View source for ISO 31-11

{{Short description|Vector and tensor}}
{{Use British (Oxford) English|date=January 2012}}
{{SpecialChars}}

'''ISO 31-11:1992''' was the part of [[International standard|international standard]] [[Physics:ISO 31|ISO 31]] that defines ''mathematical signs and symbols for use in physical sciences and technology''. It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-80000-2:2019.<ref>{{cite web|title=ISO 80000-2:2019|date= 19 May 2020 |url= https://www.iso.org/standard/64973.html |publisher=[[Organization:International Organization for Standardization|International Organization for Standardization]]|access-date=4 Oct 2021}}</ref>

Its definitions include the following:<ref name = "ISO 31-11">
{{cite book
 | first1 = Ambler | last1 = Thompson
 | first2 = Barry M | last2 = Taylor
 | url = http://physics.nist.gov/cuu/pdf/sp811.pdf
 | title = Guide for the Use of the International System of Units (SI) — NIST Special Publication 811, 2008 Edition — Second Printing
 |date=March 2008
 | publisher = NIST
 | location = Gaithersburg, MD, USA
}}</ref>

== Mathematical logic ==
{| class="wikitable"
! Sign
! Example
! Name
! Meaning and verbal equivalent
! Remarks
|- valign=top
|∧||''p'' ∧ ''q''||[[Logical conjunction|conjunction]] sign||''p'' and ''q''||
|- valign=top
|∨||''p'' ∨ ''q''||[[Logical disjunction|disjunction]] sign||''p'' or ''q'' (or both)||
|- valign=top
|¬||¬ ''p''||[[Negation|negation]] sign||negation of ''p''; not ''p''; non ''p''||
|- valign=top
|⇒||''p'' ⇒ ''q''||implication sign||if ''p'' then ''q''; ''p'' implies ''q''||Can also be written as ''q''&nbsp;⇐&nbsp;''p''. Sometimes → is used.
|- valign=top
|∀||∀''x''∈''A''&nbsp;''p''(''x'')<br>(∀''x''∈''A'')&nbsp;''p''(''x'')||universal quantifier||for every ''x'' belonging to ''A'', the proposition ''p''(''x'') is true||The "∈''A''" can be dropped where ''A'' is clear from context.
|-| valign=top
|∃||∃''x''∈''A''&nbsp;''p''(''x'')<br>(∃''x''∈''A'')&nbsp;''p''(''x'')||existential quantifier||there exists an ''x'' belonging to ''A'' for which the proposition ''p''(''x'') is true||The "∈''A''" can be dropped where ''A'' is clear from context.<br>∃! is used where exactly one ''x'' exists for which ''p''(''x'') is true.
|}

