: ${\displaystyle P(X|Y)}$ , reads "the probability of X given Y"
• determinant: ${\displaystyle |A|}$ , read "the determinant of the matrix A".^[2] When the matrix entries are written out, the determinant is denoted past surrounding the matrix entries by vertical bars instead of the usual brackets or parentheses of the matrix, as in ${\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}}$ .
• distance: ${\displaystyle P|ab}$ , cogent 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 cistron of b", though Unicode also provides special 'divides' and 'does not separate' symbols (U+2223 and U+2224: ∣, ∤)^[2]
• evaluation: ${\displaystyle f(x)|_{x=4}}$ , read "f of ten, evaluated at x equals 4" (meet subscripts at Wikibooks)
• length: ${\displaystyle |due south|}$ , read "the length of the string s"
• norm: ${\displaystyle |\mathbf {5} |}$ , read "the norm of the (greater-than-one-dimensional) vector ${\displaystyle \mathbf {five} }$ " (notation that absolute value is a ane-dimensional norm), although a double vertical bar (see below) is more than often used to avoid ambiguity.
• order: ${\displaystyle |G|}$ , read "the gild of the group Yard"
• brake: ${\displaystyle f|_{A}}$ , cogent the restriction of the function ${\displaystyle f}$ , with a domain that is a superset of ${\displaystyle A}$ , to just ${\displaystyle A}$
• set-builder notation: ${\displaystyle \{10|ten<2\}}$ , read "the prepare of 10 such that x is less than ii". Oftentimes, 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 exist used to separate variables from fixed parameters in a part, for example ${\displaystyle f(x|\mu ,\sigma )}$ , or in the notation for elliptic integrals.

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

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 prepare equally ${\displaystyle a\mid b\parallel c}$ . Run across below about LaTeX in text mode.

Physics [edit]

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

Computing [edit]

Pipe [edit]

A pipage is an inter-process communication mechanism originating in Unix, which directs the output (standard out and, optionally, standard mistake) of 1 process to the input (standard in) of another. In this mode, a series of commands can exist "piped" together, giving users the power to quickly perform complex multi-stage processing from the command line or as office of a Unix shell script ("fustigate file"). In most Unix shells (command interpreters), this is represented by the vertical bar grapheme. For example:

grep -i 'blair' filename.log | more

where the output from the grep process (all lines containing 'blair') is piped to the more than process (which allows a command line user to read through results ane page at a fourth dimension).

The aforementioned "pipe" characteristic is besides found in later versions of DOS and Microsoft Windows.

This usage has led to the character itself being called "piping".

Disjunction [edit]

In many programming languages, the vertical bar is used to designate the logic operation or, either bitwise or or logical or.

Specifically, in C and other languages following C syntax conventions, such as C++, Perl, Java and C#, a | b denotes a bitwise or; whereas a double vertical bar a || b denotes a (brusk-circuited) logical or. Since the character was originally not available in all code pages and keyboard layouts, ANSI C can transcribe information technology in course of the trigraph ??!, which, outside string literals, is equivalent to the | graphic symbol.

In regular expression syntax, the vertical bar again indicates logical or (alternation). For case: the Unix command grep -East 'fu|bar' matches lines containing 'fu' or 'bar'.

Concatenation [edit]

The double vertical bar operator "||" denotes cord concatenation in PL/I, standard ANSI SQL, and theoretical computer scientific discipline (especially cryptography).

Delimiter [edit]

Although not as common as commas or tabs, the vertical bar can be used as a delimiter in a flat file. Examples of a pipe-delimited standard data format are LEDES 1998B and HL7. Information technology is frequently used because vertical confined are typically uncommon in the data itself.

Similarly, the vertical bar may encounter employ as a delimiter for regular expression operations (e.g. in sed). This is useful when the regular expression contains instances of the more common frontwards slash (/) delimiter; using a vertical bar eliminates the need to escape all instances of the forrard slash. However, this makes the bar unusable as the regular expression "culling" operator.

Backus–Naur form [edit]

In Backus–Naur form, an expression consists of sequences of symbols and/or sequences separated by '|', indicating a choice, the whole beingness a possible exchange for the symbol on the left.