== Sets ==
{{MOS|section|MOS:MATHSPECIAL for \ which needs to be rewritten in LaTex-like syntax|date=January 2024}}
{| class="wikitable"
! Sign
! Example
! Meaning and verbal equivalent
! Remarks
|- valign="center"
|∈||''x'' ∈ ''A''||''x'' belongs to ''A''; ''x'' is an element of the set ''A''||
|- valign="center"
|∉||''x'' ∉ ''A''||''x'' does not belong to ''A''; ''x'' is not an element of the set ''A''||The negation stroke can also be vertical.
|- valign="center"
|∋||''A'' ∋ ''x''||the set ''A'' contains ''x'' (as an element)||same meaning as ''x'' ∈ ''A''
|- valign="center"
|∌||''A'' ∌ ''x''||the set ''A'' does not contain ''x'' (as an element)||same meaning as ''x'' ∉ ''A''
|- valign="center"
|{ }||{x<sub>1</sub>, x<sub>2</sub>, ..., x<sub>''n''</sub>}||set with elements x<sub>1</sub>, x<sub>2</sub>, ..., x<sub>''n''</sub>||also {x<sub>''i''</sub> {{pipe}} ''i''&nbsp;∈&nbsp;''I''}, where ''I'' denotes a set of indices
|- valign="center"
|{ {{pipe}} }||{''x'' ∈ ''A'' {{pipe}} ''p''(''x'')}||set of those elements of ''A'' for which the proposition ''p''(''x'') is true||Example: {''x'' ∈ <math>\mathbb{R}</math> {{pipe}} ''x'' > 5}<br>The ∈''A'' can be dropped where this set is clear from the context.
|- valign="center"
|card||card(''A'')||number of elements in ''A''; cardinal of ''A''||
|- valign="center"
|∖||''A'' ∖ ''B''|| [[Complement (set theory)|difference between ''A'' and ''B'']]; ''A'' minus ''B''||The set of elements which belong to ''A'' but not to ''B''.<br>''A'' ∖ ''B'' = { ''x'' {{pipe}} ''x''&nbsp;∈&nbsp;''A'' ∧ ''x''&nbsp;∉&nbsp;''B'' }<br>''A''&nbsp;−&nbsp;''B'' can also be used.
|- valign="center"
|∅|| ||the empty set||
|- valign="center"
|<math>\mathbb{N}</math>|| ||the set of [[Natural number|natural number]]s; the set of positive integers and zero||<math>\mathbb{N}</math> = {0, 1, 2, 3, ...}<br>Exclusion of zero is denoted by an asterisk:<br><math>\mathbb{N}</math><sup>*</sup> = {1, 2, 3, ...}<br><math>\mathbb{N}</math><sub>''k''</sub> = {0, 1, 2, 3, ..., ''k''&nbsp;−&nbsp;1}
|- valign="center"
|<math>\mathbb{Z}</math>|| ||the set of [[Integer|integer]]s||<math>\mathbb{Z}</math> = {..., −3, −2, −1, 0, 1, 2, 3, ...}<br>
<math>\mathbb{Z}</math><sup>*</sup> = <math>\mathbb{Z}</math> ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...}
|- valign="center"
|<math>\mathbb{Q}</math>|| ||the set of rational numbers||<math>\mathbb{Q}</math><sup>*</sup> = <math>\mathbb{Q}</math> ∖ {0}
|- valign="center"
|<math>\mathbb{R}</math>|| ||the set of real numbers||<math>\mathbb{R}</math><sup>*</sup> = <math>\mathbb{R}</math> ∖ {0}
|- valign="center"
|<math>\mathbb{C}</math>|| ||the set of complex numbers||<math>\mathbb{C}</math><sup>*</sup> = <math>\mathbb{C}</math> ∖ {0}
|- valign="center"
|[,]||[''a'',''b'']||closed interval in <math>\mathbb{R}</math> from ''a'' (included) to ''b'' (included)||[''a'',''b''] = {''x'' ∈ <math>\mathbb{R}</math> {{pipe}} ''a'' ≤ ''x'' ≤ ''b''}
|- valign="center"
|],]<br>(,]||]''a'',''b'']<br>(''a'',''b'']||left half-open interval in <math>\mathbb{R}</math> from ''a'' (excluded) to ''b'' (included)||]''a'',''b''] = {''x'' ∈ <math>\mathbb{R}</math> {{pipe}} ''a'' < ''x'' ≤ ''b''}
|- valign="center"
|[,[<br>[,)||[''a'',''b''[<br>[''a'',''b'')||right half-open interval in <math>\mathbb{R}</math> from ''a'' (included) to ''b'' (excluded)||[''a'',''b''[ = {''x'' ∈ <math>\mathbb{R}</math> {{pipe}} ''a'' ≤ ''x'' < ''b''}
|- valign="center"
|],[<br>(,)||]''a'',''b''[<br>(''a'',''b'')||open interval in <math>\mathbb{R}</math> from ''a'' (excluded) to ''b'' (excluded)||]''a'',''b''[ = {''x'' ∈ <math>\mathbb{R}</math> {{pipe}} ''a'' < ''x'' < ''b''}
|- valign="center"
|⊆||''B'' ⊆ ''A''||''B'' is included in ''A''; ''B'' is a subset of ''A''||Every element of ''B'' belongs to ''A''. ⊂ is also used.
|- valign="center"
|⊂||''B'' ⊂ ''A''||''B'' is properly included in ''A''; ''B'' is a proper subset of ''A''|| Every element of ''B'' belongs to ''A'', but ''B'' is not equal to ''A''. If ⊂ is used for "included", then ⊊ should be used for "properly included".
|- valign="center"
|⊈||''C'' ⊈ ''A''||''C'' is not included in ''A''; ''C'' is not a subset of ''A''||⊄ is also used.
|- valign="center"
|⊇||''A'' ⊇ ''B''||''A'' includes ''B'' (as subset)||''A'' contains every element of ''B''. ⊃ is also used. ''B'' ⊆ ''A'' means the same as ''A'' ⊇ ''B''.
|- valign="center"
|⊃||''A'' ⊃ ''B''.||''A'' includes ''B'' properly. ||''A'' contains every element of ''B'', but ''A'' is not equal to ''B''. If ⊃ is used for "includes", then ⊋ should be used for "includes properly".
|- valign="center"
|⊉||''A'' ⊉ ''C''||''A'' does not include ''C'' (as subset)||⊅ is also used. ''A''&nbsp;⊉&nbsp;''C'' means the same as ''C''&nbsp;⊈&nbsp;''A''.
|- valign="center"
|∪||''A'' ∪ ''B''||union of ''A'' and ''B''||The set of elements which belong to ''A'' or to ''B'' or to both ''A'' and ''B''.<br>''A''&nbsp;∪&nbsp;''B'' = { ''x'' {{pipe}} ''x''&nbsp;∈&nbsp;''A'' ∨ ''x''&nbsp;∈&nbsp;''B'' }
|- valign="center"
|⋃||<math>\bigcup_{i=1}^n A_i</math>||union of a collection of sets||<math>\bigcup_{i=1}^n A_i=A_1\cup A_2\cup\ldots\cup A_n</math>, the set of elements belonging to at least one of the sets {{math|''A''<sub>1</sub>, ..., ''A''<sub>n</sub>}}. <math>\bigcup{}_{i=1}^n</math> and <math>\bigcup_{i\in I}</math>, <math>\bigcup{}_{i \in I}</math> are also used, where ''I'' denotes a set of indices.
|- valign="center"
|∩||''A'' ∩ ''B''||intersection of ''A'' and ''B''||The set of elements which belong to both ''A'' and ''B''.<br>''A''&nbsp;∩&nbsp;''B'' = { ''x'' {{pipe}} ''x''&nbsp;∈&nbsp;''A'' ∧ ''x''&nbsp;∈&nbsp;''B'' }
|- valign="center"
|⋂||<math>\bigcap_{i=1}^n A_i</math>||intersection of a collection of sets||<math>\bigcap_{i=1}^n A_i=A_1\cap A_2\cap\ldots\cap A_n</math>, the set of elements belonging to all sets {{math|''A''<sub>1</sub>, ..., ''A''<sub>n</sub>}}. <math>\bigcap{}_{i=1}^n</math> and <math>\bigcap_{i\in I}</math>, <math>\bigcap{}_{i \in I}</math> are also used, where ''I'' denotes a set of indices.
|- valign="center"
|∁||∁<sub>''A''</sub>''B''||complement of subset ''B'' of ''A''||The set of those elements of ''A'' which do not belong to the subset ''B''. The symbol ''A'' is often omitted if the set ''A'' is clear from context. Also ∁<sub>''A''</sub>''B'' = ''A''&nbsp;∖&nbsp;''B''.
|- valign="center"
|(,)||(''a'', ''b'')||ordered pair ''a'', ''b''; couple ''a'', ''b''||(''a'',&nbsp;''b'') = (''c'',&nbsp;''d'') if and only if ''a''&nbsp;=&nbsp;''c'' and ''b''&nbsp;=&nbsp;''d''.<br>⟨''a'',&nbsp;''b''⟩ is also used.
|- valign="center"
|(,...,)||(''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;...,&nbsp;''a''<sub>''n''</sub>)||ordered ''n''-[[Tuple|tuple]]||⟨''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;...,&nbsp;''a''<sub>''n''</sub>⟩ is also used.
|- valign="center"
|×||''A'' × ''B''||cartesian product of ''A'' and ''B''||The set of ordered pairs (''a'',&nbsp;''b'') such that ''a''&nbsp;∈&nbsp;''A'' and ''b''&nbsp;∈&nbsp;''B''.<br>''A''&nbsp;×&nbsp;''B'' = { (''a'',&nbsp;''b'') {{pipe}} ''a''&nbsp;∈&nbsp;''A'' ∧ ''b''&nbsp;∈&nbsp;''B'' }<br>''A'' × ''A'' × ⋯ × ''A'' is denoted by ''A''<sup>''n''</sup>, where ''n'' is the number of factors in the product.
|- valign="center"
|Δ||Δ<sub>''A''</sub>||set of pairs (''a'',&nbsp;''a'') ∈ ''A''&nbsp;×&nbsp;''A'' where ''a''&nbsp;∈&nbsp;''A''; diagonal of the set ''A''&nbsp;×&nbsp;''A''||Δ<sub>''A''</sub> = { (''a'',&nbsp;''a'') {{pipe}} ''a''&nbsp;∈&nbsp;''A'' }<br>id<sub>''A''</sub> is also used.
|}