                        <            personal-name            >            ::=            <            name            >            |            <            initial            >          

Concurrency operator [edit]

In calculi of communicating processes (like pi-calculus), the vertical bar is used to indicate that processes execute in parallel.

APL [edit]

The pipe in APL is the modulo or residual office between 2 operands and the absolute value function side by side to i operand.

List comprehensions [edit]

The vertical bar is used for list comprehensions in some functional languages, e.g. Haskell and Erlang. Compare prepare-architect note.

Text markup [edit]

The vertical bar is used as a special character in lightweight markup languages, notably MediaWiki'south Wikitext (in the templates and internal links).

In LaTeX text mode, the vertical bar produces an em dash (—). The \textbar command can be used to produce a vertical bar.

Phonetics and orthography [edit]

In the Khoisan languages and the International Phonetic Alphabet, the vertical bar is used to write the dental click (ǀ). A double vertical bar is used to write the alveolar lateral click (ǁ). Since these are technically letters, they have their own Unicode code points in the Latin Extended-B range: U+01C0 for the single bar and U+01C1 for the double bar.

Some Northwest and Northeast Caucasian languages written in the Cyrillic script take a vertical bar called palochka (Russian: палочка , lit. 'lilliputian stick'), indicating the preceding consonant is an ejective.

Longer single and double vertical confined are used to mark prosodic boundaries in the IPA.

Literature [edit]

Punctuation [edit]

In medieval European manuscripts, a single vertical bar was a common variant of the virgula ⟨/⟩ used as a menses, scratch comma,^[7] and caesura mark.^[seven]

In Sanskrit and other Indian languages, a single vertical mark, a danda, has a similar office as a period (full stop). Two bars || (a 'double danda') is the equivalent of a pilcrow in marking the end of a stanza, paragraph or section. The danda has its ain Unicode code betoken, U+0964.

Poetry [edit]

A double vertical bar ⟨||⟩ or ⟨ǁ⟩ is the standard caesura mark in English language literary criticism and analysis. It marks the strong suspension or caesura common to many forms of poetry, specially Old English verse. It is also traditionally used to mark the division betwixt lines of verse printed equally prose (the style preferred by Oxford University Press), though information technology is at present often replaced by the frontwards slash.

Notation [edit]

In the Geneva Bible and early printings of the King James Version, a double vertical bar is used to marker margin notes that contain an culling translation from the original text. These margin notes ever begin with the conjunction "Or". In later printings of the King James Version, the double vertical bar is irregularly used to mark any comment in the margins.

Music scoring [edit]

In music, when writing chord sheets, single vertical bars associated with a colon (|: A / / / :|) represents the outset and end of a section (e.thousand. Intro, Interlude, Verse, Chorus) of music.^{[ commendation needed ]} Unmarried confined can also represent the kickoff and cease of measures (|: A / / / | D / / / | E / / / :|). A double vertical bar associated with a colon tin can stand for the repeat of a given department (||: A / / / :|| - play twice).^{[ citation needed ]}

Encoding [edit]

Solid vertical bar vs broken bar [edit]

Many early video terminals and dot-matrix printers rendered the vertical bar character as the allograph broken bar ¦. This may accept been to distinguish the grapheme from the lower-instance 'Fifty' and the upper-case 'I' on these limited-resolution devices, and to make a vertical line of them look more than like a horizontal line of dashes. Information technology was also (briefly) part of the ASCII standard.

An initial draft for a 7-fleck character ready that was published past the X3.2 subcommittee for Coded Graphic symbol Sets and Data Format on June 8, 1961, was the kickoff to include the vertical bar in a standard set up. The bar was intended to be used equally the representation for the logical OR symbol.^[8] A subsequent draft on May 12, 1966, places the vertical bar in cavalcade seven alongside regional entry codepoints, and formed the footing for the original typhoon proposal used by the International Standards System.^[8] This draft received opposition from an IBM user group known as SHARE, with its chairman, H. West. Nelson, writing a letter to the American Standards Association titled "The Proposed revised American Standard Code for Information Interchange does Not meet the needs of computer programmers!"; in this letter, he argues that no characters inside the international subset designated at columns 2-5 of the character set would be able to adequately represent logical OR and logical Non in languages such as IBM'south PL/I universally on all platforms.^[9] Equally a compromise, a requirement was introduced where the exclamation marker (!) and circumflex (^) would display as logical OR (|) and logical Not (¬) respectively in use cases such equally programming, while outside of these apply cases they would stand for their original typographic symbols:

It may be desirable to apply distinctive styling to facilitate their employ for specific purposes as, for instance, to stylize the graphics in code positions 2/1 and 5/14 to those oftentimes associated with logical OR (|) and logical Non (¬) respectively.