== Miscellaneous signs and symbols ==
{| class="wikitable"
! colspan="2" | Sign
! rowspan="2" | Example
! rowspan="2" | Meaning and verbal equivalent
! rowspan="2" | Remarks
|-
! HTML !! TeX
|- valign=top
|≝ || <math>\stackrel{\mathrm{def}}{=}</math>||''a'' ≝ ''b''||''a'' is by definition equal to ''b''&nbsp;<ref name="ISO 31-11"/>||:= is also used
|- valign=top
| = || <math>=</math> ||''a'' = ''b'' ||''a'' equals ''b''||≡ may be used to emphasize that a particular equality is an identity.
|- valign=top
| ≠ || <math>\ne</math> ||''a'' ≠ ''b''||''a'' is not equal to ''b''||<math>a \not\equiv b</math> may be used to emphasize that ''a'' is not identically equal to ''b''.
|- valign=top
| ≙ || <math>\stackrel{\wedge}{=}</math> ||''a'' ≙ ''b''||''a'' corresponds to ''b''||On a 1:10<sup>6</sup> map: 1&nbsp;cm ≙ 10&nbsp;km.
|- valign=top
| ≈ || <math>\approx</math> ||''a'' ≈ ''b''||''a'' is approximately equal to ''b''||The symbol ≃ is reserved for "is asymptotically equal to".
|- valign=top
|∼<br>∝||<math>\begin{matrix} \sim \\ \propto \end{matrix}</math>||''a'' ∼ ''b''<br>''a'' ∝ ''b''||''a'' is proportional to ''b''||
|- valign=top
| < || {{tmath|<}} ||''a'' < ''b''||''a'' is less than ''b''||
|- valign=top
| > || {{tmath|>}} ||''a'' > ''b''||''a'' is greater than ''b''||
|- valign=top
| ≤ || <math>\le</math> ||''a'' ≤ ''b''||''a'' is less than or equal to ''b''||The symbol ≦ is also used.
|- valign=top
| ≥ || <math>\ge</math> ||''a'' ≥ ''b''||''a'' is greater than or equal to ''b''||The symbol ≧ is also used.
|- valign=top
| ≪ || <math>\ll</math> ||''a'' ≪ ''b''||''a'' is much less than ''b''||
|- valign=top
| ≫ || <math>\gg</math> ||''a'' ≫ ''b''||''a'' is much greater than ''b''||
|- valign=top
| ∞ || <math>\infty</math> || ||infinity||
|- valign=top
|()<br>[]<br>{}<br>{{angbr}}||<math>\begin{matrix}() \\ {[]} \\ \{\} \\ \langle \rangle \end{matrix}</math>||<math>\begin{matrix} {(a+b)c} \\ {[a+b]c} \\ {\{a+b\}c} \\ {\langle a+b \rangle c} \end{matrix}</math>||''ac'' + ''bc'', parentheses<br>''ac'' + ''bc'', square brackets<br>''ac'' + ''bc'', braces<br>''ac'' + ''bc'', angle brackets||In ordinary algebra, the sequence of <math>(), [], \{\}, \langle \rangle </math> in order of nesting is not standardized. Special uses are made of <math>(), [], \{\}, \langle \rangle </math> in particular fields.
|- valign=top
| ∥ || <math>\|</math> ||AB ∥ CD||the line AB is parallel to the line CD||
|- valign=top
| ⊥ || <math>\perp</math>||AB ⊥ CD||the line AB is perpendicular to the line CD<ref>If the perpendicular symbol, ⟂, does not display correctly, it is similar to ⊥ (up tack: sometimes meaning orthogonal to) and it also appears similar to ⏊ (the dentistry symbol light up and horizontal)</ref> ||
|}

== Operations ==
{| class="wikitable"
! Sign
! Example
! Meaning and verbal equivalent
! Remarks
|- valign=top
| +||''a'' + ''b''||''a'' plus ''b''||
|- valign=top
|−||''a'' − ''b''||''a'' minus ''b''||
|- valign=top
|±||''a'' ± ''b''||''a'' plus or minus ''b''||
|- valign=top
|∓||''a'' ∓ ''b''||''a'' minus or plus ''b''||−(''a''&nbsp;±&nbsp;''b'') = −''a''&nbsp;∓&nbsp;''b''
|}

== Functions ==
{| class="wikitable"
! Example
! Meaning and verbal equivalent
! Remarks
|- valign=top
|''f'' : ''D'' → ''C''||function ''f'' has domain ''D'' and codomain ''C'' || Used to explicitly define the domain and codomain of a function.
|- valign=top
|''f''(''S'')||{ ''f''(''x'') {{pipe}} ''x'' ∈ ''S'' }||Set of all possible outputs in the codomain when given inputs from ''S'', a subset of the domain of ''f''.
|}

== Exponential and logarithmic functions ==
{| class="wikitable"
! Example
! Meaning and verbal equivalent
! Remarks
|- valign=top
|[[E (mathematical constant)|e]]||base of natural logarithms||e = 2.718 28...
|- valign=top
|e<sup>''x''</sup>||[[Exponential function|exponential function]] to the [[Base (exponentiation)|base]] e of ''x''||
|- valign=top
|log<sub>''a''</sub>''x''||[[Logarithm|logarithm]] to the base ''a'' of ''x''||
|- valign=top
|lb ''x''||[[Binary logarithm|binary logarithm]] (to the base 2) of ''x''||lb ''x'' = log<sub>2</sub>''x''
|- valign=top
|ln ''x''||[[Natural logarithm|natural logarithm]] (to the base e) of ''x''||ln ''x'' = log<sub>e</sub>''x''
|- valign=top
|lg ''x''||[[Common logarithm|common logarithm]] (to the base 10) of ''x''||lg ''x'' = log<sub>10</sub>''x''
|}

== Circular and hyperbolic functions ==
{| class="wikitable"
! Example
! Meaning and verbal equivalent
! Remarks
|- valign=top
|π||ratio of the [[Circumference|circumference]] of a [[Circle|circle]] to its [[Diameter|diameter]]||π =~ {{val|3.14159}}
|}

== Complex numbers ==
{| class="wikitable"
! Example
! Meaning and verbal equivalent
! Remarks
|- valign=top
|i, j||[[Imaginary unit|imaginary unit]]; i{{sup|2}} = −1||In electrotechnology, j is generally used.
|- valign=top
|Re ''z''||real part of ''z''
|rowspan=2|''z'' = ''x''&nbsp;+&nbsp;i''y'', where ''x'' = Re&nbsp;''z'' and ''y'' = Im&nbsp;''z''
|- valign=top
|Im ''z''||imaginary part of ''z''
|- valign=top
|{{pipe}}''z''{{pipe}}||[[Absolute value|absolute value]] of ''z''; modulus of ''z''||mod ''z'' is also used
|- valign=top
|arg ''z''||argument of ''z''; phase of ''z''||''z'' = ''r''e{{sup|i''φ''}}, where ''r'' = {{pipe}}''z''{{pipe}} and ''φ'' = arg&nbsp;''z'', i.e. Re&nbsp;''z'' = ''r''&nbsp;cos&nbsp;''φ'' and Im&nbsp;''z'' = ''r''&nbsp;sin&nbsp;''φ''
|- valign=top
|''z''{{i sup|*}}||(complex) conjugate of ''z''||sometimes a bar above ''z'' is used instead of ''z''<sup>*</sup>
|- valign=top
|sgn ''z''||[[Sign function|signum]] ''z''||sgn ''z'' = ''z''&nbsp;/&nbsp;{{pipe}}''z''{{pipe}} = exp(i&nbsp;arg&nbsp;''z'') for ''z''&nbsp;≠&nbsp;0, sgn&nbsp;0 =&nbsp;0
|}