—X3.2 document X3.2/475^[10]

The original vertical bar encoded at 0x7C in the original May 12, 1966 draft was and so broken every bit ¦, so information technology could not be dislocated with the unbroken logical OR. In the 1967 revision of ASCII, along with the equivalent ISO 464 code published the same year, the code point was defined to be a broken vertical bar, and the exclamation mark character was immune to be rendered as a solid vertical bar.^{[eleven] [12]} Nevertheless, the 1977 revision (ANSI X.3-1977) undid the changes made in the 1967 revision, enforcing that the circumflex could no longer be stylised as a logical Non symbol, the exclamation mark besides no longer assuasive stylisation as a vertical bar, and defining the code betoken originally set to the cleaved bar as a solid vertical bar instead;^[11] the same changes were also reverted in ISO 646-1973 published iv years prior.

Some variants of EBCDIC included both versions of the character equally different code points. The broad implementation of the extended ASCII ISO/IEC 8859 series in the 1990s too fabricated a distinction between the two forms. This was preserved in Unicode as a dissever character at U+00A6 BROKEN BAR (the term "parted rule" is used sometimes in Unicode documentation). Some fonts draw the characters the same (both are solid vertical bars, or both are broken vertical bars).^{[xiii] [ failed verification ]} The cleaved bar does not appear to take any clearly identified uses distinct from those of the vertical bar.^[xiv] In non-computing utilise — for example in mathematics, physics and general typography — the broken bar is non an acceptable substitute for the vertical bar.

US International keyboard showing broken bar on the keycap, fifty-fifty though typing shift+that key produces the solid vertical bar.

Total character set of IBM's Code folio 437 rendered in VGA, which displays the broken bar glyph for codepoint 7C, despite the 1977 revision to ASCII

Many keyboards with US or US-International layout display the broken bar on a keycap even though the solid vertical bar character is produced in modern operating systems. This includes many German QWERTZ keyboards. This is a legacy of keyboards manufactured during the 1980s and 1990s for IBM PC compatible computers featuring the cleaved bar, every bit such computers used IBM's 8-bit Code page 437 graphic symbol prepare based on ASCII, which continued to display the glyph for the broken bar at codepoint 7C on displays from MDA (1981) to VGA (1987) despite the changes made to ASCII in 1977.

The broken bar character can be typed (depending on the layout) as AltGr+` or AltGr+6 or AltGr+⇧ Shift+Right \ on Windows and Compose ! ^ on Linux. It can exist inserted into HTML as ¦

In some dictionaries, the broken bar is used to mark stress that may exist either primary or secondary. That is, [¦ba] covers the pronunciations [ˈba] and [ˌba].^[15]

Unicode code points [edit]

These glyphs are encoded in Unicode as follows:

• U+007C | VERTICAL LINE (&verbar;, &vert;, &VerticalLine;) (single vertical line)
• U+00A6 ¦ BROKEN BAR (&brvbar;) (unmarried broken line)
• U+2016 ‖ DOUBLE VERTICAL LINE (&Verbar;, &Vert;) (double vertical line ( ${\displaystyle \|}$ ): used in pairs to point norm)
• U+FF5C ｜ FULLWIDTH VERTICAL LINE (Fullwidth form)
• U+2225 ∥ PARALLEL TO (&DoubleVerticalBar;, &par;, &parallel;, &shortparallel;, &spar;)
• U+01C0 ǀ LATIN LETTER DENTAL CLICK
• U+01C1 ǁ LATIN LETTER LATERAL CLICK
• U+2223 ∣ DIVIDES (&mid;, &shortmid;, &smid;, &VerticalBar;)
• U+2502 │ BOX DRAWINGS Low-cal VERTICAL (&boxv;) (and various other box drawing characters in the range U+2500 to U+257F)
• U+0964 । DEVANAGARI DANDA
• U+0965 ॥ DEVANAGARI DOUBLE DANDA