== Matrices ==
{| class="wikitable"
! Example
! Meaning and verbal equivalent
! Remarks
|- valign=top
|'''''A'''''||[[Matrix (mathematics)|matrix]] '''''A'''''||
|}

== Coordinate systems ==
{| class="wikitable"
! Coordinates
! Position vector and its differential
! Name of coordinate system
! Remarks
|- valign=top
| ''x'', ''y'', ''z''
| [''x''&nbsp;''y''&nbsp;''z'']; [''dx''&nbsp;''dy''&nbsp;''dz'']
| [[Cartesian coordinate system|cartesian]]
| ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub> for the coordinates and '''''e'''''<sub>1</sub>, '''''e'''''<sub>2</sub>, '''''e'''''<sub>3</sub> for the base vectors are also used. This notation easily generalizes to ''n''-dimensional space. '''''e'''''<sub>''x''</sub>, '''''e'''''<sub>''y''</sub>, '''''e'''''<sub>''z''</sub> form an orthonormal right-handed system. For the base vectors, '''''i''''', '''''j''''', '''''k''''' are also used.
|- valign=top
| ''ρ'', ''φ'', ''z''
| [''x'',&nbsp;''y'',&nbsp;''z''] = [''ρ''&nbsp;cos(''φ''),&nbsp;''ρ''&nbsp;sin(''φ''),&nbsp;''z'']
| [[Cylindrical coordinate system|cylindrical]]
| '''''e'''''<sub>''ρ''</sub>(''φ''), '''''e'''''<sub>''φ''</sub>(''φ''), '''''e'''''<sub>''z''</sub> form an orthonormal right-handed system. lf ''z'' = 0, then ''ρ'' and ''φ'' are the polar coordinates.
|- valign=top
| ''r'', ''θ'', ''φ''
| [''x'', ''y'', ''z''] = ''r''[sin(''θ'')cos(''φ''),&nbsp;sin(''θ'')sin(''φ''),&nbsp;cos(''θ'')]
| [[Spherical coordinate system|spherical]]||'''''e'''''<sub>''r''</sub>(''θ'',''φ''), '''''e'''''<sub>''θ''</sub>(''θ'',''φ''),'''''e'''''<sub>''φ''</sub>(''φ'') form an orthonormal right-handed system.
|}

== Vectors and tensors ==
{| class="wikitable"
! Example
! Meaning and verbal equivalent
! Remarks
|- valign=top
|'''''a'''''<br><math>\vec a</math>||vector '''''a'''''||Instead of italic boldface, vectors can also be indicated by an arrow above the letter symbol. Any vector '''''a''''' can be multiplied by a [[Scalar (mathematics)|scalar]] ''k'', i.e. ''k'''a'''''.
|}

== Special functions ==
{| class="wikitable"
! Example
! Meaning and verbal equivalent
! Remarks
|- valign=top
|{{math|''J''<sub>''l''</sub>(''x'')}}||cylindrical [[Bessel function]]s (of the first kind)||...
|}

== See also ==
* Mathematical symbols
* [[Mathematical notation]]

== References and notes ==
{{reflist}}

{{ISO standards}}

[[Category:Mathematical symbols]]
[[Category:Mathematical notation]]

{{Sourceattribution|ISO 31-11}}