Code pages and other historical encodings [edit]

Code pages, ASCII, ISO/IEC, EBCDIC, Shift-JIS, etc.	Vertical bar ('|')	Broken bar ('¦')
ASCII, CP437, CP667, CP720, CP737, CP790, CP819, CP852, CP855, CP860, CP861, CP862, CP865, CP866, CP867, CP869, CP872, CP895, CP932, CP991	124 (7Ch)	none
CP775		167 (A7h)
CP850, CP857, CP858		221 (DDh)
CP863		160 (A0h)
CP864		219 (DBh)
ISO/IEC 8859-1, -7, -8, -nine, -13, CP1250, CP1251, CP1252, CP1253, CP1254, CP1255, CP1256, CP1257, CP1258		166 (A6h)
ISO/IEC 8859-ii, -three, -4, -5, -6, -10, -eleven, -14, -15, -16		none
EBCDIC CCSID 37	79 (4Fh)	106 (6Ah)
EBCDIC CCSID 500	187 (BBh)
JIS X 0208, JIS X 0213	Men-ku-ten 1-01-35 (7-scrap: 2143h; Shift JIS: 8162h; EUC: A1C3h)[a]

See likewise [edit]

• Bar (diacritic)
• Triple bar

Notes [edit]

1. ^ The Shift JIS and EUC encoded forms also include the ASCII vertical bar in its usual encoding (see halfwidth and fullwidth forms). The same applies when the 7-bit grade is used every bit part of ISO-2022-JP (allowing switching to and from ASCII).

References [edit]

1. ^ Raymond, Eric S. "ASCII". The Jargon File.
2. ^ ^{a b c} Weisstein, Eric W. "Unmarried Bar". mathworld.wolfram.com . Retrieved 2020-08-24 .
3. ^ Weisstein, Eric Westward. "Matrix Norm". mathworld.wolfram.com . Retrieved 2020-08-24 .
4. ^ Univalent Foundations Programme (2013). Homotopy Blazon Theory: Univalent Foundations of Mathematics (GitHub version) (PDF). Institute for Advanced Study. p. 108.
5. ^ Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics (print version). Institute for Avant-garde Study. p. 450.
6. ^ Larus Thorlacius, Thordur Jonsson (eds.), M-Theory and Quantum Geometry, Springer, 2012, p. 263.
7. ^ ^{a b} "virgula, n.", Oxford English Dictionary, 1st ed. , Oxford: Oxford University Press, 1917 .
8. ^ ^{a b} Fischer, Eric (2012). The Development of Character Codes, 1874-1968 (Thesis). Penn State University. CiteSeerX10.1.1.96.678 . Retrieved July 10, 2020.
9. ^ H. W. Nelson, letter of the alphabet to Thomas B. Steel, June viii, 1966, Honeywell Inc. X3.two Standards Subcommittee Records, 1961-1969 (CBI 67), Charles Babbage Institute, University of Minnesota, Minneapolis, box 1, binder 23.
10. ^ X3.2 document X3.2/475, Dec xiii, 1966, Honeywell Inc. X3.two Standards Subcommittee Records, 1961-1969 (CBI 67), Charles Babbage Institute, University of Minnesota, Minneapolis, box 1, folder 22.
11. ^ ^{a b} Salste, Tuomas (January 2016). "vii-flake character sets: Revisions of ASCII". Aivosto Oy. urn:nbn:fi-fe201201011004. Archived from the original on 2016-06-thirteen. Retrieved 2016-06-13 .
12. ^ Korpela, Jukka. "Graphic symbol histories - notes on some Ascii lawmaking positions". Archived from the original on 2020-03-11. Retrieved 2020-05-31 .
13. ^ Jim Price (2010-05-24). "ASCII Nautical chart: IBM PC Extended ASCII Display Characters". Retrieved 2012-02-23 .
14. ^ Jukka "Yucca" Korpela (2006-09-twenty). "Detailed descriptions of the characters". Retrieved 2012-02-23 .
15. ^ For instance, "Balearic". Merriam-Webster Lexicon. .

How To Add Dotted Vertical Line In Equation Latex,

Source: https://en.wikipedia.org/wiki/Vertical_bar

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